Multichannel Rydberg spectroscopy of complex atoms

Multichannel Rydberg spectroscopy of complex atoms

Mireille Aymar

Laboratoire Aime Cotton, Centre National de la Recherche Scientifique II, Batiment 505,91405 Orsay Cedex, France

Chris H. Greene

Department of Physics and JILA,University of Colorado, Boulder, Colorado 80309-0440

Eliane Luc-Koenig

Laboratoire Aime Cotton, Centre National de la Recherche Scientifique II, Batiment 505,91405 Orsay Cedex, France

Multichannel atomic spectra frequently exhibit such extraordinary visual complexity that they appearat first glance to be uninterpretable. The present review discusses how to unravel such spectra throughthe use of theoretical multichannel spectroscopy to extract the key dynamical implications. Moreover,this class of techniques permits a quantitative prediction or reproduction of experimental spectra forsome of the more challenging atomic systems under investigation. It is shown that multichannelspectroscopy marries the techniques of multichannel quantum-defect theory to the eigenchannelR-matrix method (or related methods). It has long been appreciated that multichannelquantum-defect theory can successfully use a collision-theory framework to interpret enormouslycomplicated Rydberg spectra. However, the capabilities of multichannel quantum-defect theory haveincreased dramatically during the past decade, through the development of nearly ab initio methodsfor the calculation of the short-range scattering parameters that control the interactions of closed andopen channels. In this review, emphasis is given to the alkaline-earth atoms, for which many differentobservables have been successfully compared with experiment over broad ranges of energy andresolution. Applications of the method to describe the photoionization spectra of more complexopen-shell atoms are also discussed. [S0034-6861(96)00504-1]

CONTENTS

I. Introduction 1016A. Origins in nuclear physics 1016B. Developments in atomic physics 1018C. Other related reviews 1019

II. Overview of the Ideas and Equations ofMultichannel Spectroscopy 1020A. Visualizing atomic photoabsorption 1021

1. Single-channel processes 10212. Two-channel processes 10213. Properties in the time domain 1022

B. Single-channel properties 1022C. Nonseparable quantum mechanics in several

dimensions 10251. Characterizing the solutions 10252. Modified form of the close-coupling method

without exchange 1026D. Long-range treatment using multichannel

quantum-defect theory 10281. Multichannel quantum-defect theory

derivations for resonant continua 10292. Bound-state properties 10323. Resonance analysis 1034

E. Frame transformations 1035F. Semiempirical analysis of multichannel spectra 1039

III. Eigenchannel R-Matrix Approach 1042A. General procedure 1042B. Specialization to two-electron systems. R-matrix

calculations in LS coupling 10441. Relationship between V(r) and the energy

levels of A+ 1044

2. Design of the basis set 1046C. Inclusion of nonperturbative spin-orbit effects:

Two-electron R-matrix calculations in jjcoupling 1047

D. Specialization to open-shell atoms. R-matrixcalculations in LS coupling 10481. Hamiltonian 10482. Constructing basis functions 1049

E. Streamlined solution of the generalizedeigensystem 1051

F. Short-range reaction matrix K and parametersof multichannel quantum-defect theory 10521. R-matrix calculations in LS coupling 10532. R-matrix calculations in LS coupling

combined with the jj-LS frametransformation 1053

3. R-matrix calculations in jj coupling 1054G. Reaction volume size, basis functions, and

fragmentation channels 1054H. Alternative streamlined solutions of the

generalized eigensystem 1057IV. Alkaline-Earth Atoms 1058

A. Discrete spectra 10591. Energy spectra 10592. Observables other than energies 1065

B. Energy positions of autoionizing levels 1067C. Autoionization widths and branching ratios 1072D. Photoionization from ground and low-lying

states 10761. Ground-state photoionization in lighter

alkaline earths Be and Mg 10762. Ground-state photoionization in heavier

alkaline earths Ca to Ra 1078

1015Reviews of Modern Physics, Vol. 68, No. 4, October 1996 0034-6861/96/68(4)/1015(109)/$26.35 © 1996 The American Physical Society

3. Photoionization from low-lying excited states 1083a. Total photoionization cross sections 1083b. Partial photoionization cross sections and

photoelectron angular distributions 1086E. Photoionization from Rydberg states 1087

1. Excitation spectra 10882. Branching ratios and angular distributions of

electrons ejected from autoionizing levels 1093V. Open-Shell Atoms 1098

A. Atoms with an open p subshell 10991. Aluminum 10992. Halogens 11003. Carbon-group atoms 11024. Chalcogens 1103

B. Open d-subshell atoms 11041. Scandium 11042. Titanium 1107

C. Concluding remarks 1109VI. Conclusions and Perspectives 1110

Acknowledgments 1116References 1116

I. INTRODUCTION

The sharpest signature of any microscopic quantum-mechanical system is its pattern of resonances. This pre-dominant importance of resonances provides a threadthat unifies much of physics from the birth of quantummechanics all the way to the present. Some of the earlystudies of resonances were conducted in the field ofnuclear physics, where the first questions were fairlysimple: Why are there resonances in the first place?What shapes can be expected in spectra exhibiting reso-nances? What information do those shapes conveyabout the internal dynamics of the system? The answersto these questions emerged from the work of Bohr(1936), Breit and Wigner (1936), Breit (1959), and oth-ers.This review concentrates on showing how resonant

spectra of many-electron atoms provide a highly de-tailed window on the complicated, nonseparable mo-tions of atomic electrons. Capabilities of current theoryhave grown so rapidly in the past decade that tremen-dously complicated experimental spectra can now be ac-curately described quantitatively from first principles, orfor heavy atoms from nearly first principles. Despite ourfocus on atomic spectra, many of the concepts have po-tentially wider applicability to other fields in physics andchemistry. Immediate examples include low-energy (upto a few MeV) nuclear scattering and photofragmenta-tion processes, molecular photoionization and photodis-sociation, low-energy collisions (up to a few eV) be-tween an atom or molecule and an electron or anotheratom, or any resonant reactive scattering process such asAB1C→A1BC , to name only a few.Whereas the earliest studies treated resonances phe-

nomenologically on a state-by-state basis, the emphasishere will be on viewing a family of resonances withsimular character, along with an adjoining fragmentationcontinuum, as consisting of one channel. This view ofthe resonance physics thus links immediately to ascattering-theory point of view, as the indices labeling

channels are the same indices occurring in the scatteringmatrix. For instance, the set of all s states (with l50) ofatomic hydrogen, from the 1s state up to the ionizationthreshold (e50), and including the continuum at e.0,constitutes a single channel. In most complex atoms,several interacting channels are present simultaneously,and their spectra are correspondingly multichannel incharacter. In fact, the only real difference between thechannel description of resonances and the ‘‘conven-tional’’ form of scattering theory is the use of matrices(such as the collision or scattering matrix S) with indicesreferring to closed channels (in which fragmentation allthe way to asymptopia is energetically forbidden). Thisclosed-channel description of resonances will be re-ferred to in the following as the multichannel spectros-copy viewpoint. Elsewhere it has been called thequantum-defect theory viewpoint, reflecting the fact thatthe formulation of Seaton (1966, 1983) is the context inwhich closed channels arise most naturally and persis-tently. However, similar ideas originated in Wigner’snuclear studies (Wigner, 1946a, 1946b) long beforequantum-defect theory was formulated [as stressed byLane (1986)]. For this reason, and also because the term‘‘quantum defect’’ masks the generality of this physicaldescription to contexts far wider than atomic Rydbergspectra, we prefer the more descriptive term, multichan-nel spectroscopy.The quantum-defect theory description of atomic

Rydberg-state resonances arose from the work of Ham(1955) and Seaton (1955, 1966, 1983). Fano (1970) con-tributed the important and general concept of frametransformation. This dramatically broadened the scopeof quantum-defect methods, as is manifested by applica-tions ranging from the work of Jungen and co-workers(Jungen and Dill, 1980; Raoult and Jungen, 1981; Jun-gen, 1984) on molecular Rydberg spectra to that of Wa-tanabe and Komine (1991) and of O’Mahony and Mota-Furtado (1991) on the amazing Rydberg spectra ofatomic hydrogen in a magnetic field.One of the goals of this review is to convey the unify-

ing interpretive power of multichannel spectroscopictheory. The success of this theory is twofold. First, mul-tichannel spectroscopy offers a powerful practical ap-proach to the calculation of real spectra having extremecomplexity. In this respect there are effectively no com-peting theoretical methods at present that rival thepower of quantum-defect ideas, at least in the realm ofautoionizing atomic spectra. Second, multichannel spec-troscopy provides a complete way of visualizing thequalitative physics involved, and in particular a way ofinterpreting dense and seemingly irregular resonantspectra, allowing simple conclusions to be drawn aboutthe underlying dynamics of the system.

A. Origins in nuclear physics

In the historical development of nuclear physics, thefirst step toward understanding resonant spectra was thedevelopment of semiempirical formulas, such as the for-mula of Breit and Wigner (1936), which described a

1016 M. Aymar, C. H. Greene, and E. Luc-Koenig: Multichannel Rydberg spectroscopy . . .

Rev. Mod. Phys., Vol. 68, No. 4, October 1996

Lorentzian-shaped enhancement of the cross section forany reactive process in the vicinity of the resonance.These were initially based on physically plausible as-sumptions, such as Bohr’s concept (Bohr, 1936) of thecompound nucleus, in which a reactive process had twoindependent stages: formation of a compound (i.e., reso-nant) system, followed by its fragmentation into reactionproducts. This phenomenological approach successfullyaccounted for the existence of resonances and for theirsymmetrically-peaked profiles. Even earlier, Fano(1935) had shown, in the seemingly different physicalcontext of atomic autoionization, how a discrete stateinteracting with a continuum produced a resonance pro-file that could exhibit a dip or an asymmetrical shape.Fano (1961) reformulated the approach much later togive a sharper derivation of these asymmetrical profilesarising from any type of discrete-continuum interaction.Equivalent formulations of discrete-continuum interac-tions carried out in different fields include the so-called‘‘Anderson model’’ in condensed-matter physics [see,for example, Sec. 4.2 of Mahan (1981)] and the ‘‘unifiedtheory of nuclear reactions’’ of Feshbach (1958). A sim-plification of this type of phenomenological theory, andultimately a key limitation, was its assumption of asingle isolated resonance state (or, at any rate, a finitenumber of such states) whose interactions with fragmen-tation continua had to be characterized on a state-by-state basis.Wigner (1946a, 1946b), Wigner and Eisenbud (1947),

and Teichmann and Wigner (1952) created the first sys-tematic formulation capable of treating general resonantprocesses in reactive systems. Their work has generallygone under the name of R-matrix theory, as it focuseson the formal properties of the logarithmic derivativematrix (or R matrix) characterizing a complicated many-particle, multichannel system within some finite radiusr0. The indices of the R matrix label the different pos-sible modes of breakup of the system, which we shall callthe fragmentation channels throughout this review.These channel indices are the same ones that label themore familiar scattering matrix S . Although the scatter-ing matrix is generally referred to as the ‘‘Heisenbergmatrix S’’ (Heisenberg, 1943), it had been introducedearlier by Wheeler (1937), who called it the matrix c .The main result of Wigner’s R-matrix formulation washis demonstration that the Hermitian matrix R(E),which is simply related to the matrix Sand is a meromor-phic (i.e., analytic except near simple poles) function ofthe energy E . Moreover, its poles on the real energy axiscan (often) be associated with resonance energies El ,i.e.,

Rij5(l

g ilg jl

El2E. (1.1)

(The actual correspondence between poles El and reso-nance states of the compound system confined to shortrange r,r0 involves a few additional subtleties.) In thispicture, each term in the summation of Eq. (1.1) wasidentified as a separate resonance, with the factors in thenumerator g il being related to the partial decay widths

of resonance l into channel i . This equation also showshow, in the immediate vicinity of any given resonance, asingle term dominates the summation, whereby the Rmatrix becomes approximately separable and so has unitrank, as does the S matrix.Equation (1.1) has been used widely in nuclear phys-

ics, and to some degree in high-energy physics, to ac-count semiempirically for resonance features in a varietyof measurements. The typical calculation attempts to fitan experimental spectrum by varying the parametersEl and g il in a least-squares sense to optimize agree-ment between calculation and experiment. This semi-empirical fitting procedure is what most nuclear physi-cists today associate with the term ‘‘R-matrix theory,’’even though R-matrix theory has come to mean some-thing rather different in atomic and molecular physics,namely, the explicit numerical calculation of the R ma-trix. In the semiempirical approach, the R matrix is re-garded as containing all the relevant information aboutthe reaction zone within r,r0, where all inelasticitiesoccur. This reaction zone is thus regarded as a ‘‘blackbox’’ and most of the physics within it is neither under-stood in detail nor is plausibly calculable.The absence of a detailed microscopic formulation

that could predict the R matrix from first principles wasnot regarded as a problem in nuclear physics for manyyears. The attitude in the community was reflected in aquote from the influential textbook of Blatt andWeisskopf (1952): ‘‘We do not really need to know allthe details of the motion within the compound nucleusin order to obtain all relevant information about nuclearreactions. A complete description of the internal motionwithin the compound nucleus is neither necessary nordesirable.’’ In the present review, we emphasize thatsuch a complete description is necessary if the goal is toperform spectroscopically useful calculations of complexspectra. We show below that in many cases it does notrequire laborious calculations. Moreover, the completedescription is desirable in the sense that one therebyattains far greater confidence in the underlying theoryand also great predictive power.A few attempts were made, mainly in the 1960s, to

determine the R matrix by explicit calculation, notablyby Lane and Thomas (1958), by Lane and Robson (1966,1969) and Chatwin (1970), and in effect by Brueckner(1955), among others. After a specific model of thenucleon-nucleon interaction potential was adopted,these methods gave R matrices to varying degrees ofaccuracy and showed that, at least in principle, one cancalculate the detailed microscopic properties without be-ing restricted by the difficulty and nonuniqueness of fit-ting the R matrix to reproduce experimental spectra.Such explicit calculations were not pursued very widelyin nuclear physics, however. The poor theoretical under-standing of the nucleon-nucleon interaction made it dif-ficult to interpret discrepancies between theory and ex-periment as being attributable to an unconverged orotherwise inaccurate method for solving the many-particle Schrodinger equation for Rij , or to an inad-equate description of the nucleon-nucleon interaction.

1017M. Aymar, C. H. Greene, and E. Luc-Koenig: Multichannel Rydberg spectroscopy . . .


As a result, the nuclear applications of Wigner’sR-matrix formulation never progressed very far beyondits widespread use in the semiempirical analysis of ex-perimental spectra (see, for example, Mo and Hornyak,1969). Other aspects of the semiempirical work thatwere unsatisfactory to some included the fact that thefitted parameters El and g il depend on the reaction-zone radius r0, and on the number of terms retained inthe summation in Eq. (1.1). This dependence of the Rmatrix on the radius r0 has been argued (see, for ex-ample, Feshbach, 1958) to be a somewhat unphysicalaspect of R-matrix theory, and is in contrast to thephysical scattering matrix S , which is independent ofany such radius. In fact, most of the competing formula-tions for calculating scattering or photoabsorption pro-cesses have a similar boundary either explicitly or im-plicitly present. For example, the complex Kohn (1948)variational principle, which has gained in popularity(McCurdy et al . , 1987; Miller, 1988; Robicheaux, 1989)does not seem at first glance to specify any boundaryradius. But a truncated basis set is always used to ex-pand the L2 or short-range part of the variational solu-tion. Consequently, a Kohn calculation will always havean ‘‘outermost’’ basis function that defines an effectiveradius, although it is a softer boundary radius than isused in R-matrix calculations. Similar remarks apply toother methods used currently, such as the Schwinger(1947) variational principle (for references see Watson,1988).

B. Developments in atomic physics

The first atomic calculations of multichannel spectra,aside from early configuration-interaction studies, wereconducted in the early 1960s within a framework calledthe close-coupling method (Burke and Smith, 1962).This method expands the full energy eigenfunctionsCE of an N-electron atomic complex in terms of ‘‘chan-nel functions’’ F i that obey the time-independent Schro-dinger equation for the (N21)-electron system. TheF i are multiplied by unknown solutions Fi(r) for the‘‘outermost’’ electron and are then fully antisymme-trized with respect to all electrons. The channel func-tions F i normally include spin and angular dependencesof the outermost electron also, so that they form a com-plete orthonormal set in all degrees of freedom exceptthe radial coordinate r . These close-coupling-type solu-tions are usually written as

CE5A(i

1r

F i~v!Fi~r !, (1.2)

in which the symbol A indicates the antisymmetrizationoperation. Projection onto each of the channel functionsresults in an infinite set of coupled integrodifferentialequations for the Fi(r). This set is in principle continu-ously infinite, but is normally truncated to a finite num-ber of equations on physical grounds and solved numeri-cally. Burke and McVicar (1965) used this procedure tocalculate doubly excited resonance states of helium that

had been observed by Madden and Codling (1965) inearly synchrotron radiation experiments.These early close-coupling calculations enjoyed re-

markable success. When the number of truncated chan-nels is kept small, the calculations are reasonably fast,but they rapidly become laborious as the number ofchannels increases. A disadvantage arose, however,upon direct integration of the close-coupling equationson an energy grid sufficiently fine to reproduce narrowautoionizing states. This stemmed from the fact that theentire integration had to be repeated over and over, vir-tually from the beginning, at every energy for which thesolutions were desired. To treat the extremely narrowresonances that are commonly seen in multichannelspectra, it was highly desirable to overcome this road-block.Two independent theoretical developments greatly

enhanced the power of these calculations. First, Seatonformulated the multichannel quantum-defect theory(Seaton, 1955, 1966, 1983). This formulation, to be dis-cussed at length in Sec. II, describes much of the com-plicated energy dependences of multichannel spectraanalytically. In this paper, the abbreviation MQDT willbe used for this theory. After implementation of multi-channel quantum-defect theory in the context of atomicclose-coupling calculations, the equations could then besolved on a very coarse energy mesh. The stronglyenergy-dependent spectra emerged in a subsequentrapid MQDT calculation (involving the solution of amodest-sized system of linear algebraic equations).The second key development that led to an extremely

efficient method of solving the close-coupling-type equa-tions was Burke’s implementation of the Wigner-Eisenbud R-matrix approach (Burke and Robb, 1975;Burke and Berrington, 1993). Here the basic idea wasthat the energies El and the partial decay width factorsg il of Eq. (1.1) could be obtained after a single diago-nalization of the full Hamiltonian H inside the boxr,r0. Once the eigenvectors and eigenvalues of H wereknown, Eq. (1.1) allowed a nearly instantaneous calcu-lation of the R matrix at all energies for which the basisset converged adequately. Calculations based on this abinitio technique reproduced measured multichannelphotoabsorption spectra for a number of atoms, includ-ing closed-shell species such as argon (Burke and Tay-lor, 1975), and open-shell atoms such as aluminum(Tayal and Burke, 1987). These studies made little or nouse of multichannel quantum-defect techniques, optinginstead to solve the long-range close-coupling equationsby direct numerical integration.Fano’s (1970) paper on molecular hydrogen sparked a

period of extraordinary progress in the interpretation ofatomic and molecular spectra. It combined multichannelquantum-defect theory with the novel concept of aframe transformation to extract a detailed description ofH2 rotational-channel interactions. Subsequent studiesby Jungen, Dill, and others extended these capabilitiesto include vibrational-channel interactions in diatomicmolecules (Jungen and Dill, 1980; Jungen, 1984; Greene



and Jungen, 1985; Gauyacq, 1987), and in triatomics(Child and Jungen, 1990; Stephens and Greene, 1995).This body of work was able to account for a number ofexamples of rich molecular spectra with unprecedentedquantitative accuracy. Some types of electronic-channelinteractions in diatomics were also treated successfullyin this approach (Ross and Jungen, 1994).Fano’s concept of a frame transformation was soon

adapted to the spectra of atoms (Lu, 1971; Lee and Lu,1973; Fano, 1975) and of negative ions (Rau and Fano,1971; Lee, 1975). The emphasis of these studies was thesemiempirical interpretation of experimentally observedspectra. A manageable set of energy-independent pa-rameters was adjusted to reproduce observations. Thesefits proved to have much more physical content thanmere ‘‘curve-fitting’’: the same parameters were used togenerate predictions of completely new observables(Dill, 1973; Lee, 1974b) that had not been fitted initially,and were subsequently verified by experiments. Manymore examples (Aymar, 1984a) of the predictive powerof multichannel quantum-defect analysis, both with andwithout frame-transformation techniques, established itas a crucial tool for understanding multichannel Ryd-berg spectra.While the examples just discussed demonstrated the

potential power of techniques based on multichannelquantum-defect theory, multichannel effective-rangetheory, and their generalizations, some limitations ofthis semiempirical analysis became increasingly appar-ent. When the number of channels exceeded a numberin the range 5–10, the number of fitted parameters be-came so large that the fits become nonunique and unsat-isfactory. As increasingly complex systems were sub-jected to experimental study, it became imperative todevelop ways to calculate the smooth, short-range pa-rameters of multichannel spectroscopy directly, i.e.,without relying on semiempirical fits. Fano and Lee(1973) took one of the first steps in that direction whenthey proposed the eigenchannel R-matrix method, whichLee (1974a) used to calculate argon MQDT parametersnear the ionization threshold. The relativistic random-phase approximation was extremely successful in deter-mining MQDT scattering parameters and numerous as-sociated photoabsorption observables for every rare-gasatom (Johnson et al., 1980; Lee and Johnson, 1980).The extensive development of nearly ab initio multi-

channel spectroscopy in the last decade has largely by-passed the difficulties of semiempirical quantum-defecttreatments alluded to above. This enhancement of theo-retical capabilities was driven largely by a reformulationof the eigenchannel R-matrix treatment into a nonitera-tive form (Greene, 1983; Le Rouzo and Raseev, 1984),which dramatically improved on the iterative versionproposed and implemented by Fano and Lee (1973).Another key contribution to the improved ability oftheory to describe complicated spectra to near-experimental precision has been the use of model poten-tials that approximately describe screening and polariza-tion effects of the inner closed electron shells. Forinstance, to describe Ca spectra, one first finds a model

one-electron Hamiltonian h that represents the Ca1 en-ergy spectrum to reasonable precision. Next, one ‘‘ham-mers’’ the two-electron problem variationally within thefinite radius max$r1 ,r2%<r0, typically using the modelHamiltonian for the valence-electron pair:H5h11h211/r12 . The resulting variational solutionsare now matched to a channel expansion in terms oflinear Coulomb functions (or whatever are the relevantlong-range solutions) at the surface of this reaction vol-ume. The matching coefficients then determine theshort-range scattering parameters, which can in turn beused to calculate experimental observables on an arbi-trarily fine energy mesh with great efficiency. Moreover,the parameters can then be used to classify and interpretthe spectra, which can be surprisingly complicated evenfor ‘‘simple’’ atoms such as He .The basic procedure just outlined has been applied

extensively to many systems, especially in the alkaline-earth atoms where a vast amount of data exists. Theprecision of the resulting calculations has been surpris-ingly good. Many experiments that had remained ‘‘unin-terpreted’’ are now completely understood, thanks tothese methods. Nearly ab initio calculations have pre-dicted a number of new, highly detailed spectra, achiev-ing agreement in most cases to near-spectroscopic accu-racy. In several cases, calculations based on thistheoretical scheme have improved significantly over theaccuracy of earlier experimental work. A number of ex-amples are described in Sec. IV below. These methodshave also been extended to treat the spectra of open-shell atoms, including some transition metals which ex-hibit impressive complexity. Recent applications toopen-shell atomic systems are discussed in Sec. V.Another area of current interest is the development of

techniques for the analysis of complex spectra in thetime domain rather than the energy domain. A time-domain spectrum is simply the Fourier transform of thecorresponding energy-domain spectrum. While this re-view will consider energy spectra almost exclusively,some interpretive aspects are useful to consider from thetemporal point of view. Time analysis can be performedusing an outgrowth of Wigner’s (1955) identification of

t l~E !52\dd l~E !

dE(1.3)

as the time delay in scattering associated with thepartial-wave scattering phase shift d l(E) at energy E .This concept of the time delay was generalized to mul-tichannel scattering theory by Smith (1960). Section IIbelow presents some additional generalizations that areuseful in the multichannel quantum-defect approach,some of which have not been previously published.These further generalizations relate to the interpretivevalue of Smith’s delay-time matrix in the presence ofweakly closed channels (i.e., bound-state channels).

C. Other related reviews

A number of other published reviews and mono-graphs touch on subjects relevant to this paper. Discus-



sions of multichannel quantum defect-theory can befound in the books by Gallagher (1994), by Fano andRau (1986), and by Friedrich (1990). Gallagher (1994)also discussed experimental aspects of importance formeasurements of multichannel spectra. A collection oforiginal articles on the formulation and application ofR-matrix methods in atomic physics was published byBurke and Berrington (1993). In a review with a muchmore general flavor, Gerjuoy et al. (1983) showed howto develop new variational principles for a broad rangeof applications. Buckman and Clark (1994) reviewed ex-perimental and theoretical work on negative ions, em-phasizing collision processes. Gauyacq (1987) adoptedtechniques similar to those used in Sec. II of this paper,in a pedagogical treatment of several different processesthat involve negative ions. Semiempirical applications ofmultichannel quantum-defect theory to the analysis ofexperimental atomic Rydberg spectra were reviewed byAymar (1984a). Greene and Jungen (1985) and Jungen(1988) have reviewed molecular applications ofquantum-defect techniques. Molecular studies requirethe introduction of some key additional concepts in or-der to deal with the rotational and vibrational degrees offreedom.(Our convention in denoting matrices is to underline

them. We denote the transpose matrix by the superscriptt , and the Hermitian conjugate matrix by a dagger (†);for spectroscopic notations, we use superscripts e and oto denote even and odd parity states.)

II. OVERVIEW OF THE IDEAS AND EQUATIONS OFMULTICHANNEL SPECTROSCOPY

The basic ideas of quantum-defect theory have re-mained mysterious to many, partly because its startingpoint already differs from the more widely familiarindependent-electron approach to atomic structure. Thephilosophy of the Hartree-Fock method—and its im-provements multiconfiguration Hartree-Fock orconfiguration-interaction methods—begin by assumingthat all atomic electrons move independently in a mutu-ally screened central potential. This approximation ismade throughout all of configuration space. When theatomic wave function can be accurately represented byone or at most a handful of configurations, as is fre-quently true for low-lying states, this independent-electron scheme describes the dynamics efficiently. Itsutility decreases for highly perturbed or correlated statesthat require extensive configuration mixing, especiallywhen continuum configurations play a crucial role.Multichannel spectroscopy adopts a philosophy closer

to that of the nuclear R-matrix theory of Wigner(Wigner, 1946a, 1946b). Specifically, an independent-electron-type approximation is adopted only in an‘‘outer region’’ r.r0, where, aside from perturbativeelectrostatic multipole interactions, it is virtually exact.In this large-r region, a matrix equivalent to the scatter-ing matrix S , generalized to incorporate energeticallyclosed channels, characterizes the solutions ‘‘exactly.’’As shown below in Sec. II.D, the large-r solutions can

be also characterized by a reaction matrix K connectedto the matrix S . Quantum-defect theory (Seaton, 1983)for a long-range Coulomb potential, effective-rangetheory (Fermi, 1934; Fermi and Marshall, 1947; Bethe,1949; Blatt and Jackson, 1949; Delves, 1958; Ross andShaw, 1961; Rau and Fano, 1971; Lee 1975) for long-range purely centrifugal potentials, and their variantspredict the energy dependence of these matrices S andK , permitting analytic continuation of the key scatteringinformation above or below fragmentation thresholdsand giving the shape of complex spectra in analytical orsemianalytical form. What quantum-defect theory can-not provide, by itself, are the actual values of scatteringparameters such as S or K for any specific atom, sym-metry and energy range.In simple cases having a relatively small number of

channels, one can hope to adjust the scattering param-eters semiempirically until an experimental spectrum isreproduced. Most of the early applications of multichan-nel quantum-defect theory used this semiempirical ap-proach. Even though the fitting procedure falls short ofbeing a comprehensive atomic theory, it is useful fororganizing extremely complicated spectra in terms ofrelatively few parameters. It also has nontrivial predic-tive power. Thus, for example, Dill (1973) and Lee(1974b) used parameters fitted to the bound energy lev-els and oscillator strengths of atomic xenon to predictthe angular distribution and spin polarization of photo-electrons. Indeed, the first applications of effective-range theory showed similar predictive power in relatinga low-energy scattering phase shift to the energy of abound state lying just below a fragmentation threshold.For systems having a large number of channels, more

than about five, normally there is not sufficient experi-mental information to permit an unambiguous determi-nation of the scattering parameters by using the semi-empirical fitting procedure. In the presence ofelectronically excited perturbing energy levels, like thedoubly excited states common in the alkaline-earth at-oms, the unknown energy dependence of the scatteringparameters also complicates the fitting; the resultingnonuniqueness of the fits has plagued some studies.More recently, it has become possible to calculate thescattering parameters in an ab initio or nearly abinitio calculation. Such calculations have most fre-quently been performed using R-matrix methods (Fanoand Lee, 1973; Lee 1974a; Burke and Robb, 1975;O’Mahony and Greene, 1985; Hamacher and Hinze,1989, 1991; Greene and Aymar, 1991), but they havealso been carried out using the relativistic random-phaseapproximation (Johnson et al., 1980), hyperspherical co-ordinate methods (Greene 1981; Watanabe, 1982, 1986;Fano, 1983; Tang et al., 1992), and the Schwinger varia-tional principle (Goforth et al., 1987; Goforth and Wat-son, 1992). These techniques for calculating multichan-nel scattering parameters and spectra have improveddramatically in recent years, both in their accuracy andin their efficiency.Below in Sec. II.A we attempt to remove some of the

mystery surrounding these methods by presenting the



basic qualitative ideas of multichannel spectroscopy.While it is developed in a somewhat heuristic fashion,this section does give a useful way of thinking aboutmultichannel spectra. Most of the qualitative phenom-enology present in multichannel Rydberg spectra can, inprinciple, be deduced from these pictures. Moreover, weregard this development as fundamental in the sensethat it requires hardly any approximations, in contrast towidely applied independent-electron models. The quan-titative formulation of these ideas awaits Sec. II.B. be-low, which will begin to give the flavor of their nontrivialpower and utility.

A. Visualizing atomic photoabsorption

Imagine that an atomic system swallows an incidentphoton of frequency v and in the process an atomicelectron gains kinetic energy \v . This language alreadyrelies on an independent-electron viewpoint, which isfrequently inappropriate since a single electron in amany-electron atom does not have ‘‘its own energy.’’More precisely, only the total energy E of the atom hasa definite value in the stationary final state reached byphotoabsorption, since the electrons are continually ex-changing energy with each other via their Coulombicinteraction 1/r12 . Similarly, the electrons frequently ex-change spin and orbital angular momentum with eachother, while the total squared spin SW 2 and orbital mo-mentum LW 2 have conserved eigenvalues (neglectingfine-structure effects). The possibility for exchange ofenergy and angular momentum between electrons di-minishes as the outermost electron moves outward to adistance r , owing to the decay of the inner electron’swave function and to the falloff of 1/r12 . Eventually aradius r0 is reached, such that beyond r.r0 the prob-ability of any such exchange process diminishes to a neg-ligible value. For all practical purposes the outermostelectron is now distinguishable from the electrons of theresidual ion that are confined to the region r,r0.

1. Single-channel processes

Figure 1(a) gives a picture of the photoabsorptionprocess reaching a singly excited Rydberg state of a two-electron atom. The inner electron is confined to r1,r0,in a stationary state of energy E1, by the steeper (un-screened) attraction to the nucleus. When electron 2moves beyond this radius r0, its own energy e is con-served at the value e5E2E1, determined by the atomicfinal-state energy E . Classically, the outermost electronis reflected from the outer turning point if e,0, while itescapes uninhibited to infinity if e.0. The former pro-cess is called photoabsorption to a discrete state, whilethe latter is called direct or nonresonant photoioniza-tion. Quantum mechanics modifies this statement in theusual way. Specifically, in the case where the outermostelectron is reflected classically (at e,0) only a discreteset of energies is allowed. These are energies for whichthe Schrodinger wave function decays exponentially atr2→` , and they are known to obey a Rydberg formula

Enl5E12Ry

~n2m l!2 . (2.1)

Here Ry is the Rydberg constant and the positive inte-ger n is the principal quantum number. The parameterm l is a dimensionless quantity, roughly constant in en-ergy but dependent on the atomic-state quantum num-bers such as l . Historically m l has been called the quan-tum defect. Section II.B describes the origin of thisenergy expression and the reinterpretation of pm l as ascattering phase shift.

2. Two-channel processes

A less trivial situation is depicted in Fig. 1(b), wherenow the inner electron e1

2 can be in one of two differentstates, with energies E1 or E2. This two-channel proto-

FIG. 1. Coulombic potentials (V) and total energies (E) foran electron outside an ionic core (shaded zone): (a) single-channel photoabsorption process with total energy E corre-sponding to a situation in which the electron is bound withrespect to the ionization limit E1; (b) two-channel photoion-ization process for final energy E between the first and secondionization limits E1 and E2. The radial position of the electronis denoted by r , in atomic units.



type atom begins to show the main features characteris-tic of more complex atoms. Again, the radius r0 marks aboundary beyond which the exchange of energy and an-gular momentum can be neglected. Below the first ion-ization threshold E1, all energy eigenstates are discretebound levels. Except for the ground state, each normallyhas a small natural linewidth due to spontaneous emis-sion that we ignore for the purpose of this review.Between the first and second thresholds E1,E,E2,

all energies are allowed eigenvalues of the Hamiltonian,but the continuum is ‘‘lumpy’’ in that the density ofstates is enhanced at a sequence of autoionizing Fesh-bach resonances converging to the upper threshold. Theresonance energies approximately obey a Rydberg for-mula

EnLSres 5E22

Ry

~n2mLS!2, (2.2)

where mLS is nearly energy independent. The widths ofthese resonances decay with principal quantum numberroughly as n23, which can be represented by incorporat-ing a nearly constant imaginary part into the quantumdefect mLS (Seaton, 1969). We show below how thisimaginary part of m can be extracted in terms of basicscattering information, mainly the scattering probabilityuS21u2 for the outer electron to induce a transition be-tween inner-electron states E2 and E1 in a single colli-sion.Above the upper threshold, E.E2, the continuum

becomes smooth and is no longer lumpy since there areno more Feshbach resonances. Because two continuumchannels are open, however, two physically distinguish-able final states are reached. These simply add incoher-ently to the total photoionization cross section, but therelative strength of their contributions can in principlebe sorted out experimentally. This could be accom-plished, for instance, by measuring the state of the ionicresidue following the photoionization, or by resolvingthe detected photoelectron energy in a spectrometer.The branching ratio of cross sections s2 /s1 into thesetwo alternative channels is generally a smooth, nearlyflat function of energy. Observation of a sharp departurefrom near constancy of s2 /s1 implies the presence of ashape resonance or else perhaps the presence of a thirdchannel contributing a Feshbach resonance. (Long-range field effects can cause s2 /s1 to vary even in theabsence of resonances, when the long-range potential isnon-Coulombic.)In the discrete energy range below the first threshold,

E,E1, another common phenomenon occurs. The lev-els belonging to the two different channels can interfereand mutually perturb each other. A simple way to seewhether a particular bound level lies within this ‘‘per-turbed’’ energy regime is to extrapolate the sequence ofautoionizing Feshbach resonances down from the regionE1,E,E2 into the range E,E1. The ‘‘influence’’ ofeach perturber extends over an energy range roughlyequal to the width it would have were it an autoionizingstate, namely

GnLSres '

4RyIm~mLS!

@n2Re~mLS!#3. (2.3)

While the energy spectrum remains discrete for E,E1,all levels E that fall within the range EnLS

res 6 12GnLS

res willbe perturbed. The perturbation causes the photoabsorp-tion intensity to depart from the usual n23 Rydberg scal-ing law and it induces a shift in the energy of levels lyingwithin this width. The energy perturbation takes a verywell-defined form: the quantum defects of all bound lev-els, referred to the lower threshold, rise by one unitacross the perturber’s width.

3. Properties in the time domain

Viewed in real time, for instance when a subpicosec-ond laser pulse excites the atom, there are three charac-teristic time scales in the internal dynamics of a two-channel atom. Two of these are the Rydberg orbitalperiods t152p(2e1 /Ry)

23/2 in channel 1 andt252p(2e2 /Ry)

23/2 in channel 2. Notice that a greatdisparity in these periods arises at an energy just belowthe threshold E1, where t1@t2. The third relevant timescale is the average time required for a Rydberg electronin channel 2 to scatter into channel 1. This is essentiallythe autoionization lifetime t125uS12u22t2. Time-dependent wave-packet calculations show that when afast laser pulse coherently excites several autoionizingstates just below the threshold E2, the ejected electronsdo not emerge in a smooth exponentially decaying curvesuch as exp(2t/t12). Instead, a sequence of autoionizingelectron pulses is seen, with the temporal separation be-tween electron pulses equal to the Rydberg orbital pe-riod t2 and their relative heights governed by the antici-pated exponential decay factor. This reflects a keyproperty of multichannel Rydberg states, always to bekept in mind: energy exchange between the Rydbergelectron and other electrons is possible only once perorbital period when the Rydberg electron enters the re-action zone r,r0. Another point to bear in mind is thatthe reaction zone does not have a sharply defined radiusr0. Rather, it should be thought of as being smeared outto some extent, even though some methods (such asR-matrix method) choose a specific value.

B. Single-channel properties

Theoretical techniques commonly denoted as‘‘quantum-defect theories’’ stress particularly the factthat Feshbach-type resonances represent excitations ofclosed channels. Other names for this general class oftechniques include multichannel effective-range theory(Ross and Shaw, 1961; Lee, 1975), and the specific for-mulation of R-matrix theory developed by Teichmannand Wigner (1952) to include closed channels. We de-velop below the quantitative formulation that corre-sponds to the viewpoint described in the previous sub-section. We begin by treating motion in a single channel



and then describe, in the following sections, the consid-erations appropriate to an arbitrary number of interact-ing channels.When a system is confined to a single relevant chan-

nel, this implies the existence of just one possible two-body breakup mode. Beyond some distance r0 we sup-pose that the system has a well-defined effective localpotential energy v(r) in the fragmentation coordinater . Here we deal explicitly only with a long-range Cou-lomb potential, as the generalization to other potentialsis fairly straightforward and has been described else-where. The effective potential for a Rydberg electron inany channel depends on its orbital quantum number l .In atomic units,

v~r !5l~ l11 !

2r221r. (2.4)

In the region of configuration space r.r0 where thispotential energy adequately describes that of the atomicelectron, the atomic wave function, rescaled by r , canalways be written as a linear combination of two linearlyindependent solutions (f ,g) of the second-order radialSchrodinger equation 2 1

2u9(r)1v(r)u(r)5eu(r). Con-siderable flexibility exists in choosing (f ,g), but follow-ing Seaton (1966, 1983), some simplicity and conve-nience are gained by choosing them such that

~1 ! f→0 as r→0;

~2 ! f→~2/pk !1/2sinS kr11klnr1h D as r→`

for e[ 12k

2.0, which includes the logarithmic phase ac-cumulation appropriate for an attractive Coulomb fieldof unit charge, and a long-range phase shift h that isgiven explicitly for the Coulomb field in Eq. (2.39) be-low.

~3 ! g→2~2/pk !1/2cosS kr11klnr1h D as

r→` for e[ 12 k

2.0.

(4) (f ,g) for e,0 join smoothly onto the positive en-ergy solutions obeying conditions (1)–(3).Briefly, (1) is desirable so that a state with very high

orbital angular momentum l in any atom will reduce toone solution fel alone without requiring any superposi-tion with gel . Condition (2) ensures that f is energy nor-malized, i.e., ^feufe8&5d(e2e8). Condition (3) guaran-tees that (f ,g) are always linearly independent to themaximum extent. Finally, condition (4) ensures that anyrapid energy variations in the calculated scattering pa-rameters reflect dynamics of the atom rather than thearbitrary choice made for the reference functions. Westress that these criteria are only adopted to obtainmaximum simplicity in the eventual formulas to be de-rived below. There is nothing magical or mysteriousabout these stipulations, as in fact any two linearly inde-pendent solutions of the radial long-range equationwould serve as well as (f ,g). In fact it is sometimes use-ful to discuss another pair of outgoing/incoming radial

waves f6[(2g6if)/A2, where i5A21. (Avoidthroughout this section, the confusion between i5A21and the channel index i .)The analytical properties of the solutions (f ,g) are

detailed completely by Seaton (1983) [who calls them(s ,2c)] and by Greene et al. (1979, 1982), although thederivations involve somewhat laborious properties ofthe confluent hypergeometric equation. The main resultsneeded below follow far more simply from approximateWKB (see, for example Messiah, 1958) solutions of theradial Coulombic Schrodinger equation, along the linessketched by Greene et al. (1982). In particular, in theclassically allowed range with positive kinetic energy, wehave

fel~r !5S 2pk~r ! D 1/2sinS Eark~r8!dr81

14

p D , (2.5a)

gel~r !52S 2pk~r ! D 1/2cosS Eark~r8!dr81

14

p D . (2.5b)

Here the Langer (1937) corrected local wave vector is

k~r !5S 2e2~ l1 1

2 !2

r212rD 1/2, (2.6)

and a is the inner classical turning point, i.e., the zero ofk(r) that lies closest to the nucleus. For positive ener-gies, the above expressions for (f ,g) are valid all theway to r→` . At e,0, however, the standard WKB con-nection formulas must be applied to give the corre-sponding solutions beyond the outermost turning point,r@b . After setting k[(22e)1/251/n , where n is the ef-fective quantum number, the asymptotic forms of thesesolutions can be written as

fel~r !→~pk!21/2~sinbD21r2nekr2cosbDrne2kr!,(2.7a)

gel~r !→2~pk!21/2~cosbD21r2nekr1sinbDrne2kr!,(2.7b)

where b and D are constants depending on e and l . Inthe WKB approximation, for a Coulomb potential ofunit charge, they can be evaluated explicitly and aregiven by Greene et al. (1982). Of greatest interest forour purposes is the phase parameter

b512

p1Ea

bk~r !dr5p~n2l !. (2.8)

This phase (divided by p) measures the number of deBroglie half wavelengths in the radial solution betweenr50 and r5` .The asymptotic expressions (2.7) derived from a

WKB approach are, remarkably enough, identical to theform of Seaton’s (1966, 1983) exact nonrelativistic solu-tions. Moreover, the WKB approximation to b in Eq.(2.8) coincides precisely with the full quantum solution.Only the parameter D in Eq. (2.7) is changed. Milne’s(1930) phase-amplitude method provides a different ex-act quantum treatment, which can be viewed (Greeneet al., 1982) as an alternative to Seaton’s (1983) deriva-



tions based on complicated special-function properties.The Milne approach readily shows how a phase param-eter such as b and an amplitude such as D arise forsolutions of the time-independent Schrodinger equationwith any one-dimensional potential. This general pointof view permits us to generalize Seaton’s basic approachto any long-range reference field. A detailed descriptionof these generalizations would take us far afield from theaims of this review, but an interested reader can finddiscussions in the existing literature (Greene et al., 1979,1982; Mies and Julienne, 1984; Fano and Rau, 1986).For many purposes, the interaction between an elec-

tron and a closed-shell atom or ion can be described bya local, central potential V(r). Such a model-potentialdescription has considerable utility and some limitations,both of which are discussed below in Sec. III. Considerthe problem of finding energy levels of an electron withpartial wave l moving in this potential field. The exactradial solution, rescaled by r in order to eliminate thefirst derivative term in the usual way, obeys

212F9~r !1S l~ l11 !

2r21V~r ! DF~r !5eF~r !. (2.9)

To within an arbitrary proportionality constant, the so-lution at any chosen energy e is uniquely specified byregularity at the origin r50, although numerical meth-ods may be required to solve Eq. (2.9) for F(r). Thisdifferential equation coincides with the equation deter-mining (f ,g), because the effective potential in Eq. (2.4)was understood to coincide with the true atomic effec-tive potential in Eq. (2.9), at sufficiently large distancesr.r0. This, combined with the fact that a second-orderdifferential equation has just two linearly independentsolutions, ensures that for r.r0 the atomic solutionF(r) is a linear combination of (f ,g) with constant co-efficients. Characterizing the two coefficients by an am-plitude Nl and a phase pm l , we have

Fel~r !5Nl@fel~r !cospm l2gel~r !sinpm l# , r.r0 .(2.10)

If F(r) has been determined by numerically integratingEq. (2.9) from r50 to r5r0 or beyond, the phase shiftpm l can be calculated from a ratio of radial Wronskians:

tanpm l5fF82f8F

gF82g8FUr>r0

. (2.11)

The real constants Nl and m l depend on energy, but forthe most part this dependence is very smooth. Boundaryconditions at r→` still remain to be applied. At anynegative energy e,0, insertion of the asymptotic forms(2.7) into Eq. (2.10) shows that the solutions Fel(r) di-verge exponentially at r→` ,

Fel~r !→Nl~pk!21/2@sin~b1pm l!D21r2nekr

2cos~b1pm l!Drne2kr# . (2.12)

Accordingly, the bound-state quantization condition,needed to kill the exponential growth at infinity, readssimply

sin~b1pm l!50, (2.13)

or n2l1m l5nr , an integer. Defining the principalquantum number n[nr1l , the allowed energy values fitthe empirical formula proposed by Rydberg (1889), inatomic units (a.u.),

enl521

2~n2m l!2 . (2.14)

The term ‘‘quantum defect’’ for m l stems from the viewthat m l measures how ‘‘defective’’ an alkali atom is com-pared to the ‘‘perfect’’ atom (from the perspective of the‘‘old’’ quantum theory) hydrogen, whose quantum de-fects all vanish. A single key result initiated the so-called‘‘quantum-defect theory’’ of Ham (1955), Seaton (1955,1958, 1966, 1983), and their successors: the smooth man-ner in which below-threshold bound-state quantum de-fects evolve with increasing energy into above-thresholdscattering phase shifts d l5pm l . Seaton (1958) furthershowed that m l suffers from a mild nonanalyticity rightat the ionization threshold e50, but this singularity is soweak that it is inconsequential for an attractive Cou-lomb potential. Some further remarks on a few difficultproperties of Coulomb functions are deferred to Sec.II.D below.For other long-range potentials, such as the purely

centrifugal potential or the repulsive Coulomb potentialso important in nuclear physics, the energy-normalizedsolutions (f ,g) become highly energy dependent atsmall distances. This strong energy dependence can befactored out analytically by expressing (f ,g) in terms oftwo new solutions (f0,g0) that are (entire) analytic func-tions of the energy at any finite radius. It is convenient(and permissible) to choose f0(r) to be proportional tothe regular solution f(r), but g0(r) need not be propor-tional to g(r) in general. This point has been discussedelsewhere (Greene et al., 1979, 1982; Seaton, 1983; Fanoand Rau, 1986). In case the analytic solutions areneeded, the transformation between (f0,g0) and (f ,g)requires two r-independent parameters A and G that arefunctions of e and l :

fel~r !5Ael1/2fel

0 ~r !, (2.15a)

gel~r !5Ael21/2@gel

0 ~r !1Gelfel0 ~r !# . (2.15b)

Rau (1988) has derived many revealing interrelation-ships between all of these parameters at positive andnegative energies.ASIDE: For the special problem of a purely centrifu-

gal long-range potential, these solutions are givenin terms of spherical Bessel functions:fel(r)5(2k/p)1/2rj l(kr), gel(r)5(2k/p)1/2rnl(kr). Thetransformation (2.15) to analytic solutions (f0,g0) in-volves coefficients Ael5k2l11 and Gel50 in this simplecase. In examples such as this, for which (f ,g) dependstrongly on the energy at small distances, the phase shiftpm l

0 of F(r) relative to (f0,g0) is a smooth, analyticfunction of energy across the relevant fragmentationthreshold. Thus, taking l50 as a specific example, onesees at once that the tangent of the analytic phase shift



tanpm l05S f l0F82f l

08F

gl0F82gl

08FD U

r>r0

(2.16)

determines the energy of a high-lying s-wave boundstate at e,0. The analog to the Rydberg formula, appli-cable for zero long-range potential and l50, is readilyderived to be tanpm l50

0 1k2150, or

e52 12 k252 1

2 cot2pm l50

0 , (2.17)

provided tanpm l500 is negative and ‘‘large’’ in absolute

value. The physical phase shift d l50 entering the scatter-ing matrix Sl5exp(2idl) in this s-wave example becomes

d l505pm l505arctan~ktanpm l500 !, (2.18a)

and

Sl50511iktanpm l50

0

12iktanpm l500 . (2.18b)

Equations (2.16)–(2.18) constitute the basic formulas ofsingle-channel effective-range theory, which relatesbound-state properties to low-energy scattering proper-ties. The connection can be made complete by identify-ing the zero energy value of tanpm l50

0 as the usual ‘‘scat-tering length,’’ while its energy derivative is related tothe ‘‘effective range.’’ In this sense, the generalized formof quantum-defect theory encompasses both single-channel and multichannel effective-range theories asspecial cases (Blatt and Jackson, 1949; Ross and Shaw,1961).

C. Nonseparable quantum mechanics in severaldimensions

An enormous class of physical problems of currentinterest requires the solution of a nonseparable, time-independent Schrodinger equation in more than one di-mension. No single theoretical approach is known thatcan solve all such problems. In fact for many systems,such as the double continuum states of two electronsinteracting with each other and with another chargedparticle, only relatively crude approximations are pres-ently available. Moreover, even these crude solutionstypically apply over small ranges of energy, angular mo-mentum, and other quantum numbers.Dramatic progress has been achieved in recent years

in developing powerful techniques capable of handlingphysical systems of unprecedented complexity, providedtheir fragmentation involves no more than one coordi-nate at a time. A famous problem of this category, whichhas only been solved quite recently (Iu et al . , 1991;O’Mahony and Mota-Furtado, 1991; Watanabe andKomine, 1991), is the near-threshold photoionizationspectrum of hydrogen or of an alkali atom in an uniformmagnetic field. In that three-dimensional problem, thefragmentation coordinate is uzu, the projection of theelectron-nucleus separation rW onto the magnetic field di-rection BW 5Bz . Motion in the two remaining coordi-nates (r ,f) or (x ,y) is bounded at any finite energy,

and is therefore quantized. (Here we suppose that theusual choice of vector potential has been made, AW

5 12BW 3rW .) Also in the atomic two-electron problem, at

energies below the threshold for double escape, a singlefragmentation coordinate can be identified, either thedistance r2 of the outermost electron from the nucleusor the hyperspherical radius R5Ar121r2

2 (Fano, 1983).Another example that, in contrast, seems to involvemore than one coordinate is Jungen’s (1984) descriptionof competing ionization and dissociation channels inH2; because only one of these processes occurs in anygiven asymptotic fragmentation channel, this last prob-lem fits into the same category as well.

1. Characterizing the solutions

We restrict our attention now to solving thestationary-state Schrodinger equation for such systemswhere one fragmentation coordinate, to be written asr , is assumed to be identifiable. As Fano (1981) hasstressed, a generalized hyperspherical radius or momentof inertia coordinate can serve quite generally in thiscapacity. Other choices may be more computationallyconvenient depending on the problem at hand and onthe solution method adopted. The main point we needfor now is that the d-dimensional configuration spacecan be separated into mutually orthogonal coordinates:one fragmentation coordinate r and d21 ‘‘surface’’ co-ordinates v . The coordinates v are interpreted to in-clude the spin degrees of freedom as well as true spatialcoordinates.Because the multidimensional Schrodinger equation is

homogeneous, with up to second-order derivatives,knowledge of a particular rescaled solution $rcb(r ,v)%and its radial derivative ]$rcb(r ,v)%/]r uniquely deter-mines cb(r ,v) at all large distances r>r0. The subscriptb distinguishes different linearly-independent solutionsthat are degenerate in energy, angular momentum, par-ity (denoted by p), and any other ‘‘exact’’ quantumnumbers. [Note that, throughout this section, the indexb should not be confused with the quantum-defect-theory phase parameter of Eq. (2.8).] In fact it would beinconvenient to tabulate the wave function and normalderivative at the continuously infinite set of points vrepresenting this surface. Fortunately, such a table is notreally needed anyway for the following reason: a dis-crete set of functions F i(v) can always be found thatspans the fixed r surface. These functions might be, asdescribed in this section, a set of r-independent eigen-states [frequently appropriate at r→` as in the close-coupling representation (Burke and Smith, 1962)], or aset of ‘‘adiabatic’’ eigenfunctions of the fixed-r Hamil-tonian (Born and Oppenheimer, 1927).In any case, the surface Fourier decompositions of

$rcb(r ,v)% and ]$rcb(r ,v)%/]r in terms of this basiscan for practical purposes be truncated to a finite num-ber N at any chosen energy, on physical grounds. In thespecial case that F i(v) are independent of r , we canwrite the decompositions as



cb~r ,v!5(i

1rF i~v!Fib~r !, (2.19a)

]@rcb~r ,v!#

]r5(

iF i~v!F8ib~r !. (2.19b)

Knowledge of Fib(r) and Fib8 (r) in Eqs. (2.19), alongwith the Schrodinger equation itself, provides enoughinformation to uniquely determine cb at all distanceslarger than r0. We focus on this large-r region becausethat is where the scattering information will eventuallybe extracted.Quite generally, if the Fourier decompositions (2.19)

of the Schrodinger solutions regular at r→0 involve Nterms or channels i , one expects to find also N linearly-independent solutions, b51, . . . ,N . Thus we can speakof Fib(r) as representing an (N3N) solution matrix.This is a square matrix prior to imposing boundary con-ditions at r→` , each column of which represents a sepa-rate independent solution as indicated by Eq. (2.19).The ith row gives the projection of the ith surface har-monic or channel function F i(v) onto each independentsolution. The phases of Fib and F i can be chosen suchthat Fib(r) is a real matrix for almost all problems inatomic and molecular physics.Any linear combination of the degenerate Schro-

dinger eigenfunctions cb is also a valid eigenfunction.This implies that considerable nonuniqueness exists inthe choice of the Fib(r). Stated in mathematical terms,any constant nonsingular (N3N) linear transformationmatrix Xbg can be used to transform Fib(r) into anequally acceptable set of solutions—for example,

cg5(b

cbXbg5(i

1r

F i~v!Mig~r !, (2.20a)

where

Mig~r !5(b

Fib~r !Xbg . (2.20b)

Prior to imposing physical boundary conditions atr→` , nothing in the mathematics or physics ‘‘prefers’’one set of independent solutions $cb% over another set$cg%. However one can identify a ‘‘minimal’’ amount ofinformation regarding the solution at smaller distancesr,r0 that provides enough information to determine thescattering matrix. This follows upon recognizing that thematrix

R[F~r !@F8~r !#215M~r !@M8~r !#21 (2.21)

is invariant under transformations such as X . FollowingWigner (1946b), we call this fundamental invariantquantity the matrix R . It is real and symmetric, and willbe seen below to determine the scattering matrix S andits close relative, the reaction matrix K .

2. Modified form of the close-coupling method withoutexchange

For readers not already familiar with multichannel,multidimensional problems in quantum mechanics, the

preceding discussion may seem somewhat nebulous. Inorder to make it more concrete, we turn to a specificexample—the atomic close-coupling equations for a dis-tant electron in the field of a residual ion or neutralatom with a single valence electron. To simplify our no-tation somewhat, the distant electron is assumed tomove in a region having no overlap with the residualelectrons. This permits us to neglect all exchange effects.We further neglect fine-structure effects; this permits usto write the exact solutions in LS coupling, omitting ref-erence to the spin degrees of freedom for the sake ofbrevity. For this example a convenient choice for thefragmentation coordinate r is the distance between theoutermost electron and the nucleus. Accordingly, the setof coordinates v orthogonal to r consists of all spatialcoordinates of the inner residue, and the angular coor-dinates r[(u ,f) of the distant electron. The surfaceharmonics or channel functions F i(v) are eigenfunc-tions of the inner-electron Hamiltonian h1, the inner-electron squared orbital angular momentum operatorlW12, the distant electron squared orbital momentum lW2

2,the squared total orbital angular momentum LW 2, and thez component Lz . The detailed eigenvalue equationsread

h1F i~v!5EiF i~v!, (2.22a)

lW12F i~v!5l1i~ l1i11 !F i~v!, (2.22b)

lW22F i~v!5l2i~ l2i11 !F i~v!, (2.22c)

LW 2F i~v!5L~L11 !F i~v!, (2.22d)

LzF i~v!5MF i~v!. (2.22e)

Aside from possible degeneracies of these eigenvaluesand routine phase ambiguities, these equations uniquelydetermine the channel functions. The total Hamiltonian(that includes electrostatic interactions only) is

H5h12121r

]2

]r2r1

lW22

2r22Znet

r

1 (k51

`

r2k21r1kPk~ r1• r !. (2.23)

The term Pk() represents a Legendre polynomial andZnet is the asymptotic (screened) net charge of the resi-due that is seen by the distant electron.At this point we depart in a crucial way from the usual

formulations of the close-coupling method (Burke andSmith, 1962). Our motivation for this departure followsfrom Fig. 2, which depicts the radial portion of configu-ration space for the two electrons.Because we consider energies below the ‘‘Wannier’’

(1953) threshold for double escape, one can identify aradius r0 such that the wave function c50 whenr1.r0 and r2.r0 simultaneously. More accurately, ofcourse, the true wave function decays exponentially inthat region, but the exponential falloff is sufficientlyrapid that neglect of c altogether in region IV can bemade arbitrarily precise by increasing the value of r0. In



this spirit, we seek to construct solutions in regions IIand III alone, subject to the (nonstandard) boundarycondition that c50 at the boundary of each region withregion IV. To be specific, we treat region II only, wherer2.r1, and require that c50 for r15r0. In region II thefragmentation coordinate r coincides with r2. With thisboundary condition, the energy spectrum of the innerelectron in Eq. (2.22a) is entirely discrete. The close-coupling expansion (2.19) is likewise discrete, whereas inthe conventional close-coupling method the summation( i in Eq. (2.19) implies an integral over the inner-electron continuum energy eigenstates. Without this in-tegral the F i(v) do not form a complete set. In conven-tional calculations this continuum integral is usuallyignored or else it is approximated by a discrete summa-tion using so-called ‘‘pseudostates’’ (Burke, Gallaher,and Geltman, 1969). The critical role of continuum con-tributions can be seen from the fact that the continuumcontributes roughly half of the hydrogen atom ground-state polarizability in a second-order perturbationtheory calculation. In fact, this correct description of thepolarizability was the initial motivation for the introduc-tion of discrete pseudostates (Burke, Gallaher, andGeltman, 1969) that approximately describe the effect ofthe neglected continuum. Another recent study (Brayand Stelbovics, 1993) has developed a systematic way toincorporate continuum coupling effects by using Stur-mian functions to represent the inner-electron wavefunction. This method appears to offer similar advan-tages to the present systematic scheme for generating acomplete set of discrete solutions that can represent theinner-electron degrees of freedom.A physically relevant point to keep in mind is the fact

that the boundary condition c50 at r15r0 causes theinner electron’s radial orbitals to differ from the usualorbitals calculated over 0,r,` . This difference mightappear to be undesirable at first glance. The entire treat-ment for the present example, however, is only intended

for an electron interacting with a residual fragment stateconfined within r,r0. The modified boundary conditionconsequently introduces no further errors. More to thepoint, though, the convergence of the expansion (2.19)improves dramatically as a result of confining the inner-electron eigenstates to a ‘‘box’’ of radius r0. There isconsequently no mathematical or physical need for ad-ditional pseudostates of any type.Insertion of Eq. (2.19) into the time-independent

Schrodinger equation at energy E , followed by projec-tion onto ^F ju, leads to the following infinite but discreteset of coupled differential equations,

Fib9 ~r !12(j

@e id ij2Vij~r !#Fjb~r !50. (2.24)

Here e i[E2Ei is the asymptotic electron kinetic en-ergy in the ith channel. The potential matrix Vij(r) in-cludes a diagonal centrifugal term in addition to an elec-trostatic coupling matrix element:

Vij~r !5S l i~ l i11 !

2r22Znet

r D d ij

1 (k51

`

r2k21^F iur1kPk~ r1• r !uF j&. (2.25)

In matrix notation, Eq. (2.24) becomes more succinct,

F9~r !12@e2V~r !#F~r !50. (2.26)

Upon truncating the infinite set of second-order coupleddifferential equations (2.24) to a finite number N , onecan find, in general, 2N linearly independent solutionsat each energy E . After boundary conditions at the ori-gin are imposed, Fib(0)50, N solutions remain math-ematically acceptable prior to enforcing boundary con-ditions at asymptotic distances.We present a few more detailed aspects of the trun-

cated system of N close-coupling equations [(2.24),(2.26)]. The first important property is constancy of the‘‘Wronskian matrix’’ of two different solution matricesF and M , provided that each of them obeys Eq. (2.26).The Wronskian is defined for this system to be

W~M ,F ![MtF82M8tF . (2.27)

By matrix-multiplying Mt to Eq. (2.26) from the left,and subtracting its analog with M and F interchanged,one finds that the Wronskian of any two solution matri-ces is a constant. This statement is independent ofboundary conditions and reflects mainly the absence offirst-derivative terms in Eq. (2.26) and the symmetry ofthe potential matrix V . A second useful property of so-lutions of Eq. (2.26) is the simple first-order differentialequation obeyed by their matrix R . This can be obtainedby differentiating the definition (2.21) and then usingEq. (2.26) to eliminate the second-derivative matrix thatoccurs. The resulting equation reads

FIG. 2. Radial configuration space for two electrons with re-gions I, II, III, and IV marked.



R8~r !5112R~e2V !R , (2.28)

where 1 is the (N3N) identity matrix. (Note that theidentity matrix is denoted by 1 through this review.)This differential equation could be solved numerically,for instance, to calculate long-range multipole correc-tions to the matrix R . Ideas along the lines of the first-order nonlinear differential equation were pursued inthe context of single-channel and multichannel problemsby Calogero (1967).For a variety of reasons, including the fact that R(r)

typically has numerous poles, direct solution of Eq.(2.28) is not the most efficient way to solve for the ma-trix R . One further useful property emerges from aninspection of Eq. (2.28), namely the fact that symmetryof the matrix R at one value of r immediately guaran-tees symmetry at all r values. In most problems the ma-trix R(r) is diagonal in some region of space, often atsmall distances, which is enough to prove that it willremain real and symmetric at all radii.

D. Long-range treatment using multichannel quantum-defect theory

For the purposes of this section, we suppose that theR matrix has been calculated for an atomic systemwithin the small-r region in which all electrons overlapeach other and exchange could be important. For anatom with two valence electrons, this corresponds to re-gion I in Fig 2. Section III below discusses efficientmethods for accomplishing this small-r part of the calcu-lation in practice. Our goal now is to discuss the motionof a single electron when it emerges from this small-rregion (or R-matrix ‘‘box’’). For a two-electron atom,this amounts to solving the Schrodinger equation in re-gions II or III in Fig. 2. One possible approach is tosimply adopt the close-coupling equations without ex-change in those regions, as presented in Eq. (2.26). Sev-eral algorithms and computer programs have been pub-lished to accomplish this numerically. This directnumerical solution of the long-range coupled equationscan be rather time consuming. However, recent rapiddevelopments in computational power and in computa-tional algorithms have improved the power of this nu-merical approach for dealing with the outer-field calcu-lation.Another approach has received widespread use owing

to its great efficiency. Seaton’s multichannel quantum-defect theory can be used to solve the long-range part ofthe problem in closed analytical form, at the price ofmaking an approximation to the close-coupling equa-tions at large distances. In its simplest form, which coin-cides with Seaton’s initial formulation, this approxima-tion neglects all anisotropic multipole contributions(k.0) to the interaction between an outermost electronand the inner residue. This means neglecting the entire(k51

` in the potential matrix [Eq. (2.25)] that enters thelong-range close-coupling equations (2.26). In someproblems, where this approximation seems dubious, onecan increase somewhat the size of the box radius r0,

since we assume that all multipole interactions are cor-rectly treated within. In any case, this approximation im-plies that the channels are completely decoupled beyondr5r0. In physical terms, this amounts to neglect the scat-tering of an electron from one channel into anotherwhenever it moves in the outer region.Mathematically, the absence of off-diagonal coupling

matrix elements in the outer region (in this approxima-tion) implies that a simple attractive Coulomb plus cen-trifugal potential applies within every channel individu-ally, at all radii r.r0. This diagonal potential in eachchannel coincides with the form of Eq. (2.4), if we takeZnet51, as would be appropriate for describing states ofa neutral atom or molecule. The generalization of theseresults to arbitrarily ionized (nonrelativistic) species isstraightforward. We start from the assumed known val-ues of the wave function and derivative at r5r0, or moreconveniently their projections on the ith channel func-tion:

Fib~r0!5^^F iu~rcb!&&, (2.29a)

Fib8 ~r0!5 K K F iU ]~rcb!

]r L L . (2.29b)

Here the double bracket notation is used to denote anintegral over only the surface of the reaction volume,while single brackets will imply the usual integral overall space.These surface properties of the previously calculated

inner-region solutions serve as boundary conditions thatuniquely determine the (N3N) solution matrix Fib(r)in the outer region, in conjunction with Eq. (2.26). Spe-cifically, in view of the relevance of Eq. (2.4) as the po-tential within each channel, the ith channel componentof the bth independent solution is a linear combinationof regular and irregular Coulomb functionsf i(r)[fe il i

(r) and gi(r)[ge il i(r). This follows because

Fib(r) obeys the same differential equation as (f i ,gi),whereby

Fib~r !5f i~r !Iib2gi~r !Jib , (2.30)

and the matrices I and J are independent of r . They canbe calculated by taking the Wronskian of both sides ofEq. (2.30) with respect to f i and then with respect togi , which gives explicitly

Iib5W~gi ,Fib!/W~gi ,f i!, (2.31a)

Jib5W~f i ,Fib!/W~gi ,f i!. (2.31b)

Since these Wronskians are independent of r , they canbe evaluated at any r>r0. They are usually calculatedright at the boundary r5r0, however, where a varia-tional or other type of calculation has provided Fib andits derivative with respect to r .So far very little has been specified about the particu-

lar choice of independent solutions represented by cband the corresponding radial-channel componentsFib(r) beyond the boundary r0. Normally, these calcu-lated independent solutions do not yet obey physicallyrelevant boundary conditions. Alternative sets of lin-



early independent solutions, corresponding to alterna-tive boundary conditions at r→` , can be used to de-scribe the stationary states at a definite energy E . Theasymptotic solutions for r>r0 are characterized by ei-ther the reaction matrix K or the scattering matrix Sreferring to open and closed channels. First, we intro-duce the solutions characterized by the reaction matrixK , which is simply connected to the matrix R . Then,below in Secs. II.D.1 and II.D.2, the basic derivations ofmultichannel quantum-defect theory are presented interms of the scattering matrix S .A first standard set of linearly independent solutions

that satisfy standing-wave boundary conditions is givenby a linear transformation of the solutions F :

M~r !5F~r !~I !21, (2.32)

where the ith channel component of this i8th indepen-dent solution is

Mii8~r !5f i~r !d ii82gi~r !Kii8. (2.33)

The asymptotic solutions Mii8(r) are characterized byan (N3N) matrix K[J(I)21 which is real and symmet-ric. Because this matrix K usually includes closed-channel indices (i.e., for channels that are energeticallyforbidden as r→`), its energy dependence is normallyquite smooth and this matrix can frequently be assumedto be energy independent over limited energy intervalssuch as across an ionization threshold. We refer to K asthe ‘‘smooth short-range reaction matrix,’’ and we callthe particular set of independent solutions (2.33) the‘‘short-range reaction-matrix solutions.’’ When an at-tractive Coulomb potential describes the long-range mo-tion in every channel, K coincides with the matrix R ofSeaton (1983). Recalling expressions (2.31) and (2.21),the K and R matrices at r5r0 are related by

K5~f2f8R !~g2g8R !21. (2.34)

The radial functions f , g , f8, g8 here are the Coulombfunctions and derivatives appropriate to each channel,evaluated at r5r0 for the appropriate orbital angularmomentum l i and channel energy e i , and arranged intodiagonal matrices.Practical quantum-defect calculations are frequently

conducted using K rather than S , as this permits theentire calculation of total cross sections to be carried outusing real arithmetic. Another real representation is the

eigenchannel version of multichannel quantum-defecttheory introduced by Fano (1970), which utilizes theeigenvalues tanpma and eigenvectors Uia of the matrixk ,

Kij5(a

UiatanpmaUaj† . (2.35)

The independent eigenchannel solutions have the fol-lowing form outside the reaction volume:

Ca5A(i

1r

F i~v!Uia@f i~r !cospma2gi~r !sinpma# ,

(2.36)

which corresponds to the multichannel generalization ofEq. (2.10). In Eq. (2.36), A is the antisymmetrizationoperator. The eigenchannel wave functions Ca ,a51, . . . ,N , have a common phase shift pma in each ofthe fragmentation channels i . The eigenchannel MQDTformulation has been widely used for studying atomicand molecular spectra. The techniques for obtainingphysical solutions that remain well behaved at r→` andfor calculating the observables, discussed in numerouspapers (Fano, 1970; Lu, 1971; Lee and Lu, 1973; Fanoand Rau, 1986), will not be detailed in this review; somerelevant points will be discussed later in Secs. II.E andII.F. In particular, it will be shown that a great deal ofsemiempirical MQDT analyses are based on the eigen-channel formulation of multichannel quantum-defecttheory.

1. Multichannel quantum-defect theory derivations forresonant continua

To describe a photoabsorption (or scattering) experi-ment, carried out with sufficiently high energy resolutionthat it effectively probes stationary states at a definiteenergy E , incoming-wave (or outgoing-wave) boundaryconditions must be imposed. To find the linear combina-tions of the arbitrary Schrodinger solutions cb that obeythese boundary conditions, it is convenient to rewritethe radial solutions (f i ,gi), defined above in Sec. II.B, interms of outgoing/incoming waves f i

6

H 2ifg J 5

1

A2 ~2f16f2!. (2.37)

(In this section, take care not to confuse i5A21 and thechannel index i .) The solutions f6 are convenient, withsimple asymptotic forms:

f6~r !→H ~pk !21/2r6i/ke6ikr6ih, e5k2/2>0;

~2pk!21/2e6ib~D21r2nekr7iDrne2kr!, e52k2/2,0,(2.38)



where the long-range phase shift for a long-range attrac-tive Coulomb potential of unit charge is known analyti-cally:

h51kln~2k !1argGS l112

i

k D212lp . (2.39)

The solutions f6(r) defined here are identical to thew6(r) functions of Seaton (1983). In contrast, they areproportional, at e.0 only, to those with the same nota-tion in Greene et al . (1982) and in Greene et al . (1979).The particular independent solutions beyond r5r0 nowhave the form

Fib~r !51

iA2f i

1~r !~Iib1iJib!21

iA2f i

2~r !~Iib2iJib!.

(2.40)

In this form it is evident that there exists a linear trans-formation that will transform the solutions cb into newsolutions, each of which has an outgoing-wave compo-nent in a single channel only. We now apply from theright side of Eq. (2.30) the inverse of the matrix I1iJ ,giving a new set of linearly independent solutions:

M~r !5F~r !~I1iJ !21. (2.41)

One obtains for the ith channel component of thei8th independent solution

Mii8~r !51

iA2f i

1~r !d ii821

iA2f i

2~r !Sii8† . (2.42)

The asymptotic solutions Mii8(r) are characterized byan (N3N) matrix S†[(I2iJ)(I1iJ)21. In the specialcase that all channels are open, S† coincides, aside fromphase factors, with the Hermitian conjugate of the usualscattering matrix. (The phase factors will be given be-low.) It is the Hermitian conjugate S† that appearsrather than S itself, because these incoming-wave statesare the time-reversed versions of the outgoing-wavestates that are used to describe particle-scattering ex-periments (Breit and Bethe, 1954). This scattering ma-trix S is related to the short-range reaction matrix Kthrough the relation

S511iK

12iK. (2.43)

In fact, the omission of some phase factors, coupled withthe fact that S usually includes closed-channel indices,leads to important differences between S and the usual‘‘physical scattering matrix’’ that we denote by Sphys

throughout this paper. Like the smooth, short-range re-action matrix, this matrix S , referred to as the ‘‘smooth,short-range scattering matrix,’’ has a weak energy de-pendence. When an attractive Coulomb potential de-scribes the long-range motion in every channel, S coin-cides with the matrix x of Seaton (1983). The solutionsof Eq. (2.42) are not acceptable if one or more channelsis closed, because the radial wave functions Mii8(r) di-verge exponentially at r→` for all closed-channel com-ponents i .

It may be helpful to describe qualitatively the mean-ing of the solutions constructed in Eq. (2.42). Imagineforming a wave packet by performing an energy integralover exp(2iEt) multiplying the i8th independent solu-tion of Eq. (2.42) with an additional energy envelopereflecting the bandwidth of the exciting laser. (Thiswould be the state excited if the dipole matrix elementreaches only this particular solution from the groundstate. We should point out that this is not normally thecase, as more frequently all or most of the solutions i8will be excited coherently, with relative amplitudesgiven by a dipole matrix element di8.) The resultingwave packet describes a gedanken experiment in whichphotoabsorption occurs at time t'0, exciting a Rydbergwave packet that will be found moving outward at posi-tive times, in channel i8 only. While this outgoing wavepacket would look more or less sensible in the classicallyallowed region, it would eventually acquire unphysicalbehavior in any channels that are closed, owing to thedivergence of the large-r stationary-state solutions. Inparticular, the packet would not be reflected backwardfrom the outermost classical turning point, as one ex-pects.To find solutions with the correct physical behavior,

we must impose boundary conditions at large r . Mostimportantly, we need to superpose the stationary solu-tions given by Eq. (2.42) to determine linear combina-tions that, for closed channels, decay exponentially inthe asymptotic region. This can be accomplished rathersimply by using a compact partitioned matrix notationborrowed from Seaton (1983). The notation was devel-oped earlier in the nuclear-physics resonance studies ofTeichmann and Wigner (1952) and of Breit (1959), andthe idea is simple. At any energy E , we collect all of theopen channels (for which E>Ei) and label the entire setby a subscript o . We call the number of open channelsNo . Similarly, we collect all closed channels (E,Ei)into a set labeled c , and label their number Nc . Sincethe total number of long-range channels is assumed toremain a fixed number N over an appreciable energyrange, we have No1Nc5N . Next, we arrange the(N3N) outer-region solution matrix M(r) into openand closed partitions accordingly:

SMoo Moc

Mco Mcc D 51

iA2f1~r !

21

iA2f2~r !S S†oo S†oc

S†co S†cc D . (2.44)

In Eq. (2.44), f6(r) indicate diagonal matrices contain-ing the radial solutions appropriate to each channelalong the diagonal, whose asymptotic forms were givenin Eq. (2.38).In this partitioned matrix notation, we can simply con-

struct the solutions needed. Because there are only Noopen channels at this energy, we anticipate only No lin-early independent ‘‘physical’’ incoming-wave solutions.Each of these is a different linear combination of the



solutions M(r). We represent the transformation takingus from the N ‘‘smooth short-range’’ S-matrix solutionsM(r) to the No ‘‘physical’’ incoming-wave solutions byM(2)(r)5M B . The transformation matrix is an(N3No) constant matrix that will itself be partitioned

B[S Bo

Bc D . (2.45)

Combining these quantities gives the open- andclosed-channel components:

SMoo2

Mco2 D 5

1

iA2 S fo1Bo2fo

2~Soo† Bo1Soc

† Bc!

2fc2Sco

† Bo1~fc12fc

2Scc† !BcD .

(2.46)

Using now the asymptotic forms of f6(r) from Eq.(2.38), we find that the large-r wave function compo-nents in the closed channels approach

Mco~2 !~r !→~••• !ekre2ib@2S co

† Bo2~Scc† 2e2ib!Bc# .

(2.47)

Exponential growth in the closed channels must beeliminated by choosing the closed part Bc of the coeffi-cient matrix to be

Bc52~Scc† 2e2ib!21Sco

† Bo . (2.48)

If we choose the open-channel subspace coefficients tobe a diagonal matrix of long-range phase parametersBo5exp(2ih), the physical states obeying the incoming-wave boundary condition are obtained; these have thefollowing asymptotic form in the open channels [i.e.,Moo

(2)(r)]:

Mii8~2 !

~r !→i21~2pki!21/2~eikird ii82e2ikirSii8

†phys!.(2.49)

The Hermitian conjugate of the (No3No) physical scat-tering matrix introduced here is

S†phys5e2ih@Soo† 2Soc

† ~Scc† 2e2ib!21Sco

† #e2ih. (2.50)

The general expression for the closed-channel compo-nents in the physical solution follows from Eq. (2.48),for r.r0,

Mco~2 !~r !5

i

A2@2fc

2Sco† e2ih2~fc

12fc2Scc

† !~Scc†

2e2ib!21Sco† e2ih# . (2.51)

A somewhat more useful expression follows after intro-ducing a solution W(r ,n ,l) proportional to the Whit-taker Coulomb function (Whittaker and Watson, 1927)that decays exponentially as r→` for all negative ener-gies. This solution will be written in each channel as

W~r ,n i ,l i!5i

A2e2ib if12

i

A2eib if2. (2.52)

More precisely, our solution W(r ,n ,l) is given by

W~r ,n ,l !5n3/2@n2G~n1l11 !G~n2l !#21/2

3Wn ,l11/2S 2rn D , (2.53)

where Wn ,l11/2 is the Whittaker function, G denotes agamma function, and the multiplicative factor in front isa constant.W(r ,n ,l) has an ‘‘energy-normalized’’ ampli-tude (2/pk(r))1/2 in the classically allowed region, likethe solutions in Eq. (2.5), but it decays properly asymp-totically. For non-Coulombic long-range fields, Eq.(2.52) remains a valid generalization of the Whittaker-type function. Whereas both fc

6 diverge exponentially atinfinity in Eq. (2.51), the divergence is cancelled inM(2)(r) owing to the choice of coefficients [Eq. (2.48)].However, this is not so apparent from inspecting Eq.(2.51). We cancel the exponentially growing terms ana-lytically, giving the long-range closed-channel compo-nent wave functions in a form involving W(r ,n ,l) thatnow explicitly gives the coefficient Zco in front of theexponential decay,

Mco~2 !5W Zco , (2.54a)

where the closed-channel coefficients are given by

Zco5eib~Scc† 2e2ib!21Sco

† e2ih. (2.54b)

Next, Eqs. (2.49) and (2.54) can be combined to give thei8th energy-normalized physical incoming-wave state:

c i8~2 !

5A (iPo

1r

F i~v!i21~2pki!21/2~eikird ii8

2e2ikirSii8† phys

!

1(iPc

1r

F i~v!Wi~r ,n i ,l i!Zii8. (2.55)

The i8th such stationary state has outgoing waves as-ymptotically only in channel i8 and can be used to de-scribe a photoionization experiment that detects elec-trons emerging in channel i8. We denote by db thereduced dipole matrix elements connecting a specific ini-tial state c0 of total angular momentum J0 to the realstates cb in Eq. (2.19) having total angular momentumquantum number J , i.e.,

db5^cbuur ~1 !uuc0&. (2.56)

Viewing the db as a column vector of N real numbers,the linear transformation (2.41) determines a smooth setof N dipole matrix elements

di8S

5(b

~I1iJ ! i8b21†db . (2.57)

This column vector of amplitudes can be partitioned inthe usual way into open- and closed-channel contribu-

tions dS[(dcS

doS

).

Next, Eqs. (2.45) and (2.48) can be combined to givereduced dipole matrix elements connecting the initialstate c0 to the No physical incoming-wave states. Thesematrix elements generally exhibit very rapid energy de-pendences reflecting closed-channel resonances and aregiven by



d ~2 !5doSe2ih2dc

S~Scc† 2e2ib!21Sco

† e2ih. (2.58)

The resonances occur when the closed-channel subma-trix Scc

† 2e2ib is nearly singular; at these energies, theresonant dipole amplitudes can grow orders of magni-tude larger than the background or nonresonant ampli-tudes.Equation (2.58) gives the dipole amplitudes required

by most standard formulations of total and partialphotofragmentation cross sections, photofragment angu-lar distributions, spin polarization, alignment, orienta-tion, or various quantities that can be measured by co-incidence experiments. For the moment, we give onlythe formulas for total and partial photoionization crosssections, in the simplest case in which the magnetic sub-states of the initial state c0 are randomly populated. Thepartial photoionization cross section into channel i , ina.u., is

s i54p2va

3~2J011 !udi

~2 !u2, (2.59)

where a is the fine-structure constant and v the photonenergy. The total photoionization cross section is thesum over all partial cross sections (a sum over totalfinal-state angular momenta J is also implied):

s tot54p2va

3~2J011 !d ~2 !†d ~2 !. (2.60)

Calculations of other observables, such as scatteringcross sections, are equally straightforward using thephysical scattering matrix Sphys in Eq. (2.50).The formalism developed in this section is based on

channel expansions in terms of the energy-normalizedCoulomb functions f and g . This formalism is not themost convenient for studying the systems in which thelong-range potential experienced by the escaping elec-tron is non-Coulombic. The (f0, g0) pair describing along-range polarization potential is better adapted to thestudy of negative ions (Kim and Greene, 1989; Greene,1990a) because this base pair and the resulting phaseshifts m0 are analytic functions of the energy and exhibitno singularities at the photodetachment threshold (Wa-tanabe and Greene, 1980). In a single-channel problemthe quantum defect m0 of negative-ion bound states canbe expressed easily in terms of the parameters A, b , andG [Eqs. (2.8) and (2.15)] that characterize the long-rangepolarization potential (Greene, 1980; Watanabe andGreene, 1980; Kim and Greene, 1989). Matching theR-matrix eigenstates and their derivatives to a channelexpansion with f0 and g0 Coulomb functions gives ashort-range reaction matrix K0. This changes the equa-tions of multichannel quantum-defect theory in a man-ner described by Greene (1990a).For photoelectron energy e less than −1/2l2, the

energy-normalized Coulomb functions (fel , gel) becomecomplex and therefore inconvenient. We will show laterin Secs. III and IV that some studies of doubly excitedstates of alkaline-earth atoms were performed usingshort-range reaction matrices K0 instead of short-rangereaction matrices K . Multichannel quantum-defect

theory calculations can be carried out using R matricesthat have a fixed number of fragmentation channels atall energies considered, even when, for some channels,e<2 1

2 l2.

2. Bound-state properties

The formulas that were developed above to eliminateexponential growth in closed channels are not immedi-ately applicable to bound states because there is noopen-channel subspace. When all channels are closed,the energy-level spectrum becomes entirely discrete,whereas it is continuous if at least one channel is open.The first thing of interest in the discrete energy range issimply the position of the bound levels. For a single-channel problem the quantization of bound states wasaccomplished by Eqs. (2.13) and (2.14) above. The moregeneral multichannel version can be derived by follow-ing similar logic.We start from the radial solution matrix Mii8(r) out-

side the reaction volume, as expressed in Eq. (2.42) interms of the smooth, short-range scattering matrix Sii8.Since all channels i are now closed, f i

6(r) diverge atr→` , according to Eq. (2.38). We attempt to construct alinear combination Mii8

(n)(r)5( i8Mii8(r)Bi8(n) of the solu-

tions (2.42), chosen to cancel this exponential diver-gence. The desired superposition follows from inspec-tion of Eq. (2.47), recalling that Bo is now absent. Theresulting MQDT form of the bound-state quantizationcondition reads

~Scc† 2e2ib!B ~n !50. (2.61)

This system of linear equations has at least one non-trivial solution if and only if

detuScc† 2e2ibu50. (2.62)

The energies En at which Eq. (2.62) has a solution arethe bound-state energies of the system. Determinationof the En normally requires a numerical search for roots.In this search the fastest energy dependence is normallyconfined to the quantity e2ib. The scattering parametersScc† normally vary quite slowly with energy, and can fre-

quently be regarded as constants over small spectralranges. The column vector of superposition coefficientsB(n) is well defined only at the bound energy levels andis undefined at other energies. [It is sometimes useful toestablish continuity of the eigenvectors of Eq. (2.62)even at nonbound energies, especially when trying toclassify complex spectra. One useful method is to diag-onalize the matrix on the left-hand side at any E , givingNc eigenvalues and eigenvectors as continuous functionsof energy. The bound-state search then amounts to find-ing energies at which at least one eigenvalue vanishes.]The bound-state eigenvector is needed to calculate otherobservable properties of the bound state at energy En ,such as the probabilities of photon emission or absorp-tion. As in Eq. (2.54a), the exponentially decaying func-tion can be expressed in terms of the W(r ,n ,l) functionof Eq. (2.53),



C~n !5(i

1r

F i~v!Wi~r ,n i ,l i!Zi~n ! , (2.63)

where the closed-channel coefficient is

Zi~n !5eib iBi

~n ! . (2.64)

Normalization of the coefficients B(n) remains unspeci-fied, of course, by the homogeneous system (2.61). Thenormalization integral can be evaluated analytically us-ing a flux conservation argument, along the linessketched by Seaton (1983), Lee and Lu (1973), andGreene et al . (1979). The expression given in Eq. (6.49)of Seaton (1983) involves the energy derivative of theshort-range reaction matrix K . While the expression isformally correct, it is numerically inconvenient in somecases owing to the occurrence of poles of K(E) on thereal energy axis. In the eigenchannel formalism of mul-tichannel quantum-defect theory, this difficulty isavoided by using an alternative normalization expres-sion that involves energy derivatives of the eigenquan-tum defects ma and eigenvectors Uia of the K matrix(Lee and Lu, 1973). While this avoids differentiation ofdivergent quantities, another drawback of the resultingexpression is that derivatives of an eigenvector matrixsuch as Uia also require great care in practice owing tosharply avoided crossings between the ma .We give instead an alternative normalization condi-

tion based on the short-range scattering matrix states,which has been derived by similar techniques. The re-sulting normalization integral can be cast into a formsuitable for rapid and accurate numerical evaluation:

E ucu2dV5~2p!21B ~n !†S 2db

dE1ie22ib

dS†

dE DB ~n !

51. (2.65)

Normalization of the bound-state eigenvector is com-pleted by enforcing this requirement on the B(n) calcu-lated in the course of solving Eq. (2.61). In this form ofthe normalization constant, only smooth quantities thatare free from poles on the real energy axis are differen-tiated. Oscillator strengths for electric dipole transitionsfrom a lower level to a bound final state normalizedaccording to Eq. (2.65) can be calculated using the samesuperposition coefficients B(n) and dipole matrix ele-ments dS of Eq. (2.57). If the lower level has angularmomentum J0 and has an isotropic distribution of mag-netic substates, this oscillator strength is given by

fn52v

3~2J011 !(i uBidiSu2. (2.66)

Some insight into the qualitative meaning of the nor-malization condition (2.65) is gained by recasting it andEq. (2.61) into another form. Note first that the bound-state condition and also most of the quantum-defect ex-pressions of the preceding subsection involve the matrixdifference Scc

† 2e2ib. Here (e2ib) ii8[Sii8LR can be re-

garded as a ‘‘long-range scattering matrix,’’ giving theamplitude that an electron fired outward from the origin

into a particular channel i8 will be reflected back towardsmall distances in channel i . In the strict ‘‘quantum-defect approximation’’ that ignores channel coupling inthe outer region, this long-range scattering matrix is di-agonal. This diagonality expresses the fact, for instance,that a Rydberg electron with a definite orbital angularmomentum l does not scatter into a different orbital mo-mentum l8 in the course of its usual motion in the purelyCoulombic outer field. Processes that change the Ryd-berg electron orbital momentum are embodied of coursein the short-range scattering matrix S . But an aniso-tropic long-range field, such as an external electric ormagnetic field, produces a long-range field scatteringmatrix SLR that is nondiagonal in a spherical represen-tation.Condition (2.61) can be rewritten in another form

which shows how an equilibrium must be establishedbetween short- and long-range scattering, in order toform a standing wave at a bound state:

~Scc† 2SLR!B ~n !50. (2.67)

This expression, with a few manipulations, gives a moregeneral form for the normalization condition as well:

~2p!21B ~n !†S iSLRdSLR†dE1iS

dS†

dE DB ~n !51. (2.68)

In this form the normalization integral involves two ma-trices that are in some sense average values of Smith’s‘‘quantum time operator’’ id/dE , with the ‘‘average’’being taken over the long-range and short-range scatter-ings, respectively. These quantities are closely relatedconceptually to the time-delay matrix (Wigner, 1955;Smith, 1960). We will denote the Hermitian time-delaymatrix by Qphys. This superscript is adopted in the samespirit as the notation in Eq. (2.50), because Qphys is de-fined in terms of the physical scattering matrix,

Qphys5iSphysdSphys†

dE. (2.69)

Note in particular that Qphys has indices relating to openchannels only, whereas the matrices in the normalizationexpression (2.68) have only closed channel indices. Nev-ertheless, they have the same structure, so we identifythe first matrix in the parentheses of Eq. (2.68) as the‘‘long-range time-delay matrix’’ QLR and the second asthe ‘‘short-range time-delay matrix’’ Q .Having established the connection between the nor-

malization constant or ‘‘density of states,’’ and the long-and short-range time-delay matrices, we can give a moredetailed interpretation of the structure of Eqs. (2.65)and (2.68). In particular, the long-range contribution tothe density of states in Eq. (2.68) is given analytically foran attractive Coulomb field by p21(db/dE) ii85d ii8n i

3.For any high Rydberg state this contribution is huge andit normally dominates strongly over the short-rangetime-delay contribution. It is also worth considering theexpression (2.50) for the physical scattering matrix,where the fastest energy dependence (except possiblyfor threshold effects or shape resonances) comes from



the closed-channel phase parameter b . For this reasonwe anticipate that the strongest energy dependences ofthe scattering matrix, for systems of the kind treated inthis review, tend to come from the closed channels thathave been ‘‘eliminated’’ in the quantum-defect sense.Deflection into such channels introduces appreciabletime delays in the emergence of a scattered wave packet,when viewed in the time domain. When viewed insteadin the energy domain for stationary scattering states,such effects appear in the corresponding density ofstates or time-delay matrix Q(E). Accordingly, we in-terpret qualitatively the long-range part of the normal-ization integral (i.e., p21uBiu2db i /dE) as mainly reflect-ing the portion of the density residing within the ithclosed channel. On the other hand, the short-range part(2p)21B†Q B in Eq. (2.68) reflects density in otherchannels whose indices are not included in the Nc ex-plicit closed channels treated in the short-range scatter-ing matrix, but whose effects have been included in thecourse of solving the time-independent Schrodingerequation.

3. Resonance analysis

The poles of the matrix S occur at the complex ener-gies for which Eq. (2.62) is satisfied. Several theoreticalmethods have been proposed for determining the reso-nance positions and widths in the complex energy planefrom the poles of the complex matrix S (see, for ex-ample, Siegert, 1939; Humblet and Rosenfeld, 1961;Noble et al . , 1993).The fact that Scc

† 2e2ib becomes singular at bound lev-els can help us to understand the origin of autoionizingresonances in scattering amplitudes [Eq. (2.51)] or indipole photoabsorption amplitudes [Eq. (2.58)]. In bothof these expressions, such a singularity clearly causes thematrix inverse to become very large. At first glance, thisdivergence of the matrix inverse might seem to lead toinfinite cross sections. But unitarity of the smooth, short-range S matrix ensures that the singularity remains finiteat real energies, provided there is no closed-channel sub-space that is totally decoupled from the continuum. (Ifsuch a subspace is identifiable, its bound spectra can betreated separately as in the previous subsection.) Thefollowing discussion is adapted to resonances with non-zero decay widths.Near resonance energies, the closed part of the physi-

cal incoming-wave state of Eq. (2.55) has a significantamplitude at finite distance. Equation (2.55) shows thatthe wave-function components in closed channels takethe form of a quasistable standing-wave pattern nearthese resonant energies, whose amplitude grows to amaximum as the energy reaches the resonance center.Following Lecomte (1987), the density of states can beexpressed in terms of the closed-channel coefficientsZco of Eq. (2.54b) as:

ds5Tr ~Zoc† Zco!. (2.70a)

The partial density of states in a particular closed chan-nel i is given by

dsi5(i8

Zii82 , (2.70b)

where the summation runs over the No linearly indepen-dent solutions [Eq. (2.55)].Equations (2.70a) and (2.70b) can be adapted, as de-

scribed by Luc-Koenig, Lecomte, and Aymar (1994), tomultichannel-quantum-defect-theory calculations usingthe short-range reaction matrix K0 instead of K . Reso-nance analysis in neutral atoms is performed (Luc-Koenig, Aymar, and Lecomte, 1994; Luc-Koenig,Lecomte, and Aymar, 1994), by defining ‘‘analytic’’ par-tial densities of states in closed channels i by

dsi05(

i8~Zii8

0!2, (2.71)

where now the closed-channel component in the physi-cal incoming-wave state [Eq. (2.55)] is expressed as( iPcr

21F i(v)W(n i ,l i ,r)(A) i21/2Zii80 . The introduction

of the parameter Ai21/2 [Eq. (2.15a)] eliminates the di-

vergences occurring when the effective number n i in theclosed channel i has an integer value less than or equalto l i .Another tool for analyzing resonance properties was

provided by Smith’s (1960) physical time-delay matrixQphys of Eq. (2.69). In fact, Smith’s symbol Q reflects theanalogy he had in mind with resonating cavities that cantrap an electromagnetic or sound wave for thousands oreven millions of internal wave ‘‘bounces.’’ One keyproperty of an ‘‘isolated’’ resonance is its decay width.We consider now states whose width is dominated byautoionization, although other decay mechanisms suchas light emission or predissociation can be incorporatedwhen appropriate. The derivation of Eq. (2.69) for thephysical time-delay matrix is rather complicated as pre-sented by Smith (1960), as it involves determining theratio of the density of particles within a finite volume tothe flux of particles in and out of that volume in differ-ent channels. The derivation of Eq. (2.68), on the otherhand, shows how a quantity having the structure of thetime-delay matrix [Eq. (2.69)] arises naturally in a calcu-lation of the normalization constant, or density of statescontributed by different channels. As Smith (1960) pro-posed, this physical time-delay matrix can be used tointerpret resonances as follows. At any isolated reso-nance in scattering or photoabsorption, a single eigen-value qa of Q

phys is found to be far larger than all othereigenvalues, and is seen to trace out an approximateLorentzian as a function energy (Burke, Cooper, andOrmonde, 1969). If we denote the corresponding domi-nant eigenvector as Cia , then the relative probabilitiesfor decay of the resonance into the different possibleopen channels i are given by uCiau2. Moreover, the totaldecay width of the resonance (full width at half maxi-mum) is given by G54/max $Tr(Qphys)%'4/qa . In morecomplex situations involving overlapping resonances,the number of eigenvalues qa of Qphys having a non-negligible value amounts to the number of closed chan-nels Nc (Nc,No) that have been eliminated when con-structing S† phys in Eq. (2.50). The density of states can



be analyzed from the evolution of $ Tr (Qphys)% with theenergy E , though, the branching ratios for autoioniza-tion can no longer be simply related to the eigenvectorsof the time-delay matrix. As was shown by Dalitz andMoorhouse (1970), this quantity can be expressed interms of the energy derivatives of the eigenphase shiftstr of the physical scattering matrix (Greene and Jungen,1985),

ds85Tr ~Qphys!52(rPo

dtr

dE. (2.72)

As was shown by Lecomte (1993) and Lecomte et al .(1994), the densities of states defined by Eqs. (2.70) and(2.72) are identical when the short-range S matrix is en-ergy independent. This case corresponds to the short-range time-delay contribution being negligible com-pared to the long-range contribution. This approachpermits calculation of the total decay width when onlythe widths and positions are of interest, and not theeigenvectors of Qphys. This type of analysis of the time-delay eigenvectors and eigenvalues can be very helpfulfor interpretation of resonance properties, and in par-ticular for isolating final-state resonance effects from themeans of exciting them.For more complicated situations, as illustrated later in

Sec. IV, the characteristics and the identifications of theresonances can be obtained from the total and partialdensities of states using Eqs. (2.70)–(2.72).Another area of current interest in this field is the

development of methods for resonance analysis in thetime domain rather than in the energy domain. Thetime-domain spectrum can be calculated directly usingsemiclassical approximations to quantum mechanics.General ideas along these lines have been developedextensively by Gutzwiller (1967), and applied with suc-cess to problems in atomic and molecular physics by Duand Delos (1988a, 1988b), and numerous other groups.These techniques often end up calculating a semiclassi-cal approximation to the ‘‘density of states,’’ and can beviewed accordingly as approximations to the trace of thetime-delay matrix Qphys. These methods are particularlyuseful in the short-time limit, where peaks in the ‘‘timedomain’’ can be associated with the periods of closed,unstable classical trajectories. ‘‘Short-time’’ features in acomplex spectrum can be regarded equivalently as themain features that show up in an experiment that doesnot have particularly high-energy resolution. The analy-sis of such short-time phenomena is often the simplest inthis picture, as surprising subtlety can be extracted fromthe study of spectral features in the time domain.Resonance analysis can also be performed using an

alternative MQDT parametrization of the resonances,based on the commonly called phase-shifted MQDTmethod. This approach was introduced by Giusti-Suzorand Fano (1984) and Cooke and Cromer (1985) andthen worked out in a more rigorous form by Lecomte(1987).The development of this method has been motivated

by an increasing number of experimental data on au-

toionizing resonances of alkaline earths obtained, duringthe last two decades, using the experimental isolated-core excitation method (Cooke et al . , 1978). Several il-lustrations will be given later in Sec. IV. We brieflysketch the basic ideas of the Lecomte’s formulation. Thephase-shifted multichannel quantum-defect theory re-duces the general problem of Nc closed channels inter-acting with No open channels to a simpler problem in-volving closed channels only, the presence of the openchannels being taken into account implicitly. From theshort-range reaction matrix K one introduces a complexmatrix kcc restricted to the closed channels,

kcc5Kcc2Kco~ i1oo1Koo!21Koc . (2.73)

This amounts to imposing Siegert-type boundary condi-tions with only outgoing waves in all open channels. Asexplained by Fano and Rau (1986) and illustrated laterin Sec. IV, the real part of k can be used to calculate thepositions of the resonances and to identify them by usingMQDT bound-state techniques.The phase-shifted multichannel quantum-

defect theory is based on a phase normalization ofthe Coulomb functions that consists of replacing thestandard pair (f ,g) by a phase-shifted pair(fcospm2gsinpm, gcospm1fsinpm) followed by obtain-ing a set of phase-shifted quantum defects m i and atransformed reaction matrix kcc8 whose elements can bewritten in the form

k ij8 5rij1iRW i .RW j , (2.74)

with rii=0. As detailed by Lecomte (1987), the nondi-agonal real rij elements describe the direct coupling be-tween the closed channels and can be diagonalizedwithin a set of ‘‘degenerate’’ channels converging to thesame threshold. The Nc vectors RW i involved in theimaginary part of the k8 matrix describe the couplingbetween the Nc closed channels and Nc effective con-tinua. When the resonances pertaining to the ith chan-nel are well separated from the others, m i and uRW iu2 char-acterize the positions and scaled autoionizing widths g iof the resonances in channel i . One has, in a.u.

g i5G i~n i!352uRW iu2/p , (2.75)

where G i is the full width at half maximum for autoion-ization and n i is the effective quantum number in chan-nel i . The explicit construction of the effective continuain terms of the jj-coupled open fragmentation channelspermits the branching ratios for autoionization towardeach open channel to be calculated for each closed chan-nel. In addition, the calculation of the photoionizationspectra requires dipole matrix elements connecting theinitial level to the closed channels component and to theeffective continua, as well as one additional dipole ele-ment corresponding to the excitation of the No2Ncnoninteracting continua.

E. Frame transformations

The key quantity characterizing the scattering andphotoabsorption spectra of a multichannel system is the



smooth, short-range scattering matrix S (or the equiva-lent reaction matrix K), and a set of smooth dipole tran-sition amplitudes d . The short-range scattering matrixS reflects a great deal of the system’s dynamics; in factHeisenberg (1943) regarded the physical scattering op-erator Sphys as even more fundamental than the Hamil-tonian itself. It is certainly true that essentially any in-teraction appearing in the Hamiltonian will be reflectedat some level in the scattering matrix. For instance, theeffect of a sufficiently small term in the Hamiltonian canbe described by first-order perturbation theory, in whichthe matrix element of a perturbation V is proportionalto a corresponding element of the reaction matrix (Rod-berg and Thaler, 1967):

Kij'2p K 1r F if iuVu1r

F jf jL 1O~V2!. (2.76)

Consequently, much quantum-mechanical intuition,which is initially sharpened for most students in the con-text of energy-level calculations, can be directly carriedover and used to develop intuition about multichannelscattering processes and spectra. For example, if thedominant interactions present in the Hamiltonian arediagonal in a particular angular momentum couplingscheme, we expect the scattering matrix to be block di-agonal in that same coupling scheme. The physical con-sequences of this viewpoint were first used profitably byFano (1970) in the context of H2 photoabsorption.Fano, Jungen, and their co-workers have been instru-mental in developing and extending this picture to sys-tematically describe a broad range of problems inatomic and molecular physics. (See, for example, Jungenand Dill, 1980; Raoult and Jungen, 1981; Greene andJungen, 1985; Fano and Rau, 1986)The basic idea of the recoupling frame transformation

can be conveyed by developing a specific example. Con-sider the photoabsorption spectrum of a light atom. Be-cause spin-orbit and other fine-structure perturbationsare particularly weak in light atoms, most of the impor-tant physics is determined by the electrostatic interac-tion. Exchange is important, because Pauli antisymme-trization affects each symmetry differently. To theextent that electrostatic interaction and Pauli exchangeeffects dominate, the energy eigenstates can be charac-terized by L and S , the total orbital and spin angularmomenta respectively, of all atomic electrons, since LW 2

and SW 2 both commute with the electrostatic Hamil-tonian.One might be tempted under these circumstances to

complete the entire photoabsorption calculation assum-ing validity of LS coupling throughout. However, thebasic logic developed by Lu (1971), and more systemati-cally by Lee and Lu (1973), shows how frame transfor-mation methods describe a large class of fine-structureeffects. These effects are surprisingly important andnonperturbative in certain regions of the spectrum, evenin light atoms. For definiteness, consider photoioniza-tion of the 1Se ground state of a two-electron atom suchas beryllium, in an energy range that extends to final

states close to the Be1(2p) ionization threshold. In astandard LS-coupling approximation, an electric dipolephoton excites only 1Po final states. Focusing just on theionization channels attached to the 2p ionic state, tworelevant channels are specifically excited, namely2pns 1Po and 2pnd 1Po. Because the 2p ionic statewould (artificially) have no spin-orbit splitting in a strictLS-coupled nonrelativistic calculation, the escaping s-or d-wave electron will move outward toward the detec-tor at infinity if the final-state energy E lies above the2p threshold E2p . If the energy lies below, E,E2p , apattern of 2pns 1Po and 2pnd 1Po Rydberg levels willbe formed just below the 2p threshold.Now consider the physics of photoelectron escape

from the real ionic 2p levels, which have a fine-structuresplitting, E2p3/2

5E2p1/216.6 cm21. This splitting is tiny

on the scale of the strongest interactions present in theatomic Be Hamiltonian. For reference, the exchangesplitting between the 1Po and 3Po 2s2p levels is morethan 21 000 cm21. While the fine-structure splitting thusseems to be negligible in comparison, at final-state ener-gies close to the Be1(2p) thresholds it can have a non-perturbative effect on the asymptotic wave functions.With the atom at a final-state energy E , the electronescapes from these two possible ionic energy levelsE2p1/2,3/2

. Owing to energy conservation, the escapingelectron energy is asymptotically different depending onthe residual ionic energy,

e2p1/25E2E2p1/2, e2p3/25E2E2p3/2

. (2.77)

The effect of these different energies of photoelectronescape in the two different channels can be summarizedsuccinctly in terms of the de Broglie phase differencebetween the corresponding wave functions. A WKBanalysis, which neglects the short-range phase contribu-tions from electron correlations, gives for this phase dif-ference as a function of r :

Df5E rSA2e3/212r8

2A2e1/212r8D dr8

'A23r3/2DE . (2.78)

(For the purposes of this qualitative analysis, we ignorethe Langer-corrected centrifugal term and the small dif-ference between the classical turning points at small dis-tances.) Figure 3 shows how this small energy differenceaffects the corresponding radial (l50) wave functions ata final-state energy in between the 2p1/2 and 2p3/2threshold, such as e2p3/252e2p1/2. The key thing to notehere is the way the wave-function components attachedto the different fine-structure thresholds remain lockedtightly in phase up to a radius rcrit of about 1000 a.u. Butbeyond that radius, the two orbitals lose their phase co-herence and at larger distances they act like indepen-dent channels. Any property relating to electron motionat distances larger than 10 3 atomic units for this energyrange of atomic Be can be expected to show nonpertur-



bative effects of the spin-orbit interaction. One suchproperty is the spectroscopy of high-lying 2pns Rydberglevels just below the 2p ionic threshold. Distances of thismagnitude translate into principal quantum numbers inthe range of ncrit'A1000'30. This critical value of nmarks a transition region beyond which spin-orbit inter-action is especially strong. An energy argument confirmsthis: the difference between the 2pns 1Po and 3Po en-ergy levels should be approximately (Dm)n23 a.u. Here,Dm is the difference between singlet and triplet quantumdefects for this series, and its magnitude is expected tobe less than 1. Simple arithmetic shows that n'30 like-wise marks the transition region in energy, where theexchange interaction becomes smaller than the ionicfine-structure splitting of 6.6 cm21.The observed capability (Fig. 3) of radial fine-

structure components to remain phase locked at smalldistances provides the key toward a formulation of thesenonperturbative spin-orbit effects. The first key is to rec-ognize that LS-coupled asymptotic channels are funda-mentally incapable of describing the fine-structure-dependent phase evolution depicted in Fig. 3. Inparticular, the asymptotic phase depends on Jc , the totalangular-momentum quantum number of the ionic resi-due, with values 1

2 and32 in this example. Because JW c

2

commutes with neither LW 2 nor SW 2, LS-coupled channelscan never suffice in the region r.rcrit . The asymptoticchannel indices i must therefore include Jc . Other thanthis absolute requirement, considerable flexibility re-mains in the choice of the other channel quantum num-bers. For different systems, or for different observables,different asymptotic coupling schemes may be advanta-geous, although they will all produce the same spectra inthe frame-transformation method. Examples of typicalcoupling schemes include Jcj-coupled channels (usuallyabbreviated as jj-coupled) ui&5unc(sclc)Jc(sl)jJM&,JcK-coupled channels ui&5unc@(sclc)Jcl#KsJM& (de-noted as pair coupling, jK or jl coupling) andJcJcs-coupled channels ui&5unc@(sclc)Jcs#JcslJM&. For

these three cases, a standard angular-momentum recou-pling, diagonal in the total J and M , relates the asymp-totic channels to the LS-coupled channel functionsu i&5unc(scs)S(lcl)LJM&.The recoupling formulas needed can be found in

Sobel’man (1972) or other standard references such asZare (1988). In the case of Jcj coupling, the real or-thogonal transformation matrix Xii involves the recou-pling matrix element ^(sclc)Jc(sl)ju(scs)S(lcl)L&(J). (AKronecker delta function is implied in all channel quan-tum numbers other than Jcj ,SL .) In problems in whichthe dominant interactions are the ionic fine-structureand multipole effects on the distant electron, thei[JcK coupling scheme leads to short-range scatteringmatrices S that are more nearly diagonal. In thisscheme the recoupling matrix Xii is^@(sclc)Jcl#Ksu(scs)S(lcl)L&(J). An analogous formulais readily derived for the JcJcs coupling scheme, which isadvantageous, for instance, in calculation of photoelec-tron angular distributions because both JW c

2 and JW cs2 com-

mute with the observed ejection angle. We repeat, how-ever, that the use of all three of these coupling schemesleads to identical results in a description of fine-structureeffects by frame-transformation techniques. In fact, thetrue wave function is not assumed to be a single term inLS coupling, jj coupling, nor in any other couplingscheme. Rather, a linear combination of the differentchannels is ultimately selected in the course of solvingthe quantum-defect equations at each energy. To con-nect to a more familiar point of view, the MQDT de-scription of spin-orbit effects using a recoupling frametransformation gives results similar to an intermediate-coupling calculation in traditional spectroscopy, for lev-els lying far below all ionization thresholds. Howeverthe MQDT version automatically includes continuumand high-Rydberg interactions that are difficult to de-scribe in traditional intermediate-coupling calculations.Importantly, and perhaps surprisingly, the dynamics arecorrectly described even when fine-structure effects areignored completely when calculating the short-rangescattering matrix SLS , which is block diagonal in LScoupling.The complete frame-transformation procedure re-

quires one additional key point. In fact, the recouplingof the short-range matrix SLS just described,

Sjj5X SLSX†, (2.79a)

and of the associated dipole matrix elements on the left-hand side of Eq. (2.57),

djj5X dLS , (2.79b)

by themselves have no effect whatsoever on the calcu-lated spectra. This can be seen by inspecting Eqs. (2.58)and (2.59), as the unitary transformation drops out ofthe total cross section. Importantly, in the spirit of Fig. 3,it is crucial to use the experimental Jc-dependent ioniza-tion thresholds when calculating the long-range diagonalphase matrix e2ib needed to evaluate the MQDT formu-las (2.50) and (2.58). A further note of clarification con-cerns the dipole matrix elements dLS in Eq. (2.79b):

FIG. 3. Spin-orbit frame-transformation example with wavefunctions in two fine-structure-split Coulombic channels.Radial s orbitals of Be calculated at an energy equidistantfrom the 2p1/2 and 2p3/2 thresholds, i.e., for e2p1/2.0 ande2p3/2,0 such that e2p3/252e2p1/2.



these are understood to be coupled reduced dipole ma-trix elements determined from an LS-coupled calcula-tion, i.e., dLS[^(SL)Juud(1)uu(S0L0)J0& .On top of the changed values of e2ib i, in different

closed fine-structure channels, an equally important as-pect is the fact that use of experimental thresholds af-fects the partitioning between open and closed channelsat any chosen energy E . That is, before inclusion of fine-structure effects, the channels attached to the 2p thresh-old are either all open or all closed. Once the splittinginto 2p1/2 and 2p3/2 thresholds is built into the calcula-tion by using their experimental energies, a new regimearises at energies between the two thresholds where the2p3/2nl channels are closed while the 2p1/2el channelsare open. In this energy range, a completely new physi-cal process occurs that cannot be described in pure LScoupling, namely autoionization by a spin flip. Althoughit occurs in only a tiny energy range for this Be example,it can happen over larger energy ranges in other sys-tems. In such energy regimes the physics is affectedqualitatively and nonperturbatively by the inner-electron spin-orbit interaction.These effects are illustrated in Fig. 4, which depicts

the photoionization cross section of Be at final-state en-ergies just below the 2p1/2 threshold. Neglecting all fine-structure effects throughout the calculation leads to thequalitatively incorrect spectrum shown in Fig. 4(a). Fig-ure 4(b) shows how LS-coupled scattering and dipole

matrices, combined with a recoupling jj-LS frame trans-formation and the use of experimental ionic 2p thresh-olds in the MQDT calculation, predict a rich multichan-nel spectrum. This particular system has not beenstudied experimentally at such high resolution, but suchrecoupling frame transformations are usually quite accu-rate, as will be seen in the comparisons of Sec. IV.To interpret the dynamics involved in spectra like that

in Fig. 4(b), we return to a heuristic time-dependentviewpoint. A light pulse incident on the 1Se ground stateat time t'0 excites an outgoing wave front of purely1Po character at small distances. The outgoing2pns 1Po wave front is sheared into its two fine-structure components that accumulate radial phase atslightly different rates. If, in fact, the final-state energyreaches the small range between the fine-structurethresholds, eventually the portion of the packet in theclosed 2p3/2ns channel will be reflected back toward thenucleus, whereas the portion in the open 2p1/2es channelcan continue outward to infinity uninhibited. This differ-ential reflection, i.e., that occurring only for one fine-structure component Jc5

32, couples the other LS sym-

metries into the photoabsorption process even thoughthey are not connected to the ground state by an electricdipole matrix element. For this reason, the spectra inthis energy range are sensitive also to the 3Po symmetryscattering matrix, as the wave front returning to thenucleus will have components in all relevant LS symme-tries. In the case of the 2pnd channels, not only 1,3Po

symmetries are coupled, but also the 3Do symmetry,whose Rydberg levels converging to the 2p thresholdwould have been true bound levels for our gedanken Beatom without spin-orbit interactions. The frame trans-formation couples the 2pnd 3Do levels to the L51o

channels in the real atom, thereby permitting thesestates to autoionize.Note also that when the energy is above all thresholds

relevant in the problem—just above Be1(2p3/2) in ourexample—the frame transformation has no effect on thetotal cross section, since there are no longer any b i oc-curring in which to use the experimental thresholds. Ourtime-dependent argument shows why this is reasonable,namely because when all channels are open, no differ-ential reflection is expected to occur semiclassically, andthe wave fronts in the fine-structure channels can pro-ceed outward indefinitely to infinity. In this energy rangewhere all channels are open, the total photoionizationcross section coincides with that which would be ob-tained in a purely LS-coupled calculation. [This is notgenerally true for long-range fields other than the attrac-tive Coulomb field, owing to the energy-dependent pa-rameters A and G discussed in the context of Eq. (2.15).]Finally, the dynamics of recoupling frame transforma-tions have been discussed elsewhere by Watanabe et al.(1984), who point out that a fairly well-defined radiuscan be identified where the shearing of the 1Po wavefunction into its fine-structure components occurs. Thisis the radius at which a Landau-Zener-type transition

FIG. 4. Be photoionization spectrum near the 2pj thresholds:(a) all fine-structure effects are neglected in the eigenchannelR-matrix and MQDT calculations performed in LS coupling;(b) results obtained with scattering and dipole matrices deter-mined by LS-coupled R-matrix calculations combined with arecoupling jj-LS frame transformation and the use of experi-mental ionization 2pj thresholds in the MQDT calculations.



occurs, and it is roughly where the exponentially-decaying exchange energy is equal to the spin-orbit in-teraction energy.The preceding considerations demonstrate that effects

of the spin-orbit interaction on a highly excited Rydbergor continuum electron are felt primarily at large dis-tances. This may seem surprising initially, since one nor-mally thinks of the spin-orbit interaction as a short-range interaction, owing to the r23 dependence of thespin-orbit term in the Hamiltonian, which is strongestnear the nucleus. This apparent contradiction clarifiesthe origin of spin-orbit effects on atomic Rydberg elec-trons. They are mostly caused by the dephasing of dif-ferent spin-orbit radial components as an indirect conse-quence of energy conservation, reflecting the spin-orbitsplitting of the residual ionic core. In fact, the directeffect of the Rydberg electron’s spin-orbit term in theHamiltonian can usually be neglected. Examples in Sec.IV will probe the accuracy and limitations of this view-point.

F. Semiempirical analysis of multichannel spectra

The semiempirical approach of multichannelquantum-defect theory, in which short-range scatteringparameters are adjusted to reproduce a particular set ofexperimental data, has enjoyed remarkable success ininterpreting complex multichannel spectra. A great dealof empirical analyses were carried out using the eigen-channel MQDT formalism introduced by Fano and co-workers (Fano, 1970; Lu and Fano, 1970; Lu, 1971; Leeand Lu, 1973). The physical solutions and observableswere calculated in terms of the ma and Uia smooth scat-tering parameters obtained by diagonalizing the short-range reaction matrix of Eq. (2.35), and of the short-range dipole matrix elements Da . Compared to theprevious semiempirical analyses based on the Seaton’sformalism, two aspects of the eigenchannel MQDT for-malism greatly enhanced the power of such analyses.First, Fano and Lu (Fano, 1970; Lu and Fano, 1970; Lu,1971) introduced a graphical method, based on the useof the so-called Lu-Fano plots, which has proved ex-tremely useful in analyzing multichannel spectra. Thesecond key development resulted from the introductionof the frame transformation between the short-rangeeigenchannels a and the long-range fragmentation chan-nels i (Fano, 1970).Earlier semiempirical studies dealt with channel inter-

action between Rydberg series converging to two differ-ent ionization limits. The first calculation (Fano, 1970)concerned the channel interaction between the J51odd-parity 1snp Rydberg series of H2 converging to theN150 and N152 rotational levels of H2

1 , which wereobserved in the photoabsorption spectrum of H2. Thendiverse experimental data (energy positions and oscilla-tor strengths in the discrete and autoionizing regions)observed in the photoabsorption spectra of Xe (Lu,1971) and Ar (Lee and Lu, 1973) were interpreted in acompact form using the same procedure. The methodwas then applied to the photoabsorption spectra of

carbon-group elements Si, Ge, Sn, and Pb (Brown et al.,1977a, 1977b, 1977c; Ginter and Ginter, 1986; Ginteret al., 1986). Simultaneously, in connection with the de-velopmen t of laser spectroscopy, the method was ex-tended to handle more complex spectra involving inter-acting Rydberg series converging to more than twoionization limits and numerous perturbed Rydberg se-ries of alkaline earths were successfully interpreted(Armstrong et al . , 1977; Esherick, 1977; Lu, 1977; Ay-mar et al . , 1978; Armstrong et al . , 1979; and referencesin Aymar, 1984a). Since then, the analysis of atomic andmolecular spectra has increasingly utilized this empiricalapproach but it is out of the scope of this review toquote all the corresponding studies.Because the semiempirical fitting procedure based on

the eigenchannel formalism of multichannel quantum-defect theory is described in detail in the above-mentioned papers, we will sketch only the main pointsand begin by illustrating a Lu-Fano plot in a simple two-channel system. Figure 5 deals with the perturbation ofthe Sr 5snp 1P Rydberg series by the 4d5p 1P doublyexcited level, the lowest member of the 4dnp 1P seriesconverging to the upper 4d ionization limit. All 1Po

bound levels are represented in a compact form by plot-ting them in the plane -n5s (mod 1) against n4d . Theeffective quantum numbers n5s , and n4d relative to thespin-orbit-averaged 4s and 4d limits are connectedthrough the relation

E5I5s2Ry

n5s2 5I4d2

Ry

n4d2 , (2.80)

where Ry is the mass-corrected Rydberg constant. (Notethat E , I5s , and I4d , expressed in cm

21, are relative tothe 5s2 ground state.) All experimental points (fullpoints) except that corresponding to the lowest 5s5p

FIG. 5. Lu-Fano plot of the odd-parity 1P1o bound levels of Sr,

comparing experimental values—depicted by solid points—with the Lu-Fano plot of Eq. (2.81b), calculated using theeigenquantum defects m1, m2 and the mixing angle u fitted tothe experimental data by Armstrong et al . (1979). The theo-retical energy levels are obtained at the intersections of theLu-Fano plot and of the curve (dashed line) -n5s (mod 1)= f(n4d) defined by Eq. (2.80). The Sr 5snp 1P1 Rydberg se-ries is perturbed by the 4d5p 1P1 doubly excited level.



level lie on the Lu-Fano plot (full line). Theoretically,this curve represents the solution of Eq. (2.62), which, interms of eigenchannel short-range scattering param-eters, is

detuUiasinp~n i1ma!u50. (2.81a)

For a two-channel problem, Eq. (2.81a) gives

detS cosu sinp~n11m1! sinu sinp~n11m2!

2sinu sinp~n21m1! cosu sinp~n21m2!D 50,

(2.81b)

where m1 and m2 are the two quantum-defect param-eters and the angle u characterizes the (2 3 2) orthogo-nal Uia matrix.The theoretical curve can be plotted continuously

even though the true spectrum is discrete. Therefore,this approach determines the quantum defect even atenergies at which there is no level because Eq. (2.80) isdisregarded. The Lu-Fano plot depends only on the dy-namical smooth scattering parameters, which are nearlyindependent of the energy. Thus the Lu-Fano plot canbe viewed as a way to highlight the short-range dynam-ics common to all the levels. The theoretical energy lev-els are obtained at the intersections of the Lu-Fano plot-n5s (mod 1)= f1(n4d) defined by Eq. (2.81b) and of thecurve (dashed line) -n5s (mod 1)= f2(n4d) defined by Eq.(2.80), where -n5s and n4d depend parametrical on theenergy. The Lu-Fano plot of Fig. 5 was calculated usingthe eigenquantum defects m150.892, m250.491 and themixing angle u50.196p fitted to the experimental databy Armstrong et al . (1979). The deviation betweentheory and experiment occurring for the 5s5p level re-sults from the fact that Armstrong et al . (1979) intro-duced only a linear energy dependence for the ma

whereas, as documented in Sec. IV, a more complicatedenergy dependence exists. It should be noted here thatthe Lu-Fano plot of Fig. 5 can be extended above the5s threshold provided the effective quantum numbern5s is replaced by d5s /p where d5s represents the phaseshift in the 5sep open channel.From a practical point of view, for a two-limit prob-

lem, the intersections of the diagonal straight linen5s5n4d with the Lu-Fano curve directly give the valuesof the two ma parameters. Moreover the slope of theLu-Fano curve at each of these intersections is related tothe angle u (Fano, 1970). The gap between the twobranches gives a visual estimation of the channel inter-action strength. For an almost unperturbed 5snl Ryd-berg series characterized by an almost constant quantumdefect m , one should expect a sharp rise of -n5s (mod 1)near the perturbing level in order to keep m5n2n5sconstant. The marked departure from a step functionvisible in Fig. 5 reflects a strong channel interaction. Aquantitative measure of the channel coupling is given(Greene, 1981) by the S-matrix element

uS12u25sin22u sin2pDm , (2.82)

which gives the probability that the Rydberg electronwill scatter from one channel to the other when it col-lides with the ionic core. The fitted valuesDm5um12m2u50.401 and u50.196p are not too farfrom the values Dm50.5 and u5p/4, which correspondto the strongest possible channel interaction.Lu-Fano plots drawn through experimental points

help the experimentalists in spectral analysis of per-turbed series. It is possible to detect channel interactionsand to estimate their strength without any calculation.However, more quantitative information on the identifi-cation of interacting levels and on the strengths of chan-nel interactions requires the determination of thesmooth scattering parameters. The fitting procedureconsists of adjusting the parameters so that the theoreti-cal and experimental energies agree. The determinationof the channel coefficients Zi

(n) of Eq. (2.64), whichmeasure the fractional admixture of each fragmentationchannel i into a particular level n allows one to label thelevels. In the Sr case discussed above, the empiricalMQDT treatment has confirmed the suggestion of Gar-ton and Codling (1968) concerning the identification ofthe 4d5p 1P perturber: this level must be identified withthe 5s8p 1P level listed by Moore (1949).For systems in which spin-orbit effects cannot be ne-

glected (rare gases, heavy alkaline earths), the fragmen-tation channels should be described in jj or jK coupling.The number N of interacting channels corresponding togiven J and parity can be large. The number of scatter-ing parameters becomes important, the unitary Uia ma-trix depending on N(N21)/2 independent parameters.The use of the jj-LS frame transformation helps togreatly reduce the number of adjustable parameters.The semiempirical analyses of the Ar absorption spec-trum (Lee and Lu, 1973) and of heavy alkaline earths(Aymar et al . , 1978; Armstrong et al . , 1979 Aymar andRobaux, 1979) introduced a set of LS-coupled channelsa and factorized the Uia matrix, so thatUia5(aXiaV aa . The Xia matrix corresponding to thegeometric jj-LS transformation accounts for most of theangular recoupling effects. Assuming that the short-range Hamiltonian is almost diagonal in LS coupling,the V aa matrix elements are fitted to the experimentaldata by minimizing the departure of the V aa matrixfrom the unit matrix. From a practical point of view, theV aa matrix elements are often expressed in terms ofgeneralized Euler angles, each of them describing a ro-tation between two particular channels (Lee and Lu,1973). The empirical analysis of the odd-parity J51spectrum of Ar (Lee and Lu, 1973) showed that the jj-LS frame transformation accounts for much of the chan-nel interaction. This was confirmed by the ab initio cal-culation performed by Lee (1974a) using the iterativeeigenchannel R-matrix variational approach, the empiri-cal and ab initio scattering parameters being in goodagreement.The graphical method of Lu-Fano is most useful for

two-limit systems with a small difference between thetwo ionization limits. In such a case, the Rydberg series



converging to the lower limit I1 is periodically perturbedby members of the series converging to the higher limitI2. The clear pattern of periodicity not only in n1, butalso in n2, can be easily visualized when data are plottedwith respect to n2 (mod 1) instead of n2 itself. When theenergy dependence of the short-range scattering param-eters is negligible, all data drawn inside a -n1 (mod 1)against n2 (mod 1) unit square come to lie on a singlecurve. This curve represents in a compact form all theperturbations arising from the interaction between thechannels (see, for example, Lee and Lu, 1973, and Fanoand Rau, 1986).Such a situation occurs for Rydberg series converging

to the doublet levels of a spin-orbit split core: Rydbergseries of rare gases converging to the p5 2P3/2,1/2 thresh-olds (Lu, 1971; Lee and Lu, 1973); series of the carbon-group elements converging to the s2p 2P1/2,3/2 thresholds(Brown et al . , 1975, 1977a, 1977b, 1977c; Ginter andGinter, 1986; Ginter et al . , 1986); autoionizing Rydbergseries of Ba converging to the 5d3/2,5/2 or 6p1/2,3/2 thresh-olds (Aymar et al . , 1983; Gounand et al . , 1983). Notethat the autoionizing series can be treated as bound se-ries providing the autoionizing widths are small, thetreatment being, of course, restricted to the positions ofresonances. The advantages related to the periodicity ofperturbing levels and to the use of the jj-LS frametransformation were fully exploited in all these studies.Now we turn again to the analyses of the perturba-

tions of bound Rydberg series m0snl of alkaline-earthatoms by doubly excited levels, which are the lowestmembers of the Rydberg series converging to the(m021)d or/and m0p thresholds (m054, 5, 6 for Ca, Sr,and Ba, respectively). The semiempirical approach hashelped greatly in analyzing much energy data obtainedby laser spectroscopy measurements, allowing one toidentify the newly observed levels and to analyze quan-titatively the channel interactions. Moreover, variousobservables other than energy (lifetimes, gJ Lande fac-tors, isotope shifts, hyperfine-structure measurements)provided by high-resolution laser spectroscopy were alsointerpreted semiempirically (see Aymar, 1984a, 1984b).In spite of its clear success, the semiempirical methodencountered several difficulties:(1) At the low end of each Rydberg series, the smooth

scattering parameters acquire a strong energy depen-dence. The lowest levels of each m0snl series cannot becorrectly described empirically, even when a linear de-pendence is invoked for some parameters (see the5s5p level on Fig. 5). More critical is the fact that it thenbecomes difficult and even sometimes impossible toidentify unambiguously the low-lying perturbers by tak-ing into account their Rydberg periodicity. The main dif-ficulties arising for the (m021)d2 and m0p

2 isolatedperturbers of even-parity series will be further addressedin Sec. IV.(2) For systems involving a large number of interact-

ing Rydberg series, assumptions have to be made to re-duce the number of free parameters. They mainly con-cern the construction of the Uia matrix. As alreadymentioned, this matrix is generally constructed by as-

suming that the a channels are nearly LS coupled. As-sumptions on the strength of channel interactions permitone to reduce the number of free parameters introducedin the fitting procedure by fixing the values of someangles generating the V aa matrix at zero. In particular,the interactions between some perturbing channels areneglected.(3) The main difficulties arise from the fact that the

energy values do not contain sufficient information todetermine the complete Uia matrix. More precisely,when more than one channel converges onto a giventhreshold, energies permit one to determine only somecombinations of Uia elements that are invariant underorthogonal transformation of these channels (Lu, 1971;Aymar, 1984a, 1984b). Experiments that distinguish be-tween these channels are necessary to get the full ma-trix. Aymar (1984a, 1984b) discussed how gJ Lande fac-tors and mainly hyperfine-structure data have been usedto extend the Ba multichannel-quantum-defect-theorymodels deduced from energy data. In this way, thesinglet-triplet mixing between 6snl 1L and 3L Rydberglevels have been correctly described (Aymar, 1984a,1984b; Post et al . , 1986). In Sec. IV we will discuss thevalidity of the assumptions made in the semiempiricalMQDT method and show that the difficulties arising inthis approach are bypassed by calculating the scatteringparameters with the R-matrix method.In addition to the early studies of the photoabsorption

spectra of rare gases between the 2P3/2 and2P1/2 thresh-

olds (Lu, 1971; Lee and Lu, 1973), a few semiempiricalMQDT simulations of autoionizing resonances wereperformed with the eigenchannel MQDT formalism(see, for example, Aymar et al . , 1982; Gounand et al . ,1983). However, during the last two decades, experi-mental data on autoionizing resonances of alkalineearths have been obtained mainly by using the experi-mental isolated-core excitation method and the semi-empirical MQDT simulations of the corresponding datawere primarily performed using the phase-shiftedMQDT approach of Cooke and Cromer (1985). The al-ternative MQDT parameters, quantum defects m i andmatrix elements k ij8 of Eq. (2.74) referring to the frag-mentation channels can be more easily extracted fromthe observation than short-range eigenchannel MQDTparameters. Indeed, as is stated in Sec. II.D.3, for anisolated autoionizing Rydberg series, the positions of theresonances are characterized by the quantum defectsm i and the scaled autoionizing widths are related to thek ij8 matrix elements. An impressive number of experi-mental data on autoionizing Rydberg series were fittedusing this approach (see, for example, Xu et al . , 1986;Abutaleb, de Graff, Ubachs, and Hogervorst, 1991;Hieronymus et al., 1990; Jones et al., 1991b; de Graaffet al., 1992, and references therein). However, most ofthe MQDT models adjusted to a particular set of mea-surements introduced numerous assumptions in order toreduce the number of free parameters. A limited num-ber of closed channels was accounted for and generallyeach closed channel was assumed to be coupled to onlyone effective continuum. Moreover, the fragmentation



channels were assumed to be purely jj- or jK-coupledand the energy dependence of the MQDT parameterswas frequently neglected. These models permit a goodsimulation of the observations but have little or no pre-dictive power.

III. EIGENCHANNEL R-MATRIX APPROACH

The preceding section showed how it is useful to char-acterize complicated spectra in terms of the smoothshort-range scattering parameters of multichannelquantum-defect theory. While these have been obtainedsemiempirically by simply fitting to experimental spectrain a number of problems, the full power of this theoreti-cal formulation emerged after when it became possibleto extract these scattering parameters from ab initio ornearly ab initio calculations. In principle, a host of dif-ferent theoretical approaches can be used to calculatethe scattering matrix S and relevant dipole matrix dS. Inthe multichannel spectroscopy viewpoint, the main ideais to solve the time-independent Schrodinger equation ata chosen energy subject to scattering-type boundaryconditions at large distances, which must be applied inall open and closed channels. Such calculations havebeen performed to treat argon photoabsorption by Lee(1974a), using an iterative form of the eigenchannelR-matrix approach developed by Fano and Lee (1973).Accurate MQDT parameters were subsequently calcu-lated by Johnson et al. (1980), near the lowest ionizationthreshold(s) for all of the rare-gas atoms using the rela-tivistic random-phase approximation, with the interest-ing difference that a relativistic version of multichannelquantum-defect theory (Zilitis, 1977; Johnson andCheng, 1979; Johnson and Lin, 1979; Lee and Johnson,1980; Chang, 1993; Goldberg and Pratt, 1987) wasadopted in place of the nonrelativistic approach dis-cussed in Sec. II. These calculations successfully ac-counted for numerous photoionization experiments inthe rare-gas atoms, including integrated total and partialcross sections in addition to the angular distribution andspin polarization of the photoelectrons. Fink andJohnson (1990) used a time-dependent Dirac-Fock ap-proach that is closely related to the relativistic random-phase approximation to predict Lu-Fano plots for therare-gas atoms with J50,2 close to the lowest threshold.The nonrelativistic random-phase approximation couldsimilarly be applied to calculate the MQDT scatteringparameters and photoabsorption amplitudes includingclosed as well as open channels, but apparently no non-relativistic random-phase approximation calculationshave yet been carried out in this manner.While these and other methods such as the hyper-

spherical close-coupling approach (Greene, 1981; Wa-tanabe, 1982, 1986; Tang et al . , 1992) and the Schwingervariational principle (Goforth et al . , 1987; Goforth andWatson, 1992) are capable of solving the Schrodingerequation and extracting the desired MQDT parameters,noniterative R-matrix methods appear to be among themost efficient. The majority of the nearly ab initio calcu-lations of quantum-defect parameters have utilized a

noniterative reformulation (Greene, 1983; Le Rouzoand Raseev, 1984) of the Fano-Lee eigenchannelR-matrix method (Fano and Lee, 1973; Lee, 1974a; Ra-seev and Le Rouzo, 1983). Because it has been used soextensively and successfully in recent years for such cal-culations, this particular (noniterative eigenchannel) for-mulation of the R-matrix approach will be the focus ofour discussion in this section. [It should be noted that,like the Wigner-Eisenbud form of the R-matrix method,the eigenchannel R-matrix method was first introducedin nuclear physics to study resonance reactions (Danosand Greiner, 1966; Mahaux and Weidenmuller, 1968).]On the other hand, we stress that many other methodsare available and have comparable strengths. A notableexample is the Wigner-Eisenbud (Wigner and Eisenbud,1947) form of the R-matrix method, which has many ofthe same advantages as the eigenchannel variant, pro-vided a Buttle correction (Buttle, 1967) is introduced tooffset its poor convergence properties. This method hasbeen extensively used by the Belfast group (Burke andRobb, 1975; Berrington et al . , 1978); for references thereaders are referred to the recent compilation of Burkeand Berrington (1993). Moreover, a modified form ofthe Wigner-Eisenbud R-matrix theory (Wigner andEisenbud, 1947) that has been used by Schneider (1975,1995) appears to be formally equivalent (Robicheaux,1991; Schneider, 1995) to the noniterative eigenchannelR-matrix method if the same variational basis sets areused in the two approaches.

A. General procedure

The eigenchannel R-matrix method is a variationalapproach that determines a set of eigenstates of theSchrodinger equation within a finite reaction volume Vin configuration space. The reaction volume is a sphereof radius r0, chosen such as ri<r0, ri being the distancebetween the ith electron and the nucleus (see Sec. II andFig. 2). The fragmentation coordinate r , introduced inSec. II, is the radial coordinate of the outermost elec-tron. In a multichannel problem, the boundary condi-tions on solutions at a finite boundary, on the surfaceS enclosing the reaction volume, are not uniquely speci-fied. The word eigenchannel, which characterizes thismethod, refers to the fact that it calculates a particularset of linearly independent solutions that are the eigen-states of the R matrix at the energy E specified before-hand. This implies that each such eigenstate, for ex-ample, the bth solution (rCb) has a normal logarithmicderivative

2bb5]ln~rCb!

]r(3.1)

that is constant at every point on the surface S .The variational principle for the eigenvalues bb can

be derived easily, starting from the exact expression forthe Schrodinger energy eigenvalue that is calculated interms of exact eigensolutions within the reaction volumeonly:



E5

EV

C!~2 12 ¹2C1UC!dV

EV

C!CdV. (3.2)

Here, the integrals extend only over the reaction volumeV . The 2 1

2¹2 kinetic-energy operator is a shorthand no-

tation for 2 12¹

252 12( i¹ i

2 , where the sum runs over allelectrons. The potential-energy operator U is assumedto be Hermitian.Application of Green’s theorem to the kinetic-energy

operator integral transcribes Eq. (3.2) into

E5

EV

@ 12 ¹W C!

•¹W C1C!UC#dV2 12 E

SC!~]C/]n !dS

EV

C!CdV,

(3.3)

in which an additional integral over the reaction surfaceS now appears. Using Eq. (3.1), the expression (3.3) canbe written, at a given energy E , as an equation forb(E):

b52

EV

C!~E2H !CdV

EV

C!d~r2r0!CdV, (3.4)

where H denotes the Hamiltonian within the sphere,made Hermitian by addition of the Bloch operator(Bloch, 1957)

H5H112r

d~r2r0!]

]rr . (3.5)

Strictly speaking, in Eq. (3.4) we have derived thus faronly an identity obeyed by any exact Schrodinger eigen-state whose logarithmic derivative is constant across thereaction surface. However, by taking the first-ordervariation of this expression with respect to small devia-tions of C from an exact solution, one can demonstratethat this is a variational principle for b(E), the negativeof the logarithmic derivative. This variational principleis typically used in practical calculations as follows. Thetrial functions C are expanded in terms of a set of narbitrary (but preferably physically motivated) basisfunctions yk

C5 (k51,n

ykCk , (3.6)

where the superposition coefficients Ck are determinedby the variational principle (3.4). Since the Hamiltonianis normally real, C , Ck , and yk will be assumed to bereal without loss of generality.In the Wigner-Eisenbud formulation (Burke and

Robb, 1975; Burke and Taylor, 1975; Burke and Ber-rington, 1993), the procedure for calculating an R matrixstarts from the variational expression (3.2) and deter-mines a complete set of eigenvalues El and eigenfunc-tions Cl for any fixed value of the boundary parameter

b by solving the system of equations ]E/]Ck50. In con-trast, the eigenchannel R-matrix approach (Greene,1983, 1985, 1988; Hamacher and Hinze, 1989) uses thevariational expression (3.4) to determine a complete setof eigenvalues bb and eigenfunctions Cb for any fixedvalue of the energy E . No constraint needs to be im-posed on the basis functions yk ; in particular they neednot be orthogonal and also they need not have anyspecified (common) logarithmic derivative on the reac-tion surface. The extremum condition ]b/]Ck50 leadsto a generalized eigenvalue problem for b in the vectorspace of the coefficients Ck , namely

GCW 5LCW b . (3.7)

The matrix G is expressed as

Gkl52EVyk~E2H !yldV2E

S

1ryk

]~ryl!

]rdS . (3.8)

The matrix G can be expressed in terms of a volumeoverlap matrix O and of a Bloch-operator matrix L inaddition to the Hamiltonian,

Gkl52~EOkl2Hkl2Lkl!. (3.9)

The matrix L consists of a simple surface overlap inte-gral

Lkl5ESykyldS . (3.10)

Eigenvectors CW b and CW b8 corresponding to distinct ei-genvalues bb and bb8 are orthogonal over the reactionsurface

CW bt LCW b85Nb

2dbb8, (3.11)

where Nb is a normalization factor. On physicalgrounds, the number of eigensolutions should equal thetotal number N of open or ‘‘weakly closed’’ channels,namely the number of channels having non-negligibleamplitude on S . The remaining channels that have neg-ligible amplitude on S but may contribute appreciablyto the dynamics inside the reaction volume are referredto as ‘‘strongly closed’’ channels.If the (n3n) matrix L is nonsingular there are n in-

dependent solutions of Eq. (3.7). In practice, however,the number of independent solutions is far smaller thanthe total number of variational basis functions includedin Eq. (3.6), because L is singular. Le Rouzo and Raseev(1984) showed that the number of nontrivial eigensolu-tions of Eq. (3.7) equals the rank of the matrix L .We can clarify the concept of a channel function in-

troduced in Sec. II.C.1 in this mathematical descriptionby introducing a set of N real orthonormal surface har-monics F i(v) which span the surface S :

ES

1

r02 F i~v!F j~v!dS5d ij , (3.12)

where v is the set of coordinates orthogonal to r (seeSec. II.C.1). In other words, N , the number of channelsin which an electron can escape from V , corresponds to



the smallest number of surface harmonics needed to ad-equately represent the n basis functions on S . Each ofthe basis functions yk can be expanded on S as a linearcombination of the F i , according to

yk5 (i51,N

1r0

F i~v!uki~r0!. (3.13)

The matrix elements of L are then simply

Lkl5 (i51,N

uki~r0!uilt ~r0!. (3.14)

It is now possible to complete the eigenchannel deri-vation of the R matrix itself (Greene, 1983, 1988). Re-calling Eqs. (2.21) and (2.29), the R-matrix eigenvectorsare the projections of the (rCb) eigensolution

Cb5 (k51,n

ykCkb (3.15)

onto the ith surface harmonic

Zib5ESF i~v!~rCb!dv/Nb5 (

k51,Nuki~r0!Ckb /Nb .

(3.16a)

When Nb is chosen such that (rCb)/Nb is normalizedover the reaction surface, the eigenvector matrix Z isorthogonal. The projection of the radial derivative of(rCb) onto the ith surface harmonic is simply

Zib8 52bbZib . (3.16b)

[Note that the Zib correspond to the Fib(r0) defined inEq. (2.19)]. The R matrix in the chosen representationof surface harmonics is now given by

Rij52(b

Zibbb21Zjb , (3.17)

which is automatically symmetric.Using Eqs. (3.7), (3.16), and (3.17), the matrix R ma-

trix can be written in matrix form as

R52utG21u . (3.18)

This form of the matrix R is identical to that previouslygiven by Nesbet (1980) and by Robicheaux (1991).Robicheaux (1991) has shown how Eq. (3.17) can be

expressed in the Wigner-Eisenbud form for R(E), in-volving a sum of poles. He introduced a Hermitian ma-trix H which is the sum of the Hamiltonian H andBloch-operator L matrices. [Schneider (1995) has pre-sented an alternative derivation that starts from thetime-independent Schrodinger equation, after which theBloch operator is added and subtracted.] This matrixH can be diagonalized by an energy-independent trans-formation matrix W :

(lHklWll5(

lOklWllEl , (3.19)

where Wll is normalized by the condition(k ,lOklWklWll85dll8. [Note that the orthogonal trans-formation matrices W introduced in this section should

not be confused with the decreasing Coulomb functionof Eq. (2.52).] The R matrix determined by the varia-tional principle (3.7) and by Eq. (3.18) is now given bythe Wigner-Eisenbud-type form of Eq. (1.1):

Rij512(l

YilYjl

E2El, (3.20)

where

Yil5(kukiWkl . (3.21)

The El are the variational energy eigenvalues, whileYil is the surface projection of the lth eigenstate ontothe ith surface harmonic. (Note that the factor beforethe summation differs depending on the authors; the fac-tors 6 1, 6 1/2, and 61/r0 are encountered in the litera-ture.) Robicheaux’s derivation explicitly connects theeigenchannel and Wigner-Eisenbud R matrix methods,showing how the well-known Wigner-Eisenbud form ofthe R matrix remains generally valid, even when thebasis functions do not have a common logarithmic de-rivative on the surface. This fact has been recognized bysome authors (see, for example, Nesbet, 1980;Schneider, 1975, 1995) but it has been rarely utilized inpractical calculations. The flexibility of using basis func-tions with different surface logarithmic derivativesspeeds up convergence of the calculation (with respectto basis set size) significantly. It is critically important toinclude an energy-dependent Buttle correction (Buttle,1967) to the R matrix to obtain accurate results usingthe original Wigner-Eisenbud-type basis set with a com-mon surface logarithmic derivative. On the other hand,calculations that use a basis set with a range of logarith-mic derivatives exhibit sufficiently rapid convergencesuch that no Buttle correction is needed.

B. Specialization to two-electron systems. R-matrixcalculations in LS coupling

Neglecting the spin-orbit terms within the reactionvolume, a model Hamiltonian can be used to effectivelydescribe the two outermost (valence) electrons of analkaline-earth atom A. These two valence electronsmove primarily outside of the closed shell A11, with atotal Hamiltonian

H52 12 ¹1

22 12 ¹2

21V~r1!1V~r2!11r12

. (3.22)

The interaction of each valence electron with thenucleus and inner-shell A11 core electrons is describedby an effective potential V(r).

1. Relationship between V(r) and the energy levels of A1

In the first eigenchannel R-matrix calculations carriedout for Be and Mg (O’Mahony and Greene, 1985;O’Mahony, 1985; O’Mahony and Watanabe, 1985), anab initio l-independent Hartree-Slater potential vHS(r)was used as the model potential V(r). The Hartree-Slater potential is a relatively crude type of



independent-electron model which is adequate for manypurposes in light atoms, but for the heavier alkaline-earth atoms more sophistication is needed. In fact, thed orbitals of Ca, Sr, and Ba, as well as the f orbitals ofBa, depend sensitively on the accuracy of the potential.The potential used in Ca by Greene and Kim (1987) andby Kim and Greene (1987, 1988) included, in addition toa l-dependent Hartree-Slater screening potentialv lHS(r), an additional polarization potential:

Vl~r !5v lHS~r !2

acp

2r4$12exp@2~r/rc

l !6#%. (3.23)

The core polarization acp and the cutoff radius rcl were

adjusted so that the eigenvalues enl of the one-electronradial equation

S 212d2

dr21l~ l11 !

2r21Vl~r !2enlDunl50 (3.24)

coincided with the experimental (spin-orbit averaged)energies of Ca1. Optimal values of the parameters canbe found in the paper of Kim and Greene (1988).More recently, a number of calculations have been

carried out using a more convenient analytical modelpotential that depends on a few parameters that must bedetermined empirically. Simple l-independent potentialsof the form

V~r !521r

$21~Z22 !exp~2a1r !1a2rexp~2a3r !%

(3.25)

were used in the lighter alkaline earths, where Z is thenuclear charge and the three empirical parameters a iwere fitted to known energy levels of the alkali-like ion.The parameters values optimized for the alkaline-earthions from Be1 to Sr1 are given in Table I(a). Accuratetwo-electron spectra have been obtained, even in anatom as heavy as Sr (Aymar, 1987; Aymar et al . , 1987;Aymar and Lecomte, 1989), using such anl-independent potential. More sophisticatedl-dependent potentials including a polarization term

Vl~r !521r

$21~Z22 !exp~2a1l r !1a2

l rexp~2a3l r !%

2acp

2r4$12exp@2~r/rc

l !6#% (3.26)

are needed to describe Ba1 and Ra1 adequately (Ay-mar, 1990; Greene and Aymar, 1991). Here, acp is theexperimental dipole polarizability of the doubly chargedpositive ion (Fajans and Joos, 1924; Tessman et al . ,1953; Dalgarno, 1962; Johnson et al . , 1983). The empiri-cal parameters a i

l and rcl were adjusted until the result-

TABLE I. Semiempirical parameters describing the model potential experienced by the outermost(valence) electron in alkaline-earth atoms.

(a) Potential of Eq. (3.25)

a1 a2 a3

Be1 6.9010 8.9581 5.0798Mg1 4.2499 11.0223 2.9417Ca1 3.9557 12.8420 2.0039Sr1 3.5515 6.0373 1.4389

l (b) Potential of Eq. (3.26)

a1 a2 a3 rc0 4.0616 13.4912 2.1539 1.5736

Ca1 1 5.3368 26.2477 2.8233 1.0290acp=3.5 2 5.5262 29.2059 2.9216 1.1717

>3 5.0687 24.3421 6.2170 0.4072

0 3.4187 4.7332 1.5915 1.7965Sr1 1 3.3235 2.2539 1.5712 1.3960acp=7.5 2 3.2533 3.2330 1.5996 1.6820

> 3 5.3540 7.9517 5.6624 1.0057

0 3.0751 2.6107 1.2026 2.6004Ba1 1 3.2304 2.9561 1.1923 2.0497acp=11.4 2 3.2961 3.0248 1.2943 1.8946

> 3 3.6237 6.7416 2.0379 1.0473

0 3.7702 4.9928 1.5179 1.3691Ra1 1 3.9430 5.0552 3.6770 1.0924acp=18 2 3.7008 4.7748 1.4956 2.2784

> 3 3.8125 5.0332 2.1016 1.2707



ing Schrodinger energy eigenvalues obtained from thepotential agree with the experimental energies of thealkaline-earth ion. The values of the parameters ob-tained for Ca1, Sr1, Ba1, and Ra1, are given in TableI(b).The extensive eigenchannel R-matrix calculations car-

ried out in recent years using model potentials haveshown one major result: accurate two-electron spectraare obtained provided the potential V(r) gives accurateenergies for the one-electron ionic system. This can beunderstood from the fact that the phase of the valenceelectron orbital as the electron emerges from the ioniccore is virtually guaranteed to be correct if the model-potential energy levels agree with the experimental ioniclevels. The detailed interactions of the two electronsoutside this core depend sensitively on these phases, i.e.,on the one-electron quantum defects of each emergingelectron. The majority of recent calculations of compli-cated alkaline-earth atom spectra have used semiempir-ical model potentials such as those given by Eq. (3.26)and Table I(b). They have been successful mainly be-cause they give a good description of the ionic (alkali-like) valence energy spectra.

2. Design of the basis set

The basis set used in the variational calculations con-sists of LS-coupled independent-particle numerical ba-sis functions. An antisymmetric two-electron basis func-tion, LS coupled to form a state of definite S , L , andparity, is expressed in terms of numerical one-electronorbitals unl(r) and Fnl(r) by

yn1l1n2l2~rW1 ,rW2!5

1

A2r1r2@un1l1~r1!Fn2l2

~r2!

3Yl1l2LM~V1 ,V2!

1~21 ! l11l22L1SFn2l2~r1!un1l1~r2!

3Yl2l1LM~V1 ,V2!# . (3.27)

An alternative form for Eq. (3.27) connects more natu-rally with the quantum-defect description of the outerregion [Eq. (2.19a)],

yin25221/2@F i~r1 ,V1 ,V2!Fn2l2~r2!/r2

1~21 !SF i~r2 ,V2 ,V1!Fn2l2~r1!/r1# , (3.28)

where the channel index stands for i[$n1l1l2% andwhere

F i~r1 ,V1 ,V2!5un1l1~r1!

r1Yl1l2LM

~V1 ,V2!. (3.29)

The channels functions F i are the functions for the ionicstates including the angular wave functions of both elec-trons. (Note that, in the following, the open and weaklyclosed channels, with specific values of n1, l1, and l2, willoften be designated by n1l1nl2, the notation n1l1el2 be-ing used for open channels to emphasize that the chan-nel is open.)

In some early R-matrix calculations performed for Be,Mg (O’Mahony, 1985; O’Mahony and Greene, 1985;O’Mahony and Watanabe, 1985), and Sr (Aymar, 1987;Aymar et al . , 1987) two different kinds of orbitals wereintroduced. The un1l1 orbitals in the channel functions[Eq. (3.29)] obey Eq. (3.24) using the potential V(r)that describes the A11-e interaction. However, the Fnlorbitals were chosen to correspond instead to ‘‘polar-ized’’ orbitals. These were still determined by solvingEq. (3.24) except with a different potential more appro-priate to the A1-e interaction.The use of different types of orbitals complicates the

calculations without providing any significant accelera-tion of the convergence. After this was recognized, themore recent two-electron R-matrix calculations haveused orbitals generated from the A11-e potentialV(r), which amounts to setting Fnl[unl in Eq. (3.27).With this choice, the two-electron basis functions in Eq.(3.27) are eigenfunctions of the independent-electronHamiltonian H05H21/r12 . [Note that with this choicethe factor 1/A2 in Eq. (3.27) should be replaced by1/A2(11dn1n2d l1l2).] Because the ionic basis set issimple, convenient, and effective for two-electron sys-tems, we consider only this type in the following. In Sec.III.D below we will show that for atoms with manyopen-shell electrons, this choice is highly inferior tothe use of Lowdin’s ‘‘natural orbitals’’ (Lowdin, 1955;Lowdin and Schull, 1956) or multiconfiguration Hartree-Fock-type (Froese-Fischer, 1977) orbitals (Robicheauxand Greene, 1992, 1993a, 1993b, 1993c; Chen and Ro-bicheaux, 1994; Miecznik et al . , 1995; Miecznik andGreene, 1996).In practice, a first set of orthogonal ‘‘closed-type’’ or-

bitals denoted unlc is obtained by solving Eq. (3.24) nu-

merically subject to the boundary condition that eachorbital vanishes on the boundary r5r0. Figure 6 shows aset of closed-type l50 orbitals obtained for Ba with r0 =20 a.u. This first set is complemented by a second set of‘‘open-type’’ orbitals (unl

o ) by integrating Eq. (3.24) atseveral energies that differ from the enl

c eigenvalues ob-tained in calculating the closed-type orbitals. These or-bitals have a nonzero amplitude at r5r0, and are neitherorthogonal to one another nor to the unl

c orbitals in gen-eral. Two such s-wave open-type orbitals are shown inFig. 6.As presently implemented, the R-matrix approach

treats the escape of (at most) one electron from the re-action volume. This is enforced by using only closed-type orbitals for the channel functions F i [Eq. (3.29)].The minimum possible value that can be used for thebox size r0 is therefore governed by the requirementthat the most diffuse A1 n1l1 ionic states relevant to theenergy range under consideration must fit within the re-action volume.The two-electron basis functions yk retained in the

R-matrix calculation are grouped into closed-type andopen-type sets. Each ‘‘closed-type’’ basis function yk

c

consists of two closed-type orbitals and thus vanisheseverywhere on the reaction surface S . Therefore, all the



coefficients uki(r0) associated with the ykc functions van-

ish in the expansion (3.13). Each ‘‘open-type’’ basisfunction yk

o has a nonzero amplitude on S and is builtfrom one open-type orbital un2l2

o , for the outer electron

and one closed-type orbital un1l1c for the inner electron.

For a given channel i5$n1l1l2%, the coefficients uki(r0)associated with yk

o functions in Eq. (3.13) reduce toun2l2o (r0). A set of closed-type functions alone forms an

orthonormal set of eigenfunctions that spans the reac-tion volume. This set would be efficient only to describea solution of the Schrodinger equation which vanisheson S . The open-type basis functions yk

o are needed toallow the outermost electron to reach the surface S andto escape beyond it in an excited bound or continuumstate. The use of yk

o basis functions adds flexibility to thebasis, although the number of such basis functionsshould be kept small (typically 1–3 per channel) to avoidnumerical difficulties associated with linear dependenceof the full basis set. The nonorthogonality of the fullbasis set causes no particular difficulties for two-electronsystems. For atoms having more than two electrons inopen shells, however, the calculation of matrix elementsbecomes tremendously complicated if the orbitals arenot orthogonal, and consequently the open-type orbitalsare orthogonalized to the closed-type orbitals.No completely general algorithm for choosing the

two-electron configurations has been identified, i.e., toselect the basis functions to include in the variationalcalculation. The optimum basis set depends on details ofthe system being studied, such as the number of open

and weakly closed channels relevant to the energyrange, the size of the reaction volume, the degree ofconvergence desired, and the available computermemory. If any particular channel is to be treated asopen or weakly closed in the MQDT calculation, bothopen-type and closed-type basis functions must be in-cluded for that channel. Trials with one to three open-type basis functions per channel have found very littlesensitivity to their number. Experimentation has shownthat, in heavy alkaline earths, a surprisingly large num-ber of strongly closed channels (or equivalently, a largeset of closed-type basis functions) is required to describevarious electron correlation, relaxation, and polarizationeffects (Bartschat and Greene, 1993). In particular, theinner electron can ‘‘relax’’ or become polarized in re-sponse to the electric field of the outermost electron.This is the dominant contribution to core relaxation inthe present model since the A11 core is treated as fro-zen. Polarization effects are neglected when an electronroams beyond the reaction volume V . The closed-typebasis set should be sufficiently flexible to permit a de-scription of polarization effects within V .The choice of the basis set and of the reaction volume

size, in calculations for two-electron atoms, will be docu-mented further in Sec. III.G below.

C. Inclusion of nonperturbative spin-orbit effects:Two-electron R-matrix calculations in j j coupling

Because the eigenchannel R-matrix calculations in jjcoupling proceed along the same lines as the R-matrixcalculations in LS coupling, only the differences are out-lined here. The model Hamiltonian used in R-matrixcalculations carried out in jj coupling includes explicitlythe spin-orbit terms, which are ignored in the Hamil-tonian of Eq. (3.22),

H5212

¹122

12

¹221V~r1!1V~r2!1Vso

s1l1j1~r1!

1Vsos2l2j2~r2!1

1r12

. (3.30)

The spin-orbit interaction between each valence elec-tron and the screened nucleus can be approximated by apotential having the form

Vsoslj~r !5

a2

2sW• lW

1r

dV

dr S 12a2

2V~r ! D 22

, (3.31)

in which a is the fine-structure constant in a.u. The lastfactor in Eq. (3.31) is suggested by the Dirac equation(see Condon and Shortley, 1935). While it is not neededin perturbative spin-orbit calculations, this factor is in-cluded, as in some previous studies, in order to ensurethat solutions of the radial Schrodinger equation arewell defined near the origin, r→0. Without this factor,the r23 behavior of the spin-orbit potential near the ori-gin leads to an ill-defined mathematical behavior of theradial wave functions.The variational basis set consists of antisymmetrized

two-electron functions whose angular momenta are jj

FIG. 6. Radial one-electron ns orbitals of Ba1 used in theeigenchannel R-matrix calculations for Ba carried out with areaction volume of radius r0 = 20 a.u: (a) closed-type orbitals;(b) open-type orbitals.



coupled to form a state of definite total angular momen-tum J and parity,

yk~rW1 ,rW2!51

r1r2A2~11dn1n2d l1l2d j1j2!

3(un1l1j1~r1!un2l2j2~r2!

3$@xs1~1 !Yl1

~1 !# j1@xs2~2 !Yl2

~2 !# j2%JM

2~21 ! j11j22Jun2l2j2~r1!un1l1j1~r2!

3$@xs2~1 !Yl2

~1 !# j2@xs1~2 !Yl1

~2 !# j1%JM),

(3.32)

where the basis-vector label k is a shorthand notationfor the quantum numbers n1l1j1n2l2j2. In Eq. (3.32), theelectron spinors xs and spherical harmonics Yl are stan-dard. The j-dependent one-electron radial function unljobeys the radial Schrodinger equation

S 212d2

dr21l~ l11 !

2r21V~r !1Vso

slj~r !2enljDunlj50.

(3.33)

All the jj-coupled eigenchannel R-matrix calculationsperformed in heavy alkaline-earth atoms use for theelectron-core effective potential the empiricall-dependent model potential Vl of Eq. (3.26), also uti-lized in recent LS-coupled R-matrix calculations. In-deed, the semiempirical parameters listed in Table I(b)were adjusted to obtain optimum agreement betweenthe eigenvalues enlj of Eq. (3.33) and the experimentalj-dependent energies of the alkaline-earth ions. Ofcourse, if the spin-orbit term is neglected in the one-electron radial Schrodinger equation, one obtains enl en-ergies that agree well with the spin-orbit-averaged ener-gies of alkaline-earth ions.Some calculations (Lecomte et al . , 1994, Luc-Koenig

et al . , 1995) were performed by adding to the two-electron Hamiltonian H of Eq. (3.30) a dielectronic po-larization correction (Chisholm and Opik, 1964;Hameed, 1972) to the 1/r12 interaction. Following Victorand Laughlin (1972), the dielectronic polarization termbetween the l1l2 and l18l28 basis functions was taken to be

Vpol~2 !~rW1 ,rW2!52

acp

r12r2

2 P1~ r1• r2!@w~rcl1 ,r1!

3w~rcl18 ,r1!w~rc

l2 ,r2!w~rcl28 ,r2!#

1/4,

(3.34)

where the cutoff function is w(rcl ,r)512exp@2(r/rc

l)6#.In Eq. (3.34), P1 is a Legendre polynomial and acp thedipole polarizability of the core. The l-dependent cutoffradii are given in Table I(b).

D. Specialization to open-shell atoms. R-matrixcalculations in LS coupling

The LS-coupled eigenchannel R-matrix approach hasbeen used to calculate photoionization cross sections

and bound-states properties of some open p-shell atomsand of two open d-shell atoms, scandium and titanium.Calculations in open p-shell atoms dealt with Al(O’Mahony, 1985; Miecznik et al . , 1995), with halogens(Robicheaux and Greene, 1992, 1993a), with atoms inthe carbon group (Robicheaux and Greene, 1993b), andin the oxygen group (Chen and Robicheaux, 1994). Allof these atoms have a ground configuration of the typem0s

2m0pq, with m053 and q51 for Al. For the second-

row elements (m052), one has q52 for C, q54 for O,and q55 for F. Scandium, the simplest transition-metalatom, with ground-state configuration 3d4s2, has beenchosen to see whether the eigenchannel R-matrixmethod can reproduce the extremely complicated spec-tra of open d-shell atoms (Armstrong and Robicheaux,1993; Robicheaux and Greene, 1993c, 1993d). The accu-rate results obtained in Sc have encouraged Miecznikand Greene (1996) to undertake eigenchannelR-matrix calculations in titanium, with ground state3d24s2.

1. Hamiltonian

Most of the eigenchannel R-matrix calculations inopen-shell atoms were carried out with a HamiltonianH that does not refer to the full atomic system. As in thealkaline-earth atoms, they utilized a model Hamiltonianof the valence electrons only, and approximated the ef-fects of the closed-shell inner electrons and of thenucleus on the valence electrons through an effectivepotential V(r). Neglecting the spin-orbit terms withinthe reaction volume, the valence-electron Hamiltonian,in a.u., is

H5(i

$2 12 ¹ i

21V~ri!%1(i ,ji,j

1/rij . (3.35)

In Eq. (3.35), the sums over indices i and j run over theNv valence electrons outside the rare-gas-like inert core.One has Nv=3 for Al and Sc, Nv=4 for the carbon-groupatoms and for Ti, Nv=6 for the oxygen-group atoms, andNv=7 for the halogens.The first eigenchannel R-matrix study carried out in

an atom with more than two valence electrons con-cerned Al (O’Mahony, 1985). As in the previousR-matrix calculation carried out in Be and Mg(O’Mahony and Greene, 1985), the interaction of eachvalence electron with the closed-shell core was describedby an ab initio l-independent Hartree-Slater potential.The second calculation dealt with Si (Greene and Kim,1988). A Hartree-Slater potential with an empirical po-larization potential was used to describe the Si 41-e in-teraction.The techniques used in the recent calculations have

been improved compared to those employed in Al(O’Mahony, 1985) and Si (Greene and Kim, 1988). Anumber of calculations (Robicheaux and Greene, 1992,1993a, 1993b, 1993c, 1993d; Armstrong and Robicheaux,1993; Miecznik et al . , 1995) have been carried out usingmore sophisticated analytical one-electron model poten-



tial including an l-dependent screened Coulomb poten-tial and a polarization potential,

Vl521r$Nv1~Z2Nv!exp~2a1

l r !1a2l rexp~2a3

l r !%

1Vpol~1 !~r !, (3.36)

where the polarization potential has the form

Vpol~1 !~r !5

acp

2r4$12exp@2~r/rc!

3#%2. (3.37)

In Eq. (3.37), theoretical values were used for the dipolepolarizability acp (Johnson et al . , 1983). The parametersa il and rc were fitted to optimize agreement between the

calculated energy levels of Hv and the experimental en-ergy levels of the (Nv21)+ ion. Complete lists of theparameters for the semiempirical potentials optimized inthe open-shell atoms that were studied, as well as thereferences on the experimental levels included in thefits, can be found in the relevant papers (Robicheauxand Greene, 1992, 1993b, 1993c; Miecznik et al . , 1995).The one-electron Hamiltonian has the form:

Hv52 12 ¹21Vl~r !. (3.38)

In addition, the dielectronic polarization interaction hasbeen systematically introduced in the model Hamil-tonian used in the recent eigenchannel R-matrix calcu-lations of open-shell atoms. This consists in replacing theHamiltonian of Eq. (3.35) by

H5(i

$2 12 ¹ i

21Vl~ri!%1(i ,ji,j

~1/rij1Vpol~2 !~rW i ,rW j!!,

(3.39)

with

Vpol~2 !~rW i ,rW j!522P1~ r i . r j!@Vpol

~1 !~ri!Vpol~1 !~rj!#

1/2. (3.40)

Formally, Eq. (3.40) is identical to Eq. (3.34) given inSec. III.C above. Note that the dielectronic polarizationinteraction has not been systematically included in thecalculations in alkaline earths. However, its effect hasbeen analyzed in some particular cases (Lecomte etal . , 1994; Luc-Koenig et al . , 1995). As described belowin Sec. IV, the dielectronic polarization interaction sig-nificantly influences the Ba 5d5g autoionization widths(Luc-Koenig et al . , 1995). In contrast, the influence ofthis interaction on the dynamics of open-shell atoms hasnot been studied in detail.The calculations in open-shell atoms conducted with a

parametrized model potential emphasize the need foraccurate energy-level data for alkali-like ions. Some cal-culations for the heavier halogens Br and I and theheavier C-group atoms Ge and Sn have suffered fromthe limited amount of ionic-level information that wasavailable. Errors could be introduced in the calculationthrough the parameters of the model potential. To by-pass these difficulties, another approach was used re-cently by Chen and Robicheaux (1994) to study the

oxygen-group atoms with the eigenchannel R-matrix ap-proach. The Hamiltonian chosen by Chen and Ro-bicheaux (1994) refers to the full atomic system. How-ever it still contains the one- and two-electronpolarization terms. One has

H5(i

S 2 12 ¹ i

22Z

ri1Vpol

~1 !~ri! D1(i ,ji,j

~1/rij

1Vpol~2 !~rW i ,rW j!!. (3.41)

In Eq. (3.41), the sums over indices i and j run over allelectrons; Vpol

(1)(ri) and Vpol(2)(rW i ,rW j) have the form given

in Eqs. (3.37) and (3.40), respectively. Theoretical values(Johnson et al . , 1983) were used for the dipole polariz-ability acp and the cutoff radius rc was determined byevaluating the expectation value ^nlurunl& for orbitals ofthe outermost closed shell. Comparison of eigenchannelR-matrix calculations carried out in S with the Hamil-tonian of Eq. (3.41) referring to all 16 electrons and witha model potential and 6 electrons (Chen and Ro-bicheaux, 1994) will be discussed later in Sec. V.A full-electron description has been used also recently

by Miecznik and Greene (1996) in titanium (22 elec-trons) for calculating photoabsorption and photoioniza-tion from excited states. However, in those calculations,the one- and two-electron polarization terms were notincluded.

2. Constructing basis functions

One major complication occurs in R-matrix calcula-tions of open-shell atoms compared to those in two-valence-electron systems. This complication is related tothe construction of basis functions used for the initial-and final-state atomic wave functions. A key require-ment of any R-matrix calculation is an accurate descrip-tion of the target-state wave functions, which corre-spond to the possible wave functions of the positive ionin the studied energy range. For two-electron systems,when a semiempirical model is used to describe the in-teraction of each valence electron with the closed-shellcore, this aspect of the computation is easy since thetarget functions are simply orbitals of the one-electronionic potential. More effort goes into determining mul-tielectron target functions, which magnifies the amountof work needed to describe the full atomic dynamics.Configuration interaction is generally included in thetarget functions. It is very important in choosing targetbasis functions to try to work with the minimum numbernecessary to achieve convergence for the atomic dynam-ics because each target basis function translates to ; 10atomic functions. Different approaches have been usedin open-shell atoms to describe ionic states.First, consider the calculations that included a model

potential to account for the effects of closed shells. Inorder to reduce the number of multielectron basis func-tions needed for convergence, the many-electron func-tions are not expanded in terms of the orbitals that areeigenstates of Hv [Eq. (3.38)] but in terms of orbitalsthat cause the core states to converge. Indeed, a more



compact description of target states could be obtainedusing ‘‘natural orbitals’’ (Lowdin, 1955; Lowdin andShull, 1956; Froese Fischer, 1991). The natural orbitalscan be obtained by diagonalizing a large configuration-interaction matrix constructed from orbital solutions ofHv for the target states of interest, and superposing theorbitals. Natural orbitals can also be chosen to minimizethe total energies of selected target states. The nl natu-ral orbitals are associated with the lowest n values oc-curring for s , p , d , f , . . . symmetries. Different tech-niques were sometimes used for a given atom todetermine the nl natural orbitals associated with differ-ent l values (see, for example, Robicheaux and Greene,1992). Natural orbitals can be more diffuse or more con-tracted than spectroscopic orbitals.Because of the limited number of orbital solutions of

Eq. (3.38) introduced in determining the natural orbit-als, excited natural orbitals can have spurious nodesclose to r0. In order to eliminate this unphysical behav-ior, an l-dependent potential Vl that has these orbitalsas eigenstates is optimized. Different forms were usedfor the potential Vl . The potential used by Robicheauxand Greene (1992) has the same form as Vl [Eq. (3.36)]with Nv replaced by 2 and a i

l replaced by a il , whereas

the potential used in Sc and Al (Robicheaux andGreene, 1993c; Miecznik et al . , 1995) has the form

Vl~r !5Vl~r !112exp~2a1

l r !

r1

a2l $12exp@2a3

l r#%

r2.

(3.42)

Lists of the parameters a il optimized in each open-shell

system that was studied can be found in the correspond-ing papers (Robicheaux and Greene, 1992, 1993b, 1993c;Miecznik et al . , 1995). This potential is next used togenerate s , p , d , f , . . . orbitals with higher quantumnumbers.Although the procedure used to generate natural or-

bitals, that we have just described, proved to be highlysuccessful in previous applications (Robicheaux andGreene, 1992, 1993a, 1993b, 1993c, 1993d; Armstrongand Robicheaux, 1993; Miecznik et al . , 1995), it is notfree of difficulties. As emphasized by Miecznik andGreene (1996), the form of natural orbitals (and thus therate of convergence of configuration-interaction targetstates) depends strongly on the number of orbital solu-tions of Hv , as well as on the type and number of targetconfigurations used to obtain the natural orbitals.A different approach to describe the target states,

based on the Hartree-Fock approximation, was used byChen and Robicheaux (1994) in their full-electron calcu-lation in oxygen-group atoms and by Miecznik andGreene (1996) in titanium. Indeed, while using orbitalsolutions of Eq. (3.38), it is necessary to include a largeconfiguration-interaction basis, with singly excited, dou-bly excited, and triply excited configurations to get mod-est degree of convergence, whereas a fully variational(Hartree-Fock) calculation takes better care of such ex-citations, effectively minimizing the number of excita-tions to singly and doubly excited states. Radial orbitals

needed to calculate short-range interaction parameterswere obtained in the Hartree-Fock or multiconfigura-tion Hartree-Fock approximation (Froese Fischer, 1977,1991). The natural orbitals incorporate screening andcorrelations associated with the ionic system. More pre-cisely, in the oxygen-group atoms, Chen and Ro-bicheaux (1994) performed first a Hartree-Fock calcula-tion to obtain one-electron orbitals of the configuration-averaged ion. Next, a multiconfiguration Hartree-Fockcalculation was performed to get the lowest d and f or-bitals to describe correlation in the valence shell. A simi-lar step procedure was used in Ti by Miecznik andGreene (1996). First the inner-electron orbitals as wellas the valence 3d and 4s orbitals of Ti1 were optimizedon the 3d24s configuration, after which 4p was opti-mized on 3d24p . Next, a multiconfiguration Hartree-Fock calculation of the 3d3 2Ge term was performed togenerate 4d , 5s , 5p , and 4f correlations orbitals.Once target orbitals are obtained, higher-n orbitals

needed to represent the outgoing electron are gener-ated. In oxygen-group atoms (Chen and Robicheaux,1994) and in Ti (Miecznik and Greene, 1996), higher-norbitals were obtained by constructing a local screenedpotential Vscr(r) using the Hartree-Fock wave functionsof both the inner shells and the valence s and p orbitals.The explicit form of Vscr(r) is

Vscr~r !52Z

r1(

iwiE

0

` 1r.

Pi2~r8!dr8, (3.43)

where r. is the larger of r and r8, wi is the number ofelectrons in subshell i described by the radial functionPi, and the summation runs over all subshells of the ionground state. High-n orbitals are orthogonalized to thelow-n input orbitals.As for two-valence-electron systems, open-type orbit-

als (which have nonzero value on the R-matrix surface)and closed-type orbitals (which are zero on the bound-ary) are computed. Closed-type orbitals are automati-cally orthogonal. These orbitals are next used to con-struct open-type functions (where only one of theelectron orbitals of the many-electron basis functions isof open type) and closed-type functions (where all orbit-als are of closed type).The basis functions used for a given open-shell atom

to describe the initial-state and the different final-stateLS symmetries incorporate configuration interaction inboth the (Ne-1) electron ionic states and the full Newave functions. (The notation Ne is used for the totalnumber of electrons in order to avoid confusion with thenumber of open and weakly closed fragmentation chan-nels, referred to as N throughout this paper.) Basis func-tions of close-coupling type are constructed by attachingan l wave of appropriate symmetry onto each compo-nent of the target states. Part of these functions de-scribes relaxation of the orbitals due to the addition ofthe extra electron. As in two-valence electron systems,the open or weakly closed channels are described byclosed-type and open-type functions. Multielectron ma-trix elements are considerably easier to evaluate whenall orbitals are orthogonal to each other. Moreover, nu-



merical difficulties could occur in Gram-Schmidt or-thogonalization of two orbitals with each other and withthe closed-type orbitals. Thus, only one open-type or-bital, orthogonalized to the closed-type orbitals, is usedfor each l value.Correlation-type basis functions related to the close-

coupling-type basis as well as various other correlation-type functions are also included. Such functions areneeded, in particular, to represent the dipole polariza-tion of the ionic states. For example, for the openp-shell atoms, correlation functions of the type m0p

q12

were introduced for describing the m0s2m0p

q groundstates (with q<4) and correlation functions of the typem0sm0p

q11 for the final-state symmetries. No generalrule has been identified to choose the correlation-typefunctions in open-shell systems. Some calculations inopen p-shell atoms included only a restricted number ofcorrelation-type functions. So, for example, as stated byRobicheaux and Greene (1992), these functions seem toplay a smaller role in the halogen dynamics than they doin the dynamics of the alkaline earths, where a largenumber of strongly closed channels were found neces-sary to obtain converged results. In contrast, as shownlater in Sec. V, it was found to be important for theconvergence of Sc quantum defects (Robicheaux andGreene, 1993c) that numerous correlation-type func-tions and strongly closed channels be included. As inalkaline earths, all basis functions used to describe thecorrelation-type basis functions, as well as the initialstates, are of closed-type.A final remark concerns the choice of ionic orbitals to

describe basis functions. These basis functions do notnecessarily give the best convergence for the initial state;however, they tend to give better agreement betweenthe length and velocity gauge cross sections than basisfunctions that are chosen to give the best convergencefor the initial level.The choice of the basis functions adapted to study

each type of open-shell atoms that was investigated us-ing the eigenchannel R-matrix approach will be furtherdocumented in Sec. V.

E. Streamlined solution of the generalized eigensystem

As discussed in the previous sections, eigenchannelR-matrix calculations may involve large sets of n two- ormany-electron basis functions. For such large basis sets,the numerical solution of the generalized eigensystem[Eq. (3.7)] at many energies can become very time con-suming. One way to bypass this difficulty and improvethe efficiency of repeated R-matrix calculations at manydifferent energies is to use Robicheaux’s Wigner-Eisenbud-type (Robicheaux, 1991) reformulation of theeigenchannel R-matrix method, which was discussedabove. For instance, this formulation was used by Chenand Robicheaux (1994) to study chalcogens. This ap-proach lacks the flexibility to change the number ofweakly closed channels as the energy is varied. Specifi-cally, if one decides to add one more weakly closedchannel when the energy increases above a certain

value, it becomes necessary to enlarge the matrixH5H1L to include the relevant open-type basis func-tions, and then rediagonalize H . For all energies havingthis fixed number (N) of open and weakly closed chan-nels, the R matrix is then given analytically by Eq. (3.20)without requiring the numerical solution of anotherlarge matrix equation.An alternative ‘‘streamlined’’ reformulation (Greene

and Kim, 1988) casts the generalized eigenvalue equa-tion of the eigenchannel R-matrix method into a formthat can be rapidly solved at many energies, and whichretains the flexibility to change the number of open orweakly closed channels as often as desired, without re-quiring the solution of a new large (n3n) matrix equa-tion each time. The streamlined formulation relies onthe fact that the size of the open-type basis set (typicallyof order N to 3N) is always much smaller than theclosed-type basis set. For a calculation involving N openand weakly closed channels, the number N of nontrivialsolutions of the generalized eigensystem [Eq. (3.7)] issmaller than n by typically one or two orders of magni-tude.The streamlined reformulation partitions the basis

functions yk into no open-type (o) and nc closed-type(c) subsets, depending on whether each basis function isnonzero on S or zero on S , respectively. With this par-titioning, the matrices G and L in Eqs. (3.9) and (3.10)have the structure:

G5S Gcc Gco

Goc GooD (3.44)

and

L5S 0 0

0 LooD . (3.45)

This partitioning allows us to write Eq. (3.7) as twocoupled matrix equations,

GccCW c1GcoCW o50, (3.46a)

and

GocCW c1GooCW o5LooCW ob . (3.46b)

Using Eq. (3.46a) to eliminate CW c from Eq. (3.46b), theequation determining the eigenvalue b is now of muchsmaller dimension no3no ,

VCW o5LooCW ob , (3.47)

where

V5Goo2Goc~Gcc!21Gco . (3.48)

The closed components CW c of the eigenvector are de-duced from the open components CW o by

CW c52~Gcc!21GcoCW o . (3.49)

Here we have used the fact that the Bloch matrix Lccvanishes, and have assumed that the closed-closed por-tion of the overlap matrix O , namely Occ , is the unitmatrix. (This can be accomplished, even for a nonor-thogonal basis set, by orthogonalization of the closed-type basis functions.) The matrix Loo reduces to N sepa-



rable blocks of nonzero elements, each of these blocksbeing of rank 1. This guarantees that the number ofeigensolutions of Eq. (3.47) is N .Inversion of the matrix Gcc at many energies E is ef-

ficiently accomplished by first transforming the closedportion of the basis set into the energy-independent rep-resentation in which Hcc is diagonal, with eigenvaluesEl and orthonormal eigenvectors Xkl . The matrix V isthen explicitly

Vkk852~EOookk82Hoo

kk82Lookk8!22(

l

3~EOoc8

kl2Hoc8

kl2Loc8

kl!~EOc8o

lk82Hc8olk82Lc8o

lk8!

E2El.

(3.50)

In Eq. (3.50), the notation c8 implies that the closedportion of the two-electron basis set is now in the trans-

formed representation, i.e., Ooc8kl

5(k8Oockk8Xk8l ,

Oc8olk

5(k8Oc8ok8kXk8l , with the same transformation be-

ing applied also to the matrices H and L . The semiana-lytic energy dependence of all matrices in Eq. (3.50),combined with the much smaller number of open-typebasis functions than closed-type functions, now improvesthe speed of solving Eq. (3.47) on a fine energy meshdramatically compared to the original Eq. (3.7).To summarize, the eigenchannel R-matrix computa-

tional scheme in two-electron atoms consists of the fol-lowing steps: (1) Choice of the reaction-volume size andof the fragmentation channels. (2) Choice of the two-electron basis set. (3) Numerical calculation of the one-electron orbitals. (4) Calculation of the matrix elementsof O , H , L , and L . (5) Diagonalization of the Hamil-tonian Hcc . (6) Solution of the streamlined form of thegeneralized eigensystem (3.47) on a chosen energymesh.The eigenchannel R-matrix computation scheme in

open-shell atoms proceeds along the same lines as thatin two-electron atoms. However, major complicationsoccur, mainly in steps (2)–(4). As was emphasized inSec. D above, most difficulties occur in the constructionof basis functions used for the initial- and final-stateatomic wave functions. Effort goes into determiningmultielectron target functions and one-electron naturalorbitals. The second major complication in R-matrix cal-culations of open-shell atoms compared to those in two-valence-electron systems concerns the computation ofmatrix elements. For two-electron atoms when the two-electron basis functions are chosen to be eigenfunctionsof the independent-particle Hamiltonian H05H21/r12 , the evaluation of matrix elements is especiallystraightforward and standard. The calculation of multi-electron matrix elements is much more complicated andan efficient computer code has been developed to evalu-ate the angular parts of multielectron one- and two-particle operators (Robicheaux and Greene, 1992).The evaluation of the 1/rij matrix elements and the

diagonalization of the Hamiltonian Hcc are the most

time consuming parts of the computation. However,these need to be done only once, after which the energy-dependent R matrix is determined semianalytically at allenergies: this is the major advantage of both the eigen-channel and Wigner-Eisenbud formulations ofR-matrix theory. A second advantage of the R-matrixmethod is the faster convergence achieved by limitingthe variational calculation to as small a portion of con-figuration space as is possible.

F. Short-range reaction matrix K and parameters ofmultichannel quantum-defect theory

After the N eigenvalues bb and corresponding Nlinearly-independent eigenvectors Ckb have been ob-tained by solving Eqs. (3.47) and (3.49) at a given energyE , the solutions are matched to a linear combination ofCoulomb functions in each open or weakly closed chan-nel. Recalling Sec. II, we see that two alternative short-range reaction matrices can be determined. Matching toenergy-normalized Coulomb functions (fe2l2

, ge2l2) gives

the short-range reaction matrix K(E) [Eq. (2.34)] or theequivalent smooth, short-range scattering matrix S(E).Matching the R-matrix eigenstates to analytic Coulombfunctions (fe2l2

0 , ge2l20 ) gives a K0 short-range reaction

matrix. We restrict the following discussion to the K ma-trix, keeping in mind that the K0 short-range reactionmatrix can also be used to solve the equations of multi-channel quantum-defect theory.Because the boundary conditions at infinity, which

quantize the Rydberg energies in the closed channels,have not yet been applied, the short-range scatteringmatrix S(E), or the equivalent eigenchannel MQDT pa-rameters ma and Uia [Eq. (2.35)], vary relativelysmoothly as functions of the energy E . The full energydependence of S(E) is obtained by solving the general-ized eigensystem of Eq. (3.47) on as fine an energy meshas desired. When the spectrum is desired on an ex-tremely dense energy mesh, for instance in an energyrange that includes very high-lying Rydberg levels, it ismore efficient to first solve Eq. (3.47) on a relativelycoarse grid of selected energies. Interpolation of thesesmooth matrices then permits the equations of multi-channel quantum-defect theory [Eqs. (2.44)–(2.46)] tobe solved on an arbitrarily fine energy mesh to predictthe relevant observables.Calculation of the photoionization cross section from

a low-lying level requires the additional determinationof dipole matrix elements. The wave function of the ini-tial state that will be exposed to incident radiation canbe obtained by diagonalization of the Hamiltonian ma-trix constructed using closed-type two-electron basisfunctions only, provided the box is large enough to en-close its full probability density. The dipole matrix ele-ments dS [Eq. (2.57)] that connect the initial and finalstates can then be evaluated by restricting the integra-tion volume to the interior of the R-matrix box. Like-wise, the short-range matrix S , the short-range dipolematrix elements dS vary smoothly as functions of the



energy. Photoionization of an excited state, especially ofa Rydberg state [as in isolated-core excitation experi-ments (Cooke et al . , 1978)], requires the inclusion oflonger-range contributions to the dipole matrix ele-ments. In many cases this can be performed analytically,as is discussed in Sec. IV below.As illustrated later in Secs. IV and V, in energy do-

mains involving sharp resonances, reliable comparisonbetween theory and experiment should use a theoreticalspectrum convolved with the experimental linewidth. Toobtain a convolved cross section that can be comparedto experiment, the cross section needs to be calculatedon an extremely fine mesh before numerically convolv-ing it. This is a wasteful procedure since the short-rangeMQDT parameters do not vary much with energy. Ro-bicheaux (1993) developed a procedure to obtain a con-volved total cross section directly from the smooth scat-tering parameters that avoids explicitly performingnumerical convolution. The approximate formula de-rived for photoabsorption by neutral species is accurateto the extent that the short-range scattering parametersdo not vary over the convolution width. This precon-volved technique was used in some recent works (Chenand Robicheaux, 1994; Miecznik and Greene, 1996).Figure 7 illustrates the energy dependence of the

MQDT parameters and the implication of this depen-dence on the final results. Figure 7 compares the eigen-quantum defects ma and photoabsorption spectra of Baobtained from R-matrix calculations using r0515 a.u. or20 a.u. Figures 7(a) and 7(b) display the ma obtained forthe 1Po symmetry as functions of the energy relative tothe double-ionization limit, for a wide energy rangefrom below the 6s threshold to above the 6p thresholds.The ma vary smoothly with the energy whatever thevalue of r0, but comparison of curve (a) obtained withr0515 a.u. with curve (b) corresponding to r0520 a.u.shows that the energy dependence of the ma increaseswith r0. Such r0 dependence of the ma was encounteredin all cases we considered, but disappeared in the finalresults. Photoionization cross sections for the photoab-sorption spectrum between the 5d5/2 and 6p1/2 thresh-olds calculated, in the velocity form, by combiningLS-coupled R-matrix calculations done for the 1Po,3Po, and 3Do symmetries with the jj-LS frame transfor-mation are displayed in Fig. 7(c). The differences be-tween the full line (r0515 a.u.) and the dashed line(r0520 a.u.) are not significant, being of the same orderof magnitude as the differences between length and ve-locity results for a given r0. Note that the calculationswere performed with small two-electron basis sets, re-sulting in poor convergence of the variational calcula-tion.In the nearly ab initio theoretical multichannel spec-

troscopy, three different kinds of MQDT calculationshave been carried out using the dynamical quantities Sand dS obtained by eigenchannel R-matrix calculations.

1. R-matrix calculations in LS coupling

Calculations in LS coupling (denoted as LS calcula-tions) neglect spin-orbit effects everywhere. These cal-

culations start from the LS-coupled matrices SLS anddLSS derived from the nonrelativistic Hamiltonian [Eq.

(3.22)]. The MQDT fragmentation channels for this typeof calculation are the LS-coupled channels i5$n1l1l2%introduced in Sec. III.B.2 above. We will see in Sec. IVthat the nonrelativistic LS method is adequate for de-scribing low-resolution photoabsorption spectra that areincapable of resolving fine-structure splittings.

2. R-matrix calculations in LS coupling combined with thejj-LS frame transformation

Calculations start from the nonrelativisticLS-coupled quantities, but then apply the jj-LS frametransformation (see Sec. II.E) and perform the MQDT

FIG. 7. Comparison of eigenquantum defects ma and photo-absorption spectra obtained in Ba with LS-coupled eigenchan-nel R-matrix calculations performed using different reaction-volume sizes and combined with the jj-LS frametransformation: (a) eigenquantum defects ma for the

1P1o sym-

metry obtained with r0 = 15 a.u.; (b) eigenquantum defectsma for the 1P1

o symmetry obtained with r0 = 20 a.u.; (c) pho-toabsorption spectrum between the 5d5/2 and 6p1/2 ionizationlimits calculated using r0 = 15 a.u. (solid line) and r0 = 20 a.u.(dashed line). The vertical bars in (a) and (b) correspond tothe 6s , 5d , and 6p ionization limits. The energies are referredto the double-ionization limit.



calculation using experimental fine-structure ionizationthresholds. These calculations are denoted as jj-LS (orframe-transformation) calculations. The MQDT frag-mentations channels for this type of calculation are thejj-coupled channels i5$n1l1j1l2j2% introduced in Sec.III.C above.The approximations embodied in the jj-LS frame

transformation are accurate as long as K(E) does notvary appreciably over an energy range comparable tothe fine-structure splitting of the A1 core levels. As isvisible on Fig. 7, at the low-energy end in each channelthe MQDT parameters acquire a strong energy depen-dence. The use of experimental thresholds in stronglyclosed channels can adversely affect the MQDT calcula-tion of observables, sometimes causing spurious reso-nances to appear or physical resonances to disappear.These instabilities are due to the exponential growth ofCoulomb functions associated with the strongly closedchannels. The use of the frame transformation in thisenergy range requires extra caution. It was found to becrucial that spin-orbit-averaged theoretical energies beintroduced instead of experimental ones for the thresh-old energies associated with the strongly closed channelstreated as weakly closed and also, as documented inSecs. IV and V, with some particular weakly closedchannels supporting an isolated level at the low end ofthe channel. To minimize these difficulties it is also ad-visable to include only the weakly closed channels in-volved in the energy range under consideration. Asshown later in Secs. IV and V, the jj-LS method canquantitatively describe a class of nonperturbative spin-orbit effects even in some atoms as heavy as barium oras complicated as scandium and titanium.

3. R-matrix calculations in jj coupling

We denote as jj calculations any jj-coupled R-matrixcalculations based on the Hamiltonian that includesspin-orbit interactions both inside and outside theR-matrix box. At the present time, calculations using thejj-coupled eigenchannel R-matrix approach deal onlywith two-electron systems, and the discussion belowconcerns only alkaline earths.The jj method is needed when the ionic fine-structure

splittings become as large as 1 eV or more. The jjmethod is also recommended for calculations requiring alarge reaction volume. Indeed, the energy dependenceof MQDT parameters rapidly increases with r0, and re-sults obtained using the jj-LS method can drasticallydepend on the choice of either the experimental or thetheoretical ionization thresholds for the closed channels.Even using the jj method, some results, such as those

obtained for 6d2 and 7p2 levels of Ba (Lecomte et al . ,1994), depend slightly on the choice of the thresholdenergies although the experimental threshold energiesused in the MQDT calculations are very close to thetheoretical energies introduced in the matching proce-dure. An adaptation of the streamlined eigenchannelmethod recently developed to circumvent these difficul-ties occurring at the bottom end of Rydberg series,

where MQDT parameters acquire the strongest energydependence, is presented in Sec. III.H below.Finally, it is worth comparing the approximations

used in eigenchannel R-matrix calculations to take intoaccount relativistic effects to those used in Wigner-Eisenbud-type R-matrix calculations. To our knowl-edge, in the Wigner-Eisenbud R-matrix formalism, theframe-transformation treatment has been used forstudying electron-atom scattering processes only. So, forinstance, starting from LS-coupled reaction matrices ob-tained by LS-coupled R-matrix calculations, Saraph(1978) used the jj-LS frame transformation for comput-ing collision strengths for transitions between fine-structure levels. Another way to include relativisticterms is to reduce the fully relativistic Dirac plus Breitequations to the Breit-Pauli form (Condon and Shortley,1935) and to retain either all terms of the Breit-PauliHamiltonian or only the one-particle operators. This ap-proach was developed by Scott and Burke (1980) andwas first applied to the calculation of collision strengthsin Be-like Fe XXIII. As further documented in Secs. IVand V, the same approach has also been employed byBartschat and co-workers to investigate Sr, Ba, Zn, andHg photoionization (Bartschat and Scott, 1985a, 1985b;Bartschat et al . , 1986, 1991; Bartschat, 1987; Mende etal . , 1995). The approximations embodied in the calcula-tions that include only the one-particle terms of theBreit-Pauli Hamiltonian are comparable to those in-volved in the jj-coupled eigenchannel R-matrix calcula-tions. To go beyond the approximations underlying thejj-coupled eigenchannel R-matrix method or the Breit-Pauli R-matrix method of Scott and Burke (1980),R-matrix calculations should be based on the DiracHamiltonian as was performed early by Chang (1975,1977). This point will be further documented in Sec. VI.

G. Reaction volume size, basis functions, andfragmentation channels

This section focuses on practical considerations on theeigenchannel R-matrix technologies. Most of the calcu-lations carried out using the eigenchannel R-matrix ap-proach have dealt with alkaline earths. Moreover, thetechniques used in open-shell atoms are dependent onthe atom being studied. Accordingly, the following sec-tion is restricted to two-electron systems.Prior to each eigenchannel R-matrix calculation, it is

necessary to decide which channels should be treated asopen, weakly closed, or strongly closed in the studiedenergy range. This choice is important in order to selectthe reaction-volume boundary r0 and the two-electronfunctions to include in the variational calculation. How-ever, the choice is not unique, as will be stressed later.First, we document the choice of the reaction volume

size. As a first criterion, the reaction volume V shouldbe large enough to include the charge distributions ofthe ionic states A1 n1l1 involved in the open or weaklyclosed fragmentation channels. The majority of theR-matrix calculations performed in recent years havedealt with energies lower than the lowest A1 m0p



threshold (m0= 2, 3, 4, 5, 6 for Be, Mg, Ca, Sr, Ba, andRa, respectively) and n1l1n2l2 states corresponding toan orbital momentum l2 for the outer electron such asl2<4. Calculations in the lighter elements Be and Mghave typically used r0;9–12 a.u. Those for the heavierelements Ca to Ra have used r0;15–22 a.u. owing to thesmaller binding energies of the ion (Greene and Aymar,1991). A calculation of the Ca photoabsorption spec-trum from the 4p threshold up to nearly the 6s thresh-old (Kim and Greene, 1988) and recent calculations ofseveral autoionizing Rydberg series of Ba below the7p threshold (Luc-Koenig and Aymar, 1992; Aymar etal . , 1994; Luc-Koenig, Aymar, and Lecomte, 1994; Luc-Koenig, Lecomte, and Aymar, 1994; Aymar and Luc-Koenig, 1995; Luc-Koenig et al . , 1995; van Leeuwen etal . , 1995, 1996) use larger r0 values in the range of 30–50a.u. The largest box sizes used in calculations of this typeare around r0;100–120 a.u.; they were used in the in-vestigations of Ba near the 8s threshold, of Sr near the6f , 6g , 6h thresholds (Wood and Greene, 1994) and alsoin the study of H2 and Li2 up to the n56 thresholds(Pan et al., 1994).It is worth noting that the size of the reaction volume

does not depend solely on the extent of the ionic levelsthat must be enclosed in the reaction volume. It is alsogoverned by the value of the orbital momentum l2 cor-responding to the Rydberg or continuum electron. Forcalculations that neglect multipole interactions beyondthe reaction volume V , a larger box size r0 helps to de-scribe these interactions accurately within V . These mul-tipole effects play an especially important role in deter-mining the small quantum defects for large l2 values.Thus, for example, g-wave quantum defects are betterdescribed in Mg using r0520 a.u. than using 12 a.u.(Lindsay, Dai, et al . , 1992). In Sr also, a better descrip-tion of the 4dng levels was found using r0535 a.u. in-stead of r0520 a.u. (Goutis et al . , 1992). Recently, fromthe observation of 7sni Rydberg series of Ba (Camus,Mahon, and Pruvost, 1993), it has been shown that avery good description of the measured quantum defectsfor 7sni Rydberg series is attained using r0;50–60 a.u.,while poor results are obtained using r0530 a.u. (Aymaret al . , 1994). The highest partial waves included in suchcalculations are up to l2;10 in the Ba calculation ofWood and Greene (1994). These authors performedR-matrix calculations in jK coupling with an R-matrixbox of 100 a.u.Recently, Wood and Greene (1994) extended the

eigenchannel R-matrix approach to include effects oflong-range multipole interactions beyond the R-matrixreaction volume. More precisely, the Cb eigensolutionsand their derivatives have been propagated from r0 to alarger radius r08 by solving the close-coupling equationswithout exchange for r>r0. Higher l2 values can bestudied without increasing the size of the reaction vol-ume. l2 values up to l2512 have been investigated in Sr,by propagating the solutions from r05100 a.u. up tor08;250 a.u. Similarly, Pan et al . (1994) successfully ana-lyzed highly doubly excited states converging onto thenearly degenerate thresholds n56 in H2 and Li2.

Our aim now is to give details on the choice of thevariational basis sets and to show that this choice is de-termined by the accuracy desired in the description ofthe physical observables. We concentrate on two simpleexamples.The first example concerns an investigation of the Ca

1Po spectrum, which was studied by Greene and Kim(1987) over a range of energy from far below the 4sionization threshold up to the 4p threshold. An accuratedescription of the bound spectrum and of the photoab-sorption spectrum was obtained using small-scaleLS-coupled R-matrix calculations. The five open orweakly closed channels relevant to the autoionizing en-ergy range below the 4p threshold were: 4snp , 3dnp ,3dnf , 4pns , and 4pnd . The calculation used anR-matrix boundary of r0518 a.u., which easily enclosesthe charge distribution of the Ca1 4p state within thereaction volume. Table II lists the closed-type basis set,which consists of 88 n1l1n2l2 functions. This set wascomplemented by 15 open-type basis functions associ-ated with the open or weakly closed channels, whichresults in a total of 103 two-electron basis functions. Foreach open or weakly closed channel, three open-typebasis functions were included and another five or sixclosed-type basis functions. A total of 28 closed-type ba-sis functions were used for these MQDT fragmentationchannels, while another 60 closed-type functions wereused to describe the strongly closed channels convergingonto the Ca1 ns , np , nd , and nf thresholds lying higherthan the Ca1 4p threshold. At energies below the 4sthreshold, the 3dnf , 4pns , and 4pnd channels arestrongly closed, meaning that the energy is still belowthe first states in those channels. Accordingly, theR-matrix basis set used in this energy range did not con-tain any open-type functions for these channels. How-ever, the same set of closed-type basis functions wasused, which results in 94 two-electron basis functions.The size of the basis set to be used strongly depends

on the degree of convergence desired. Tests showed thatthe qualitative shape of the photoabsorption spectrumof Ca between the 4s and 3d thresholds could be ob-tained using only thirteen (three in the open channeland two in the weakly closed channels) basis functions(instead of 103) and r0512 a.u. (instead of 18 a.u.).However, the quality of the variational wave functions,as judged by the agreement between length and velocityresults, is greatly improved by including a large set ofstrongly closed channels. Unfortunately, there is no gen-eral criterion for choosing the basis set before doing acalculation, whereas, after having done a calculationwith a large basis, it is possible to eliminate basis func-

TABLE II. Closed-type basis set of nlml8 functions used forthe 1Po symmetry of Ca.

nsmp 4<n<8 4<m<9 +9s4p

npmd 4<n<8 3<m<8 +9p3d

ndmf 3>n<7 4<m<8 +3d9f



tions yk whose contributions are clearly negligible frominspection of the eigenvectors Ckb corresponding to thedifferent eigenvalues bb in Eq. (3.7) or Eqs. (3.47) and(3.49).The second example corresponds to the even-parity

J=0 spectra of Sr and Ba investigated below the m0pthreshold. LS-coupled R-matrix calculations were com-bined with the jj-LS frame transformation (Sec. II.E).The Sr R-matrix calculations (Kompitsas et al . , 1991)for the 1Se and 3Pe symmetries used a box size ofr0520 a.u. The 1Se calculation included three open orweakly closed channels: m0sns , (m021)dnd , andm0pnp . Table III lists the configurations included in theclosed-type basis set. It consists of 237 n1l1n2l2 configu-rations with l2<6. The additional 15 open-type basisfunctions associated with the open and weakly closedchannels give a total of 246 two-electron basis functions.Among them, only 33 are associated with the open orweakly closed channels (eight closed-type functions andthree open-type functions per channel).To achieve convergence for the energy of the strongly

correlated Ca 4p2 1S0 (Aymar and Telmini, 1991; Assi-mopoulos et al . , 1994), Sr 5p2 1S0 (Kompitsas et al . ,1991), and Ba 6p2 1S0 levels (Greene and Theodosiou,1990; Greene and Aymar, 1991; Wood et al . , 1993), itwas found essential to include a large number of closed-type n1l1n2l2 functions involving large orbital momental1 ,l2>3. This large number of functions represents highdoubly-excited-type configurations that would fre-quently be neglected (Bartschat and Greene, 1993) butplay an unexpectedly large role for these resonances un-usually sensitive to the chosen model.More generally, the LS-coupled R-matrix calculations

performed at relatively low energies below the m0pthreshold typically use 100 to 400 basis functions to de-scribe two to five open and weakly closed fragmentationchannels. When the R matrix is set up in jj coupling, thenumber of basis functions is larger, typically by about afactor of three (or four), accounting for the three (orfour) different LS symmetries generally involved in asingle J symmetry. The most recent calculations per-formed in jj coupling at higher energies (Aymar et al.,1994; Luc-Koenig, Aymar, and Lecomte, 1994; Luc-Koenig, Lecomte, and Aymar, 1994; Aymar and Luc-Koenig, 1995; Luc-Koenig et al . , 1995; van Leeuwen etal . , 1995, 1996) involve 30 to 50 fragmentation channelswhich are described by 1000 to 2500 basis functions.Some convergence-test calculations involve up to 5000basis functions. The convergence of the variational cal-culation with respect to the basis-set size is dramatically

slowed down if a value of r0 is used that is larger thannecessary. Indeed, the number of basis functions prob-ably increases in proportion to r0

2. As a rule of thumb, itis our experience that for r0< 15 a.u., good convergenceis achieved using only six closed-type basis functions perchannel. This number should be increased to eight forr0; 20 a.u. and to 15–18 for r0 ; 50 a.u.The choice of open channels to be included in the

calculation is easy because, for given symmetry and en-ergy range, there exists a finite set of open channels.R-matrix calculations of autoionizing levels located be-low the A1 (m011)p threshold explicitly introduced allrelevant open channels. In contrast, the number ofclosed channels is infinite. The choice of which weaklyclosed and strongly closed channels to include in theR-matrix calculation is not unique. In particular, it de-pends on which short-range reaction matrix K(E) orK0(E) is used in the MQDT calculations. R-matrix cal-culations performed to determine a K(E) matrix shouldtreat the channel i5$n1l1l2% as strongly closed when thephotoelectron energy is smaller than 21/2l2

2. This re-striction can be bypassed by using a K0 matrix. How-ever, as shown below, the use of a K0 matrix complicatesthe calculations.Although no fully general rule has been found to

choose the channels to be treated as weakly closed orstrongly closed, from our experimentation we recom-mend:(1) The use of matrices K rather than matrices K0.

For a given energy range, the matrix K refers to frag-mentation channels i in which the photoelectron energyis larger than 21/2l2

2. This leads to treatment of a spe-cific channel i5$n1l1l2% as strongly closed when thephotoelectron energy is smaller than 21/2l2

2. Thus,MQDT calculations over a large energy range requirethe determination of several K matrices involving differ-ent sets of fragmentation channels. It is also crucial tointroduce a large number of strongly closed channels toget converged results. MQDT calculations with a fixednumber of fragmentations channels could be conductedover a large energy range with a single K0 matrix that isable to treat all fragmentation channels throughout theenergy range independently of how the photoelectronenergy compares to 21/2l2

2. Such treatment does not af-fect the results, but it complicates the calculations byincreasing the number of fragmentation channels. Fur-thermore, in the low-energy range this causes avoidedcrossings and strong energy dependences of the ma andalso near degeneracies among some of the ma .(2) Inclusion of all weakly closed channels relevant to

the studied energy range and symmetry. Failure to in-clude all weakly closed channels can distort the final re-sults, especially when the neglected channels interactstrongly with weakly closed channels included in the cal-culation. This can induce rapid increase by one unit ofsome eigenquantum defects ma , in particular when thelowest resonances in the weakly closed channel treatedas strongly closed lie in the studied energy range. Thismay also cause adverse effects on the results. In particu-lar, the resonances in the omitted channels will lie muchtoo high in energy unless these resonances fit inside the

TABLE III. Closed-type basis set of nlml functions used forthe 1Se symmetry in Sr and Ba.

For 0<l<5 nl<n<nl17 nl11<m<nl17For l56 nl<n<nl15 nl11<m<nl15

With l50,1 ns5np5m0

l52 nd5m021l>3 nl5l11



R-matrix reaction volume (Kim and Greene, 1988;Lecomte et al., 1994, 1995). However, our experiencehas shown that treating weakly closed channels that donot interact significantly with the channels under studyas strongly closed makes no differences in many cases.Closed channels n1l1l2 with relatively high l2 values canoften be treated as strongly closed. For example, resultson 4fnf levels obtained by treating the 4fnh channels asstrongly closed (Luc-Koenig, Aymar, and Lecomte,1994) agree perfectly with those obtained by treating the4fnh channels as weakly closed (van Leeuwen et al . ,1995).(3) Caution when using a large reaction volume. Dif-

ficulties related to the exponential growth of closed-channel Coulomb solutions (for which asymptoticboundary conditions have been postponed) are ampli-fied. Moreover, it was found that when the lowest reso-nances belonging to a weakly closed channel fit insidethe R-matrix box then this channel becomes uncoupledfrom the others and its eigenquantum defect ma de-creases rapidly with energy (Kim and Greene, 1988;Lecomte et al . , 1995). An adaptation of the eigenchan-nel method developed recently to circumvent the diffi-culties occurring at the bottom end of Rydberg series,where MQDT parameters acquire the strongest energydependence, is presented in Sec. III.H below.

H. Alternative streamlined solutions of the generalizedeigensystem

An adaptation of the streamlined eigenchannelmethod was recently developed to circumvent the diffi-culties encountered in describing doubly excited statesof alkaline earths that are located at the bottom end ofRydberg series. The method developed for studying6d2 and 7p2 autoionizing levels of Ba (Lecomte et al . ,1994) reduces to identifying some levels enclosed in thereaction volume with bound states and treating their in-teractions with the open continua. The method is closein spirit to the Fano’s (1961) configuration-interactionmethod, which dealt with the same situation.First, consider the treatment of the 6d2 levels, which

are located above the 6s , 5d , and 6p thresholds, andwhich do not interact with Rydberg series converging tothresholds higher than 6p . Instead of viewing the 6d2

levels as the lowest members of Rydberg series converg-ing to 6dj thresholds and treating their interaction withthe open channels using standard MQDT techniques, areaction volume (r0;40–50 a.u.) is chosen to be largeenough to enclose the 6d2 levels. The 6dnd channelsare treated as strongly closed channels in the energyrange enclosing the 6d2 resonances. The eigenchannelR-matrix method is employed to construct a matrix Rrestricted to open channels from which a scattering ma-trix S referring to open channels only is deduced [Eqs.(2.34) and (2.43)].As in the standard streamlined eigenchannel

R-matrix method, the two-electron basis functions arepartitioned into two distinct sets. However, in contrastwith this former approach (Sec. III.E), the partition does

not depend on the type (open or closed) of the basisfunctions but on the nature of the channels they de-scribe:(i) Open channels: each of the N0 open channels con-

verging to the 6s , 5d , or 6p ionization limits is describednot only by some open-type functions but also by severalclosed-type functions. The corresponding set of two-electron functions is denoted $o%.(ii) Strongly closed channels: a great number of basis

functions describes strongly closed channels convergingonto thresholds located higher than the 6p threshold;among them there are the 6dnd channels. The corre-sponding basis functions, constructed from closed-typefunctions only, form the set denoted $ c%.Two key points follow from the new partition:(1) As in the streamlined eigenchannel R-matrix

method, the eigensystem [Eq. (3.7)] is transformed bypartitioning the matrices G , L , and O into submatricesreferring to the subsets $o% and $ c% and then trans-formed again into a representation in which the closedportion of the Hamiltonian Hcc is diagonalized using anorthogonal eigenvector matrix W . Among the eigen-states of Hcc , those corresponding to the lowest eigen-values can be identified with physical low-lying doublyexcited states such as 6d2, 7p2, 7s5g , . . . enclosedwithin the reaction volume. Corresponding states (de-noted as Fr) are grouped in a first set denoted $r%. Asecond set $p% groups the remaining eigenstates, whichcannot be identified with physical states; these eigen-states describe polarization effects. The division of theset $ c% into the sets $r% and $p%, which depends on thestudied energy range, is somewhat arbitrary, but the fi-nal results are completely independent of this division.With this new partition, some submatrices that do not

have zero values in the standard streamlined eigenchan-nel R-matrix formulation now vanish. More precisely,one has

Loc50, Ooc50, and Oco50. (3.51a)

Moreover, Goc [Eqs. (3.9) and (3.51a)] reduces to

G o c522Hoc . (3.51b)

As detailed by Lecomte et al . (1994), the generalizedeigensystem can be transformed into the form

G8xW o[@G oo8 2G or812 ~E2Er!

21Gro8 #xW o5L ooxW ob ,(3.52a)

xW r52 12 ~E2Er!

21Gro8 xW o , (3.52b)

where the matrix xW r represents the amplitudes of thefunction Fr , associated with the eigenvalues Er , in theNo eigensolutions Cb . In Eq. (3.52a), the matrix

G oo8 5G oo2G o cWcp12 ~E2Ep!21Wpc

t G c o (3.53a)

includes the contribution of the eigenstates of Hcc de-scribing polarization effects. The matrix

G or8 5G o cWcr (3.53b)



couples the physical eigenstates Fr to the set $o% de-scribing the open channels.(2) A second key point of the new partition and of the

subsequent transformation of Eq. (3.7) into Eq. (3.52) isthat, from Eqs. (3.9) and (3.51b), the matrix Gro8 is suchthat each of its elements is proportional to the matrixelement of 1/r12 between a particular state Fr and a par-ticular function of the set $o% that describes continua.As detailed by Lecomte et al . (1994), the new parti-

tion permits an easy identification of doubly excitedstates within the R-matrix box. Moreover, the construc-tion of the matrix R associated with the solutions of theeigensystem [Eq. (3.52)] gives a scattering matrix S in aform well suited to resonance analysis. This matrix isexpressed as the sum of a nonresonant part (S(0)) and aresonant part as

Sij5Sij~0 !22ip (

aa8P$r%

Tiat SE2Er2Drr1

i

2grrD

aa8

21

Ta8j .

(3.54)

The background matrix S(0) varies very slowly with en-ergy. This matrix describes the continua when the dou-bly excited levels Fr and the open channels are notcoupled, i.e., when Gro8 50; this matrix was obtainedfrom a separate R-matrix calculation involving a basisexcluding the doubly excited levels contained in the set$r%. In the resonant part, Drr and grr are shift and widthmatrices, respectively, and the matrix T represents the1/r12 interaction between the doubly excited states Fr(set $r%) included in the R-matrix box and the nonreso-nant continua.We consider now the treatment of 7p2 levels or

higher-lying nl2 levels, such as 4f2 levels, that are mixedwith neighboring levels pertaining to Rydberg seriesconverging to high-lying thresholds. The approach de-veloped to treat the 6d2 levels, where no weakly closedchannel are introduced, is no longer valid to describethese levels. An extension of the approach discussedabove allows the mixing between the 7p2 or 4f2 levelsand the Rydberg series to be accounted for (Lecomteet al . , 1994, 1995). The 6d2, 7p2, and 4f2 levels are stillassumed to be contained and identified within the reac-tion volume, i.e., the channels converging to the 6d ,7p , and 4f thresholds are still treated as strongly closedchannels. However, some channels treated as stronglyclosed in the previous formulation are now treated asweakly closed in order to describe Rydberg levels thatare not included in the reaction volume. The R-matrixcalculation gives a short-range scattering matrix refer-ring to open and closed channels. The resonances corre-spond either to the doubly excited states Fr enclosedwithin the reaction volume or to levels pertaining to theweakly closed channels. The two types of resonances canbe analyzed simultaneously, as described by Lecomteet al . (1995).

IV. ALKALINE-EARTH ATOMS

This section describes the multichannel spectroscopyof a variety of different atoms and different symmetries.

Our efforts concentrate on the alkaline-earth atomsranging from beryllium to radium, i.e., on two-electronatoms for which the long-range potential experienced bya single escaping electron is Coulombic. Another class oftwo-electron systems will not be treated in detail butonly succinctly mentioned in this review. It concernstwo-electron systems for which the long-range electron-core potential differs from a pure attractive Coulombfield. This second class includes atomic helium, the hy-drogen negative ion, and the alkali negative ions.Our goals for this section are:(1) To show, with examples, how complex spectra

provide us with a window into the nature of channelinteractions. Theoretical multichannel spectroscopy pro-vides a powerful way of analyzing the global aspects ofthe short-range dynamics that are common to the entireenergy-level and autoionizing pattern of a spectrum. Wealso develop practical graphical techniques for openingthis dynamical window.(2) To develop in detail one particularly powerful ap-

proach to the calculation of complex spectra. This ap-proach marries the techniques of multichannel spectros-copy (or multichannel quantum-defect theory) to theeigenchannel R-matrix method. We refer to this generalcomputational scheme as nearly ab initio multichannelspectroscopy; its strength is the capability to calculatethe short-range scattering parameters relating to bothopen and closed channels. Practical considerations ofthe method will be stressed and its current limitationswill be described.(3) To develop a catalog of the types of spectra that

can be observed. The catalog not only illustrates thegreat diversity of photoabsorption spectra, it also veri-fies the general applicability of the nearly ab initio ap-proach to calculate many different observables overbroad energy ranges. Section A deals with bound spec-tra. Sections B and C show how to extract informationon the autoionizing resonances without reference to theexcitation process. Section B is devoted to the energy-level structure of autoionizing spectra, and Sec. C con-cerns the autoionization widths and branching ratios.Sections D and E deal with two kinds of photoionizationspectra corresponding to different excitation schemesstarting from either a low-lying or a Rydberg level. Ac-curate results can be obtained, not only for the mostcommon observables such as energy-level positions andtotal photoionization cross sections, but also for observ-ables more sensitive to channel interactions and spin-orbit effects. Special attention will be given to aniso-tropic observables such as the photoelectron angulardistributions.(4) To demonstrate how multichannel spectroscopy

permits the extraction of regularities and differencesamong the channel interactions for different atoms andsymmetries. These are harder to extract from raw spec-tra for two reasons: (i) multichannel resonant spectraare so complicated in general that it is hard to extractanything simple from the spectrum of one atom to com-pare with that of another atom; (ii) complex spectra de-pend on other details, such as the ionic threshold ener-



gies, which can greatly modify one atom’s spectralappearance from that of a comparable atom even if thechannel-interaction parameters are identical.(5) To assess the validity of assumptions made in

semiempirical quantum-defect studies that fit MQDTparameters to reproduce experimental data. Results ob-tained by combining the R-matrix and MQDT methodsare compared to those provided by semiempirical andother ab initio approaches. These comparisons showthat the calculation of short-range scattering parametersby the eigenchannel R-matrix method greatly enhancesthe predictive power of multichannel spectroscopy. Atthe same time, we show that many semiempirical studiescarried out prior to these nearly ab initio studies wereable to unravel the spectra successfully.Some readers may prefer to skim this section to get a

flavor for the different types of spectra that are possiblefor multichannel systems. Most of the individual subsec-tions in the following are self-contained to encouragethis approach. At the same time, we have attempted tocover a substantial fraction of the work in this area inrecent years, to permit use of this section as an encyclo-pedia of alkaline-earth spectra.

A. Discrete spectra

Figure 8 compares schematic energy diagrams of all ofthe alkaline-earth atoms. Over the energy range consid-ered here, the electrons in the closed shells of the doublycharged ionic core cannot be excited. We treat only two-electron excitations in the range extending from theground state, denoted as m0s

2 (Be, m0 =2, Mg, m0 =3;Ca, m0 =4; Sr, m0 =5; Ba, m0 =6, Ra, m0 =7), to thedouble ionization limit A11. The positions of the lowest

ionization limits and their fine-structure splittings are in-dicated on Fig. 8 for each atom.The energy-level structure of the alkaline-earth atoms

differs qualitatively from the spectrum of a one-electronatom or ion, with the alkaline earths exhibiting fargreater complexity. In particular, Rydberg states of theouter valence electron attached to different states of theinner valence electron can perturb each other strongly,especially when states of the same symmetry becomenearly degenerate in energy. The resulting channel in-teractions produce great visual complexity in the valencephotoabsorption spectrum, even though only two elec-trons participate. A major difference between the‘‘lighter’’ alkaline-earth atoms Be and Mg and the‘‘heavier’’ alkaline earths Ca, Sr, Ba, and Ra is that thefirst excited state of Be1 and Mg1 is the m0p state,whereas it is the (m021)d state for the heavier alkaline-earth ions. The lightest alkaline-earth atoms Be and Mghave comparatively few doubly excited perturbing levelsin the bound-state spectrum below the first ionizationthreshold, whereas the perturbations are numerous inCa, Sr, Ba, and Ra, even at such low energies. Accord-ingly, the following discussion of the bound Rydberg se-ries perturbations focuses on the heavier alkaline-earthatoms.Eigenchannel R-matrix calculations have been carried

out both in LS and jj coupling schemes to analyze theperturbations of bound Rydberg series m0snl by low-lying doubly excited levels located below the first m0sionization limit. Not surprisingly, these perturbationsstrongly affect the energy-level structure and other ob-servables that are even more sensitive to channel mix-ing. Effects of perturbations on energy spectra and onother observables will be discussed in succession.

1. Energy spectra

Figure 9 compares the Lu-Fano plots of the odd-parity J51 bound levels for all of the ‘‘heavier’’alkaline-earth atoms: Ca, Sr, Ba, and Ra. On each Fig.9(a) to 9(d), the solid curve shows the quantum defectscalculated using the eigenchannel R-matrix method,while the points give the quantum defects of the experi-mental levels (Moore, 1949, 1952, 1958; Tomkins andErcoli, 1967; Armstrong et al . , 1979, 1980). As ex-plained in Sec. II.F, the theoretical quantum defects areshown as continuous curves, whereas the quantum de-fect of each experimental level is a discrete point. Thecontinuity of the resulting plot improves our ability tovisualize the full energy dependence of the quantum de-fects. The perturbation of the m0snp

1P Rydberg seriesof Ca @Fig. 9(a)] or Sr @Fig. 9(b)] by the(m021)dm0p

1P doubly excited state was well de-scribed by a two-channel R-matrix calculation in LScoupling (Greene and Kim, 1987; Aymar et al . , 1987).Compared to Ca and Sr, the J51o spectrum of Ba@Fig. 9(c)] displays much greater complexity. In Ba, it isno longer possible to neglect spin-orbit effects; thesehave been accounted for approximately in the calcula-tion through the jj-LS frame transformation. A quanti-tative description of the J51o bound spectrum of Ba

FIG. 8. Schematic energy diagram of the alkaline-earth atomsBe to Ra, displaying the lowest ionization limits with theirfine-structure splittings (in cm21) and the double-ionizationlimits. For each atom, the energy is relative to the m0s

2 groundstate.



was achieved (Aymar, 1990) in an LS-coupledR-matrix calculation for the 1Po, 3Po, and 3Do symme-tries. The treatment included three channels (6snp ,5dnp , and 5dnf) for the 1P and 3P symmetries and two

channels (5dnp and 5dnf) for the 3D symmetry. TheLS-coupled reaction matrices were recoupled into asingle jj-coupled matrix of dimension 8 for J51. Thisproblem involves three ionization limits (6s , 5d3/2 ,

FIG. 9. Lu-Fano plots of the odd-parity J51 bound levels of Ca to Ra, comparing experimental values, depicted by solid points,with eigenchannel R-matrix calculations (solid curves and open squares). (a),(b) Results on the m0snp

1P Rydberg series of (a)Ca and (b) Sr perturbed by the (m021)dm0p

1P level were obtained in LS coupling; the dashed curve for Ca is the empiricalMQDT fit of Armstrong et al . (1979). (c) The LS-coupled R-matrix result for Ba combined with the jj-LS frame transformation;the levels (1) to (8) correspond to 5dnp perturbers of the 6snp 1P1 and

3P1 Rydberg series, the 5d6p1P1 being labeled by (3)

and the 5d7p 1P1 by (6); the 5d8p1P1 level is diluted into high-lying 6snp 1P1 levels; the three 5d4f levels cause the rapid

increases of the quantum defects for 3.75<n5d5/2<4. (d) The jj-coupled R-matrix result for Ra for the 7snp 1P1 and3P1 Rydberg

series perturbed by the 6d7p J51 levels [for assignment of the levels (1) to (7) see Greene and Aymar (1991)]. (From Aymaret al . , 1987; Greene and Kim, 1987; Aymar, 1990, and Greene and Aymar, 1991).



5d5/2). For this reason Fig. 9(c) is the projection of athree-dimensional Lu-Fano plot onto the two-dimensional plane of -n6s (mod 1) versus n5d5/2. A char-acteristic feature of the energy diagram of Fig. 8 is thesmaller gap between the 6s and 5d thresholds in Bacompared to the corresponding gaps between the m0sand (m021)d thresholds in Ca, Sr, and even Ra. Forthis reason, in Ba, a large number (twelve) of doublyexcited levels, which are the lower members of the5dnp (6<n<8) and 5dnf (n54) Rydberg series, per-turb the 6snp 1P1 and

3P1 Rydberg series. Figure 9(d)shows that the Ra Lu-Fano plot, obtained using a 13-channel jj-coupled R-matrix calculation (Greene andAymar, 1991) looks simpler than the Ba plot. In fact,only the three Ra 6d7p J51 doubly excited levels arelocated below the 7s limit and thus perturb the 7snp1P1 and

3P1 Rydberg series. The comparison of calcu-lated (open squares) and observed (solid points) levels issatisfying for the lower- and higher-lying 7snp 1P1 lev-els, whereas some discrepancies occur in the intermedi-ate energy range. In particular, the theoretical level la-beled by T cannot be associated with any experimentallevel. It remains unclear whether these discrepancies aredue to some inaccuracy of the calculation, or to an ex-perimental error or misclassification. As expected,poorer agreement is obtained between theory and ex-periment for the higher-lying 7snp 1P1 levels usingLS-coupled eigenchannel R-matrix calculations com-bined with the jj-LS frame transformation (see Greeneand Aymar, 1991).Perturbations of the m0snp J51 series have been

analyzed previously for every alkaline-earth atom fromCa to Ra using empirical MQDT models fitted to experi-ment (Armstrong et al . , 1979, 1980). The dashed curvefor Ca @Fig. 9(a)] is the MQDT least-squares fit of Arm-strong et al . (1979). The nearly ab initio R-matrix cal-culations improve upon the description of the low-lyinglevels achieved in those empirical studies. This improve-ment derives from the correct description of the energydependence of the short-range scattering parameters; inmost semiempirical fits an oversimplified energy depen-dence has been assumed, in order to minimize the num-ber of fitted parameters.The Ca and Sr Lu-Fano plots are remarkably similar

to each other, while Ba and Ra differ. Most of the dif-ferences between the lightest and heaviest atoms derivefrom two effects: from the different number of doublyexcited levels located below the first ionization limit andfrom spin-orbit effects. The short-range scattering pa-rameters exhibit strong similarities among all atomsfrom Ca to Ra. Below the m0s threshold, the J51o

spectra are dominated by the mixing between them0snp

1P and (m021)dnp 1P channels, which is re-flected by the large curvature of the Lu-Fano plots of Ca@Fig. 9(a)] and Sr @Fig. 9(b)]. Similar curvature is appar-ent for Ba @Fig. 9(c)] around n5d3/2;2.5 (5d6p 1P per-turber) and n5d3/2 ;3.5 (5d7p 1P perturber), and for Ra@Fig. 9(d)] around n6d3/2;2.7 (6d7p 1P perturber).These similarities are even more evident in Fig. 10,

where the eigenquantum defects ma [Figs. 10(a), 10(c),10(e), and 10(g)] and mixing angles u [Figs. 10(b), 10(d),10(f), and 10(h)] from R-matrix calculations (Aymar etal . , 1987; Greene and Kim, 1987; Aymar, 1990; Greeneand Aymar, 1991) are displayed for the m0snp

1P and(m021)dnp 1P channels of Ca to Ra versus the energy.The Ca and Sr results were obtained from a two-channeltreatment, whereas those for Ba and Ra were derivedfrom calculations involving a larger number of channels.The dashed curves in Figs. 10(e) and 10(f) for Ba showthe empirical smooth scattering parameters fitted byArmstrong et al . (1979). The assumption that these pa-rameters have a linear energy dependence, frequentlymade in empirical studies, is seen to be quantitativelyvalid only over a restricted energy range near the thresh-old; far below the m0s threshold, the energy dependencebecomes much more complicated (see also Fig. 7). Be-cause R-matrix calculations carried out in the differentatoms use different r0 values and different numbers ofinteracting channels, a detailed comparison between thedifferent curves (a), (c), (e), and (g) displaying the en-ergy dependence of m1 and m2 is not very meaningful.However, let us note that the values of Dm5m12m2(mod 1) near the lowest threshold are comparable foreach element: 0.36 (Ca), 0.36 (Sr), 0.25 (Ba), 0.33 (Ra).Even more striking similarities are evident for the

two-channel mixing angle u @Figs. 10(b), 10(d), 10(f),10(h)]. For all elements, u evolves from zero at low en-ergy to a maximum ;0.20p around the m0s thresholdwhere the two channels are strongly mixed. RecallingSec. II.F, note that the strongest possible mixing corre-sponds to u5p/4 and Dm= 0.5. This near invariance ofthe strength of channel mixing was previously discussedby Armstrong et al . (1979, 1980) and by Wynne andArmstrong (1979). These authors showed that all 1Po

bound-state data for Ca, Sr, and Ba can be compactlyrepresented by using the two common parametersu50.19p and Dm50.40 and a third parameterm5m12Dm/2 that is different for each atom. These em-pirical values obtained with a two-channel treatmentagree well with the R-matrix predictions, although morepoorly for Ba and Ra. These discrepancies for the heavi-est alkaline-earth atoms primarily reflect the fact thatmore than two channels are actually involved in the dy-namics.One additional paper should be mentioned here.

Armstrong et al . (1981) carried out ab initio calcula-tions of short-range scattering parameters in alkalineearths using a local-density approximation for the ex-change correlation potential. Their calculated two-channel interaction parameters for the 1Po series agreewith the fitted parameters for Ca and Ba, while for Srthe agreement is poorer. That study also showed howthe channel-interaction angle u varies with energy.We turn now to the even-parity J50 spectra of Ca, Sr,

and Ba. Calculated Lu-Fano plots displayed in Fig. 11are compared with experimental levels (Moore, 1949,1952, 1958; Armstrong et al . , 1977; Esherick, 1977; Ay-mar et al . , 1978; Sugar and Corliss, 1985). For Ca @Fig.11(a)] and Sr [Fig. 11(b)], the curves (solid lines) wereobtained with LS-coupled eigenchannel R-matrix calcu-



lations combined with the jj-LS frame transformation.Ca results are from Aymar and Telmini (1991); Sr re-sults are derived from those of Aymar et al . (1987) andof Kompitsas et al . (1991). These R-matrix calculationsinvolve the following five LS-coupled channels:@m0sns , m0pnp , and (m021)dnd 1S], @m0pnp and(m021)dnd 3P]. The Ba Lu-Fano plot @Fig. 11(c)] wasobtained from a five-channel jj-coupled eigenchannelR-matrix calculation (Greene and Aymar, 1991). TheLu-Fano plots of Ca @Fig. 11(a)] and Sr @Fig. 11(b)] ex-tend across the first ionization limit, while the Ba plot@Fig. 11(c)] is restricted to the bound spectrum. Theplots of Fig. 11 are projections of four-dimensional sur-faces onto the -nm0s

(mod 1) versus n(m021)d3/2plane.

The Ca and Sr Lu-Fano plots above the ionization limitdisplay the physical ‘‘collision’’ eigenphase shifts ob-tained by diagonalization of the physical scattering ma-trix [Eq. (2.50)]. Each sharp rise of the phase shift isassociated with an autoionizing resonance. The size ofeach avoided crossing between the branches of the Lu-Fano plots in Fig. 11 can be interpreted as reflecting thechannel-interaction strength between either the m0pnp(1S0,

3P0) channels or else the (m021)dnd(1S0,

3P0) channels and the m0sns1S0 Rydberg series

(or m0ses1S0 continuum). The J50e spectra of Ca [Fig.

11(a)] and Sr [Fig. 11(b)], which are immediately seen tobe similar, are strongly dominated by the mixing be-tween singlet channels. In fact, in Ca the interaction be-tween the singlet and triplet channels is almost negli-

gible in the energy range of Fig. 11(a), which means thatit is adequate to neglect their coupling altogether. Thisapproximation is no longer valid for Ca at higher ener-gies very close to the 3dj spin-orbit-split thresholds, aswill be documented in Sec. IV.B. Figure 11(b) showsthat neglect of singlet-triplet mixing for this symmetry inSr is only valid below the 5s threshold.For the 3P0 symmetry of Ca and Sr, two perturbers

m0p2 (labeled by A) and (m021)d2 (labeled by B) are

located below the m0s threshold. Concerning the 1S0symmetry of Ca and Sr, R-matrix calculations haveshown that the levels referred to as P0 and P08 involvecomparable admixtures of the (m021)d2 and m0p

2 1Sconfigurations. Multiconfiguration Hartree-Fock calcu-lations also demonstrated that the P0 level is a strongmixture of those configurations (Froese Fischer andHansen, 1981, 1985; Vaeck et al . , 1988). Recently Brageand Froese Fischer (1994a) have applied a nonvaria-tional Spline-Galerkin method to reinvestigate a largenumber of 4snl Rydberg series of Ca. They used atomicwave functions in a close-coupling form, including pseu-dostates. Radial functions for the channel functions andfor the pseudostates were taken from Ca1 Hartree-Fockcalculations, while a B-spline expansion was used to rep-resent the radial functions of the outer electrons. Core-polarization effects were also incorporated. A very accu-rate description of 4sns 1S0 Rydberg levels up ton523 was achieved. They also found nearly equalweights for the 3dnd and 4pnp 1S0 channels in the P0

FIG. 10. Eigenquantum defectsma and mixing angle u for them0snp

1P (1) and(m021)dnp 1P (2) channels ofCa to Ra below the m0s ioniza-tion limit, as functions of theenergy E referred to thedouble-ionization limit. Thecurves (full lines) are derivedfrom LS-coupled eigenchannelR-matrix calculations. Thedashed lines for Ba correspondto the empirical MQDT fit ofArmstrong et al . (1979). (FromAymar et al . , 1987; Greene andKim, 1987; Aymar, 1990, andGreene and Aymar, 1991).



level. For levels such as these, which exhibit such strongelectron correlations, classifications based onindependent-electron quantum numbers should not betaken too seriously. With this qualification in mind, thetrends of the (m021)dnd autoionizing series for Ca andSr investigated with the R-matrix method (see Sec. IV.Bbelow) suggest that the labels given in the earlier litera-ture (Moore, 1949, 1952) should be reversed. This leadsto the identification of P0 as the (m021)d2 level andP08 as the m0p

2 level. As discussed below in Sec. IV.B,the upper level P08 shows up as an autoionization fea-ture, which was observed in the J50e autoionizing spec-tra of Ca (Bolovinos et al . , 1992) and Sr (Kompitsaset al . , 1991). In both Ca and Sr, the position and widthof that m0p

2 1S0 (P08) resonance agree well withR-matrix results (Aymar and Telmini, 1991; Kompitsaset al . , 1991). Below the m0s threshold, the calculationsaccurately reproduce the perturbations of the m0sns

1S0 Rydberg series by the P0 doubly excited state. Asimilar description of the 4sns 1S0 perturbed Rydbergseries of Ca was obtained by Osani et al . (1991), using aWigner-Eisenbud-type formulation of the R-matrixmethod.Figure 11(c) shows that more J50e doubly excited

states appear below the first ionization limit in Ba(namely six) than in either Ca and Sr (three), much likethe situation discussed for the J51o spectrum. In addi-tion to the 5d2 3P0, 5d

2 1S0, and 6p2 3P0 doubly ex-cited levels, homologous to the three perturbers appear-ing in Ca and Sr, there are three other 5dnd doublyexcited levels. The 5d2 1S0 level (labeled by 2) lies be-low the 6s threshold, while other R-matrix calculationsshow that the 6p2 1S0 level of Ba is a broad autoionizingresonance, although we postpone discussion of thispoint. The admixture of the (m021)d2 and m0p

2 1S0configurations were found to be much smaller than in Ca

FIG. 11. Lu-Fano plots of the even-parity J50 levels of Ca to Ba comparing experimental values depicted by solid points witheigenchannel R-matrix calculations (solid curves). (a),(b) For (a) Ca and (b) Sr the plots were obtained with LS-coupled eigen-channel R-matrix calculations combined with the jj-LS frame transformation. Below the first limit, (a) and (b) illustrate theperturbation of the m0sns

1S0 Rydberg series by doubly excited levels: (A) m0p2 3P0, (B) (m021)d2 3P0, with P0 and P08

corresponding to admixtures of the (m021)d2 and m0p2 1S configurations. Above the first limit located at n3d3/252.83 in Ca and

n4d3/252.74 in Sr, (a) and (b) display the eigenphase shifts, the vertical arrows, at the top of the figures, indicating the positions ofautoionizing resonances. (c) The Ba result obtained with jj-coupled eigenchannel R-matrix calculation displays the perturbation ofthe 6sns 1S0 Rydberg series by doubly excited levels: (1) 5d

2 3P , (2) 5d2 1S , (3) 6p2 3P , (4) 5d6d 3P , (5) 5d6d 1S , (6) 5d7d3P . [The Ca figure is from Aymar and Telmini (1991) and the Ba figure from Greene and Aymar (1991).]



and Sr; so, for instance, the weight of 5dnd channels forthe 5d2 1S0 is ;75%. Except in the lowest energy range,the R-matrix calculation correctly reproduces the per-turbation of the 6sns 1S0 Rydberg series by doubly ex-cited levels.Figure 12 displays Lu-Fano plots of the even-parity

J52 bound levels of Ca, Sr, and Ba. On each Fig. 12(a)–12(c), the quantum defects of experimental levels(Moore, 1949, 1952, 1958; Armstrong et al . , 1977; Esh-erick, 1977; Aymar et al . , 1978; Sugar and Corliss, 1985)are depicted by points, which can be compared withquantum defects calculated in eigenchannel R-matrixcalculations (solid curves). Results from LS-coupledR-matrix calculations for the 1De levels of Ca (Aymarand Telmini, 1991) and of Sr (Aymar et al . , 1987) aredisplayed in Figs. 12(a) and 12(b). The calculations in-cluded four interacting MQDT channels: m0snd ,(m021)dnd , (m021)dns , and m0pnp . The low-lyingP2 state perturbs the m0snd

1D series, while the sharprises of the quantum defects near the ionization limitreflect perturbations by the (m021)d(m011)s and P28doubly excited states, where m0 5 3, 4, and 5 for Ca, Sr,and Ba, respectively. As for the J50e case, similar re-sults were obtained by Osani et al . (1991). These resultsalso agree with multiconfiguration Hartree-Fock predic-tions (Froese Fischer and Hansen, 1981, 1985; Vaecket al . , 1988) and with the recent Spline-Galerkin calcu-lation of Brage and Froese Fischer (1994a, 1994b). TheP2 and P28 levels involve a strong admixture of the(m021)d2 and m0p

2 1D configurations. Also, followingthe case for J50e, the labels previously given in the lit-erature (Moore, 1949, 1952) should be reversed. Specifi-cally we recommend identifying P2 as the (m021)d2

level and P28 as the m0p2 level. In both Ca and Sr, the

(m021)d(m011)s perturber is spread out over manym0snd Rydberg levels. In Sr, the resonance associatedwith the 5p2 level was observed (Esherick, 1977) justabove the threshold. In Ca, the 4p2 (P28) level is pre-dicted (Aymar and Telmini, 1991) to be centered almostexactly on the 4s ionization limit. The character of thisperturber is so diluted among the high 4snd Rydberglevels that it does not make sense to identify any indi-vidual quantum state as this P28 level.The barium J52e states display greater complexity.

Figure 12(c) shows the perturbations of the 6snd 1D2and 3D2 Rydberg series by 16 doubly excited levels as-sociated with the 5dns , 5dnd , and 6p2 configurations.These theoretical results were based on an 11-channeljj-coupled R-matrix calculation (Greene and Aymar,1991) that included the channels 6s1/2nd3/2 , 6s1/2nd5/2 ,5d3/2ns1/2 , 5d3/2nd3/2 , 5d3/2nd5/2 , 5d5/2ns1/2 ,5d5/2nd3/2 , 5d5/2nd5/2 , 6p1/2np3/2 , 6p3/2np1/2 , and6p3/2np3/2 . Except for some discrepancies, theory is inclear agreement with the experimental quantum defects,accounting for the great complexity of the Ba spectralpattern. In contrast with Ca and Sr, the Ba 5d2 (labeledby 1) and 6p2 1D2 (labeled by 2) levels are both boundand, as in the 1D2 case, only weakly admixed together.

Returning to the J50e,2e spectra, the jj-coupledR-matrix results for Ba presented in Figs. 11 and 12have been compared (Greene and Aymar, 1991) withLS-coupled R-matrix calculations followed by a jj-LS

FIG. 12. Lu-Fano plots of the even-parity 1D bound levels ofCa and Sr and J52e bound levels of Ba comparing experimen-tal values, depicted by solid points, with eigenchannelR-matrix calculations (solid curves). (a),(b) LS-coupledR-matrix results for (a) Ca and (b) Sr display the perturbationof the m0snd

1D series by the P2 and P28 levels which corre-spond to admixture of the (m021)d2 and m0p

2 1D configu-rations and by the (m021)d(m011)s 1D level. (c) Thejj-coupled R-matrix result for the 6snd 1D2 and

3D2 Rydbergseries of Ba perturbed by 5dnl and 6p2 J52 levels; in particu-lar: (1) 5d2 1D2, (2) 6p

2 1D2, and (3) 5d7d 1D2. [The Cafigure is from Aymar and Telmini (1991) and the Ba figurefrom Greene and Aymar (1991).]



frame transformation. As explained above in Sec. II.F,use of the frame transformation near the bottom end ofa Rydberg series requires some caution. Because theJ50e,2e spectra involve the low-lying 6p2 levels, the Kmatrices deduced from LS-coupled R-matrix calcula-tions were used to conduct separate MQDT test calcu-lations, using either experimental j-dependent 6p1/2,3/2threshold energies or theoretical spin-orbit-averaged6p threshold energies. The jj-LS calculation requiredthe use of theoretical threshold energy in order to give acorrect description of the whole J52e spectrum, in con-trast with the J50e, whose spectrum is much less sensi-tive to the threshold energies utilized. This extreme sen-sitivity to threshold energies is an undesirableconsequence of carrying out a frame transformation us-ing quantum defects that vary rapidly with energy;it reflects a limitation of current implementations ofthe frame-transformation calculations. For both theJ50e and J52e cases, neglect of the Ba1 6p ionic-corefine structure adversely affects the accuracy of theframe-transformation calculation compared to thejj-calculation, but only for the 6p2 levels.Empirical MQDT analyses of the J50e,2e spectra

were previously carried out in Ca (Armstrong et al . ,1977; Wynne and Armstrong, 1979), Sr (Esherick, 1977;Wynne and Armstrong, 1979), and Ba (Aymar et al . ,1978; Aymar and Robaux, 1979) with the purpose ofinterpreting newly observed data. There, the experimen-tal levels were exactly on the fitted Lu-Fano curves,whereas some deviations are visible on Figs. 11 and 12.However, there are two improvements of theseR-matrix studies over the empirical fits. First, the lowestlevels, lying in the energy range in which smooth scat-tering parameters acquire a strong energy dependence,are for the most part correctly described, whereas em-pirical studies generally disregarded them. The second,more striking, improvement concerns the assignment oflevels. As previously pointed out by Wynne and Arm-strong (1979) and by Froese Fischer and Hansen (1981),the empirical MQDT method developed to take into ac-count the effects of periodic perturbation fails to iden-tify isolated perturbers such as the P0 and P2 levels oc-curring in Ca and Sr. Previous empirical MQDT fits ofthe J50e,2e spectra of Ca and Sr (Armstrong et al . ,1977; Esherick, 1977; Wynne and Armstrong, 1979)made some hypotheses for the perturber identificationsand entirely neglected the m0p

2-(m021)d2 interaction.The R-matrix approach bypasses these difficulties en-countered in Ca and Sr. A detailed comparison betweenR-matrix and fitted short-range scattering parametersobtained in Ca was given by Aymar and Telmini (1991).The J52e Ba spectrum is better suited to empirical

MQDT studies than are Ca and Sr because each 5dnlseries supports several bound levels, while the 6p2 per-turbers are still considered as ‘‘isolated’’ perturbers. Theidentifications deduced from MQDT fits (Aymar etal . , 1978; Aymar and Robaux, 1979) were completelyconfirmed for the J50e spectrum, while the 6p2 per-

turbers and some neighboring levels were misclassifiedin the J52e case. Greene and Aymar (1991) discussedthese misclassifications.Spectral regularities along the 1S0

e and 1D2e series of

Ca, Sr, and Ba were also previously analyzed by Wynneand Armstrong (1979). In contrast with the 1P1

o case,neither for the 1D2

e , nor for the 1S0e spectra was it pos-

sible to fit the level positions for all atoms using a com-mon set of short-range scattering parameters. In particu-lar, the parameters fitted to Ca and Sr completely failedto reproduce the Ba data. R-matrix calculations confirmthese conclusions obtained empirically. All even-parityspectra are dominated by a strong mixing between thesinglet m0pnp and (m021)dnd channels, but the evo-lution of this mixing with energy is different for eachatom. The differences in the dynamics of electron corre-lations in odd- and even-parity bound singlet spectra isprobably related to the presence, in the latter spectra, ofthe strongly correlated singlet m0p

2 and (m021)d2 lev-els involving two electrons in the same orbit.

2. Observables other than energies

A good description of an energy spectrum is certainlynecessary, but not sufficient, to confirm that channel in-teractions have been treated accurately. Other observ-ables test the accuracy of the wave functions more sen-sitively.The hyperfine structure of perturbed Rydberg levels,

for instance, is quite sensitive to state mixing (for refer-ences see Aymar, 1984a, 1984b). Hyperfine structure ob-served in high-lying 6snd 1,3D2 Rydberg levels of Ba,perturbed by the 5d7d 1D2 level [labeled by 3 in Fig.12(c)] has been analyzed in terms of Rydberg singlet-triplet mixing. Just below the 6s threshold, the zero-order wave function (associated with the Hamiltonianwithout hyperfine effects) of each J52e level can bewritten to a good approximation as

C~n !5anu6snd 1D2&1bnu6snd 3D2&1gnu5d7d 1D2&,(4.1)

where the an , bn , and gn expansion coefficients can berelated to the closed-channel coefficients Zi

(n) [see Eq.(2.64)] in the wave function of each level n . (Note thatbn in this context should not be confused with the angu-lar distribution asymmetry parameter, nor with theclosed-channel quantum-defect-theory phase parameterof Sec. II.)As explained by Eliel and Hogervorst (1983),

Rinneberg and Neukammer (1982, 1983), and by Aymar(1984a, 1984b), the hyperfine energies can be calculatedas a function of the an , bn , and gn mixing coefficients.Using the gn coefficients inferred from semiempiricalMQDT analysis of energy levels or from measurementsof lifetime, isotope shifts and Lande factors, Eliel andHogervorst (1983) and Rinneberg and Neukammer(1982, 1983) extracted the an and bn coefficients by fit-ting the calculated energies to their experimental results.Figure 13 presents the evolution of the coefficient bn asa function of the principal quantum number n , for levels



identified as singlets (i.e., those levels with an2>bn

2). Thedot and plus symbols marked are derived from hyperfinestructure measurements of Eliel and Hogervorst (1983),and from those of Rinneberg and Neukammer (1982,1983), respectively. The dashed and solid curves depictbn values obtained with 11-channel LS- and jj-coupledR-matrix calculations (Greene and Aymar, 1991). (Notethat the jj-coupled R-matrix calculation corresponds tothe calculation giving the Lu-Fano plot of Fig. 12(c).)The theoretical positions of the 5d7d 1D2 level (circledopen circle) are somewhat higher than the experimentalposition, which causes both theoretical curves to beshifted slightly toward high-n values. However, it is clearthat the experimental behavior is better reproduced bythe jj-coupled R-matrix results than by the jj-LS andframe-transformation calculation.An interesting and little-probed energy regime occurs

for the very high-lying m0snl Rydberg levels of oddalkaline-earth isotopes, when the hyperfine splitting ofthe inner m0s electron becomes comparable to the en-ergy separation ('1/n3 a.u.) between adjacent Rydberglevels. In this energy range, the spectrum is truly multi-channel in character, and the hyperfine interaction af-fects the Rydberg-electron motion nonperturbatively.Even though the nature of the threshold splitting is dif-ferent from the fine-structure-split thresholds discussedin Sec. II.E, the MQDT description can be immediatelygeneralized to treat this case. This extension of theframe-transformation method to treat hyperfine struc-ture has been carried out by Sun, Lu, and co-workers(Sun and Lu, 1988; Sun, 1989; Sun et al . , 1989). The Sr

5sns and 5snd hyperfine structures observed byBeigang et al . (1983) and by Beigang (1985) for stateswith n@ 100 have been successfully analyzed with thehelp of semiempirical MQDT models based on this hy-perfine frame-transformation method (Sun and Lu,1988; Sun et al . , 1989).We return now to the J51o spectrum of Ba, in the

energy range close to the lowest ionization thresholdwhere the 5d8p 1P1 doubly excited level is diluted intohigh-lying 6snp Rydberg levels @n5d3/2>4.1 on Fig.9(c)]. In this range, the oscillator strengths [Eq. (2.66)]observed for the 6s2 1S0 → 6snp 1P1 transitions (Par-kinson et al . , 1976; Connerade et al . , 1988) deviate dra-matically from the simple n6s

23 law expected for an un-perturbed series. The modulation of log10(n6s

3 f) near the5d8p 1P1 perturber arises from the redistribution of theperturbing level’s intensity among the high-lying 6snp1P1 levels as well as into the adjacent continuum, with amaximum close to the 6s threshold. Figure 14 comparesthe experimental data (crosses) obtained by Conneradeet al . (1988) with results (dots) obtained by combiningLS-coupled R-matrix calculations with the jj-LS frametransformation (Aymar, 1990). Theory reproduces theintensity minimum near n;24,25 as well as the oscillatorstrengths of high-lying levels, but it overestimates theoscillator strengths for the lowest levels considered here.The modulation of Rydberg-level oscillator strengthsnear an isolated perturber is reasonably well describedby the standard Fano line-shape formula (Fano, 1961),even though this formula was derived for one discrete

FIG. 13. Rydberg singlet-triplet mixing coefficients b [Eq.(4.1)] for the high-lying 6snd 1D2 levels of Ba as functions ofthe principal quantum n in the energy range perturbed by the5d7d 1D2 level. The jj-LS (dashed line) and jj (full line)results are compared with values deduced from hyperfine-structure measurements: d Eliel and Hogervorst (1983), +Rinneberg and Neukammer (1982, 1983). For the 5d7d 1D2level, the mixing coefficient (circled +) obtained by Rinnebergand Neukammer (1982, 1983) is compared with the theoreticalpredictions, depicted by concentric open circles. (From Greeneand Aymar, 1991.)

FIG. 14. Oscillator strengths of the 6s2 1S0 → 6snp 1P1 tran-sitions of Ba. The log10(n6s

3 f) values are plotted as functions ofthe effective quantum number n6s relative to the 6s ionizationlimit. The experimental results (+) of Connerade et al . (1988)are compared with the velocity results (d) obtained by com-bining LS-coupled R-matrix calculations with the jj-LS frametransformation. The 6snp 1P1 levels are labeled by their prin-cipal quantum number n . (From Aymar, 1990.)



state interacting with a continuum, because the Rydberglevels are so closely spaced that they resemble a con-tinuum (Connerade, 1992).A combined theoretical and experimental investiga-

tion of several oscillator strengths connecting discretelevels of Sr was carried out by Werij et al. (1992). Thisproject showed that a reliable description of many tran-sitions can be achieved using LS-coupled eigenchannelR-matrix and quantum-defect calculations. Not surpris-ingly, theory describes the stronger transitions more ef-fectively than the weaker ones, since the weak transi-tions tend to have competing contributions of oppositesign in the configuration-interaction expansion. In theindependent-electron model, no single electron transi-tion from the ground state can give an absorption oscil-lator strength larger than 1. For the main resonancetransition of Sr, namely 5s2 1S→5s5p 1Po, the absorp-tion oscillator strength is approximately f'1.9, whichimplies that both valence electrons participate stronglyin the transition. For most of the singlet transitions, thecalculated oscillator strengths (Werij et al., 1992) agreereasonably well with multiconfiguration Hartree-Fockcalculations of Vaeck et al. (1988). No similarly exhaus-tive multiconfiguration Hartree-Fock study is availablefor the triplet manifold for comparison with theR-matrix values.

B. Energy positions of autoionizing levels

It can be difficult to fully interpret a spectrum of au-toionizing states from study of the experimental or theo-retical photoionization cross section alone. Complexitiesassociated with autoionizing levels can remain hidden,and frequently even the resonance positions cannot beextracted. Moreover, interferences between the discreteand continuum excitation amplitudes often result inasymmetric Beutler-Fano profiles (Fano, 1961), whichfurther complicates the pattern of autoionization and itsrelation to the true ‘‘resonance energies.’’ In some cases,resonances fail to appear in a particular cross sectionbecause they have vanishing transition strengths. An-other difficulty that plagues the analysis of complex mul-tichannel systems is the overlapping nature of the reso-nance pattern. It is crucial for theoretical multichannelspectroscopy to develop the ability to extract resonancepositions, widths, and classifications that are indepen-dent of the excitation process.In Sec. II.F, it was mentioned that a Lu-Fano plot can

be extended into the energy range above an ionizationthreshold to gain useful information about autoionizingresonances; examples of this are shown in the right-handportions of Figs. 11(a) and 11(b). Each rise of one phaseshift d5pm by one cycle implies the existence of oneresonant state. As usual, the magnitude of the gap at anavoided crossing indicates the strength of interactionsbetween closed and open channels, which determinesthe resonance widths. However, the quantitative deter-mination of the positions and widths of autoionizingresonances from such a plot is possible only in relatively

simple cases involving a small number of channels (see,for example, Giusti-Suzor and Fano, 1984).For more complicated situations involving several in-

teracting open and closed channels (see Sec. II.D.3), theresonance energies and classifications can be obtainedby using MQDT techniques for treating bound-statespectra, starting from an effective reaction matrix Keffwhose open-channel components have been ‘‘elimi-nated’’ in some manner. For example, this matrix maycorrespond to the real part of the k-matrix introduced inEq. (2.73). In this way resonances have been analyzed inCa, Sr, and Ba (Kompitsas et al . , 1990, 1991; Aymar andTelmini, 1991; Gounand et al . , 1991; Luc-Koenig andAymar, 1991, 1992; Goutis et al . , 1992; Assimopouloset al . , 1994; Aymar et al . , 1994; Luc-Koenig, Aymar,and Lecomte, 1994; Luc-Koenig, Bolovinos, et al . , 1994;Luc-Koenig, Lecomte, and Aymar, 1994). The effectivereaction matrix Keff restricted to the subspace of closedchannels immediately permits a determination of theresonance positions and of the corresponding channeldecompositions Zi

(n) [see Eq. (2.64)]. These coefficientsdetermine the dominant closed channel i to which aresonance belongs. The interactions among closed chan-nels can also be analyzed graphically using Lu-Fanoplots. This entire procedure based on the construction ofan effective reaction matrix is (generally) adequate tolocate the position of a resonance with an error compa-rable to its autoionization width. Another method forextracting resonance properties based on the time-delay(or density-of-states) matrix gives in addition the auto-ionization widths; this method will be described in Sec.IV.C below.The procedure of the preceding paragraph has been

applied to interpret autoionizing Rydberg series of Ca,Sr, and Ba converging to the ionic fine-structure levels(m021)d3/2 and (m021)d5/2 . The analysis provides in-formation about the perturbers of these series that cor-respond to the lower-lying members of Rydberg seriesconverging to the m0p1/2,3/2 thresholds. For illustration,we concentrate on the even-parity autoionizing J50e

levels.Figure 15 displays Ca Lu-Fano plots obtained with

two effective matrices Keff deduced from two separateLS-coupled eigenchannel R-matrix calculations carriedout for the 1Se and 3Pe symmetries (Aymar andTelmini, 1991). The reaction matrix K

3P has no con-tinuum channel in this energy range; it can thus be useddirectly to analyze the resonances since when the 1Se-3Pe mixing is disregarded, all states are bound. For the1Se symmetry, a closed-channel matrix K eff

1S was first de-duced from the complete reaction matrix K

1Sconnectingthe open channel 4ses and the two closed channels3dnd and 4pnp . The predicted resonance energies areplotted in the -n3d (mod 1) versus n4p plane, the effec-tive quantum numbers n3d and n4p having been calcu-lated using spin-orbit-averaged ionization limits. The4pnp perturbers of the 3dnd Rydberg series are easy toidentify. Each perturbing level causes a sharp rise of-n3d for the relevant branch of the Lu-Fano curve. The



discussion of Fig. 11(a) in Sec. IV.A.1 considered theclassification of the P08 state, which is a strong admixtureof the 3d2 and 4p2 singlet configurations but is some-what better identified as the 4p2 1S state. This identifi-cation follows in part from the fact that P08 lies on anearly vertical branch of the 1S Lu-Fano plot. The rapidincrease of -n3d along the 3P branch, just below the3d threshold at n4p;3, reflects a perturbation of the3dnd 3P series by a 4p5p state that is centered almostright on the 3d threshold. The character of this 4p5p3P perturber is entirely diluted among the high-lyingmembers of the 3dnd 3P series. A similar procedurehas been used to identify the J52e 4pnp levels thatperturb the 3dnd and 3dns series.Initial survey calculations in LS coupling were fol-

lowed by more complete jj-LS frame-transformationtreatments of the J50e,2e spectra that incorporatedfine-structure effects (Aymar and Telmini, 1991). Whenfine structure is included, the energies of the 4pnp per-turbers remain close to the predictions of theLS-coupled calculations. Close to the 3d3/2 threshold,however, the high-lying 3dns ,nd levels are strongly af-fected, especially the 3dnd J50 levels. Recently,R-matrix predictions for the Ca J50e,2e spectra havebeen confirmed by the experiment of Bolovinos et al .(1992). These authors reported the first observation ofthe broad 4p2 1S0 (P08) autoionizing resonance, whosewidth is around 80 cm21. Figure 16 shows that the Lu-Fano plot of experimental (open squares) low-lying3dnd J50 levels and of the 4p2 1S0 level (labeled byA) agrees with the R-matrix predictions (solid line andpoints) of Aymar and Telmini (1991). The energy levels

are now plotted in the -n3d3/2(mod 1) versus n3d5/2 planewhich makes 4pnp perturbers more difficult to discern.This shows the interest of carrying out preliminary cal-culations in LS coupling. The MQDT treatment in-volves four closed fragmentation channels: 3d3/2nd3/2 ,3d5/2nd5/2 , 4p1/2np1/2 , and 4p3/2np3/2 . Neither Fig. 16nor the complete Lu-Fano plot of J50e levels of Calocated between the 4s and 3d3/2 limits (2<n3d5/2<42),show the nearly horizontal line that typically representsan ‘‘unperturbed’’ 3d3/2nd3/2 Rydberg series. In fact, theperiodicity in n3d5/2, expected for the two-channel prob-lem involving the two 3dnd series, is absent over thewhole energy range. The periodicity is destroyed by the4p2 1S0 (P08) perturber and by the 4p5p

3P0 level closeto the 3d3/2 threshold @see Fig. 16]. Accordingly, most ofthe predicted energy levels cannot be clearly attributedeither to 3d3/2nd3/2 or to 3d5/2nd5/2 series. Moreover, inCa various even-parity 3dnl levels with 0<J<3, whichhave been recorded using a two-step laser experiment,were successfully accounted by jj-coupled eigenchannelR-matrix calculations (Assimopoulos et al . , 1994; Luc-Koenig, Bolovinos, et al . , 1994).The valence-electron spectrum of atomic strontium

appears in most regions to be very similar to the calciumspectrum. Figure 17 displays a case in which some dif-ferences can be seen, namely for the even-parity J50autoionizing levels of Sr located below the 4d3/2 thresh-old, i.e., in the energy range analogous to that consid-ered for Ca in Fig. 16. For Sr, in contrast with Ca, theexperiment and the R-matrix calculations were con-ducted hand in hand (Kompitsas et al., 1991). MQDTanalysis of an effective closed-channel reaction matrixKeff helped greatly to interpret and identify the autoion-izing resonances recorded in the even-parity spectra ofSr, for J50e (Kompitsas et al . , 1991) and also forJ51e and J52e (Goutis et al . , 1992). The experimentalJ50e data in Fig. 17 (solid points) again agree with the

FIG. 15. Lu-Fano plot of the energy positions of even-parityautoionizing levels of Ca between the 4s and 3d ionizationlimits. The solid lines and points were obtained from two sepa-rate LS-coupled eigenchannel R-matrix calculations carriedout for the 1Se and 3Pe symmetries. The quantities n3d andn4p are the effective quantum numbers relative to the spin-orbit-averaged Ca1 3d and 4p ionization limits. The 3dnd1S Rydberg series is perturbed by the P08 level, which corre-sponds to an admixture of the 4p2 and 3d2 1S configurations.The rapid increase of 2n3d near n4p;3 reflects the perturba-tion of the 3dnd 3P Rydberg series by the 4p5p 3P level,which is diluted among high-lying 3dnd 3P Rydberg levels.(From Aymar and Telmini, 1991.)

FIG. 16. Lu-Fano plot of the energy positions of even-parityJ50 autoionizing levels of Ca between the 4s and 3d3/2 ion-ization limits. The experimental data of Bolovinos et al .(1992) (open squares) are compared with the results of Aymarand Telmini (1991) (solid points and curves) obtained withLS-coupled eigenchannel R-matrix calculations combinedwith the jj-LS frame transformation. The open square labeledA depicts the 4p2 1S0 perturber of the 3d3/2nd3/2 and3d5/2nd5/2 series.



LS-coupled R-matrix calculations used in conjunctionwith the jj-LS frame transformation (Kompitsas et al.,1991). In the energy range close to the 4d3/2 threshold(n4d5/2>10), the autoionizing structures correspond to a4d3/2nd3/2 Rydberg series perturbed periodically by the4d5/2nd5/2 levels. A two-channel (4d3/2nd3/2 ,4d5/2nd5/2) MQDT model that would be appropriate fora description of the high-lying J50e levels obviouslyfails to reproduce the data observed in the lower energyrange. At lower energies, members of the 4dnd seriesare affected by the presence of the 5p2 1S0 and 5p6p3P0 perturbing levels, marked respectively as (A) and(B) in Fig. 17. The former level, denoted P08 in Sec.III.A.1, corresponds to a broad autoionizing profile(with a width of 230 cm21) that was observed experi-mentally using two-photon laser excitation from the5s2 ground state (Kompitsas et al . , 1991). The latter3P0 level has not yet been observed. However, its effecton the positions of neighboring 4dnd J50 levels is quitevisible in Fig. 17. This level also affects the oscillatorstrengths of the 4d5p 1P1 → 4dnd J50 transitions.The perturbations of 4dnl autoionizing Rydberg seriesby several 5p6p levels show still greater complexity forthe J51e and J52e symmetries and also display goodagreement between theory and experiment (Goutis etal . , 1992).Figures 16 and 17 contrast the behavior of the

(m021)dnd J50e series for Ca and Sr. For both atoms,these series are affected by the m0p

2 1S0 perturber lo-cated among low-lying (m021)dnd levels. The(m021)dnd series of Ca and Sr are also affected by them0p(m011)p 3P0 perturber. The evident differencesbetween Ca (Fig. 16) and Sr (Fig. 17) result from the factthe 4p5p 3P0 level of Ca is diluted into 3dnd high-lyingRydberg levels, whereas the 5p6p 3P0 perturber is welllocalized in Sr. The low-lying Ba 5dnd J50 levels inter-

act with the 6p2 1S0 perturber (Aymar et al . , 1982; Ay-mar et al . , 1983; Greene and Aymar, 1991), whereas theBa 6p7p 3P0 level is located above the 5d threshold(Camus et al . , 1983). Because of the differences in thelocation of the m0p

2 1S0 and m0p(m011)p 3P0 levels,channel mixing near the (m021)d3/2 threshold differsconsiderably in Ca, Sr, and Ba.Strong mixing, similar to that observed in the Ca

J50e spectrum, has been found in the 4fnf series of Ba.The strongly coupled Rydberg series are the 4f5/2nf7/2and 4f7/2nf5/5 with J54, 5, or 6, which converge to twodifferent fine-structure levels (Luc-Koenig and Aymar,1992; Luc-Koenig, Aymar, and Lecomte, 1994). Figure18 displays the Lu-Fano plot of energy positions for thehigh-lying 4fnf J56 autoionizing levels of Ba. Theframe-transformation result (solid curves and points) ofLuc-Koenig and Aymar (1992) is compared to the ex-perimental data (open squares) of de Graaff et al.(1992). This experiment probed the 4fnf series using atwo-step laser-excitation scheme starting from the 5d2

metastable levels and using the 5dnf autoionizing reso-nances as intermediate levels. The theoretical energiesin Fig. 18 agree well with the experimental data. Foreach level, the theoretical energy lies within the width ofthe recorded resonance.To reproduce the energy positions of high-lying 4fnf

J56 levels, R-matrix calculations were conducted forthe 1Ie symmetry and for the 3Le symmetries withL55, 6, and 7. The frame transformation gave ajj-coupled reaction matrix describing the coupling of 33channels converging to the 6s , 5dj , 6pj , 7s , 6dj , and4f j thresholds. From this full reaction matrix, a smallereffective reaction matrix Keff was formed involving onlythe three 4fnf closed channels. This permitted a simpli-fied analysis of the 4fnf J56 levels located between the6d5/2 and 4f5/2 limits.The slant lines on Fig. 18 correspond to the series

4f5/2nf7/2 and 4f7/2nf5/2 , which interact strongly, whereasthe vertical lines are associated with the nearly unper-

FIG. 17. Lu-Fano plot of the energy positions of the J50even-parity autoionizing levels of Sr between the 5s and4d3/2 ionization limits. The experimental data (solid points) arecompared with the results (solid curves) obtained withLS-coupled eigenchannel R-matrix calculations combinedwith the jj-LS frame transformation. The open squares corre-spond to unobserved levels. Concerning the perturbers of the4d3/2nd3/2 and 4d5/2nd5/2 Rydberg series, the point A depictsthe experimental data for the 5p2 1S0 level and B the predic-tion for the 5p6p 3P0 level. (From Kompitsas et al . , 1991.)

FIG. 18. Lu-Fano plot of the energy positions of the 4fnfJ=6 autoionizing levels of Ba, comparing the experimentaldata (open squares) of de Graaff et al . (1992) withLS-coupled R-matrix calculations combined with the jj-LSframe transformation (solid curves and points). (From Luc-Koenig and Aymar, 1992.)



turbed 4f7/2nf7/2 series. The behavior of the stronglymixed 4f5/2nf7/2 and 4f7/2nf5/2 J56 series can be ana-lyzed using a two-channel model involving the eigen-channels 4fnf 1I and 3I . The rotation angle that char-acterizes the orthogonal Uia matrix in this two-channelmodel is approximately equal to u12;p/4 and the slantbranches of the Lu-Fano plot reflect the fact that thedifference uDmu5um12m2u of the eigenquantum defectsamounts to 0.5, i.e., has the maximum value. This makesit impossible to classify the levels as belonging to anyone series. Moreover, the channel mixing is highly en-ergy dependent, with significant changes in mixing coef-ficients frequently found even in two successive levels.In the higher-energy range close to the 4f5/2 threshold,the 4f7/2nf5/2 J56 levels are completely diluted amongthe 4f5/2nf7/2 states. Similarly strong mixing was foundbetween the 4f5/2nf7/2 and 4f7/2nf5/2 J54 or J55 series;this was again related to a difference between the eigen-quantum defects for the two channels 4fnf 1,3LJ5L closeto 0.5 (Luc-Koenig, Aymar, and Lecomte, 1994). Foreach final-state symmetry in the range J=4–6, the largevalue of uDmu arises from the exchange electrostatic in-teraction and is typical of the nln8l channel mixing.Strong mixing between the Ca 3dnd 1LJ5L and3LJ5L J53 channels was also observed and exhibited inthe corresponding Lu-Fano plot (Luc-Koenig, Bolovi-nos, et al., 1994).In contrast, for the Ca J50e symmetry (Fig. 16),

where the coupling of 3dnd is also maximum, it is the3dnd-4pnp mixing that indirectly (but strongly) couplesthe eigenchannels 3dnd 1S and 3P , even though theyare not directly coupled by the electrostatic interaction.In this case, the channel mixing angle is also u12;p/4,but uDmu amounts to only 0.21. This translates [Eq.(2.82)] into a 40% probability that the Rydberg electronwill scatter from one fine-structure channel to anotherwhen it collides with the Ca1 3d ionic core.Returning to the 4fnf levels of Ba, we note that the

resonance energy positions calculated by Luc-Koenigand Aymar (1992) agreed well with experimental data ofde Graaff et al . (1992). This agreement provided someof the first evidence that the R-matrix and MQDT tech-niques are capable of handling excited autoionizing lev-els of Ba lying above the 6p threshold. As discussedbelow in Sec. IV.D.2, another calculation in a similarenergy range had been carried out previously by Kimand Greene (1988) concerning the Ca photoabsorptionspectrum up to the 6s threshold, but no experimentaldata are yet available to check those Ca predictions. Thegood agreement between theory and experiment for theenergy positions of the Ba 4fnf levels with J56, J54,and J55 is somewhat surprising for such a complexcase. As in some of the lower-energy spectra alreadydiscussed, the spectrum is dominated by strong electroncorrelations and strong spin-orbit effects; this higher-energy range involves a much larger number of interact-ing channels; in addition the centrifugal barrier, sensi-tively controls the nature of the Ba f orbitals.More recently, eigenchannel R-matrix studies were

performed on several autoionizing Rydberg series of Ba

in the energy range between the 6p and 7p thresholds(Aymar et al . , 1994; Lecomte et al . , 1994, 1995; Luc-Koenig, Aymar, and Lecomte, 1994; Luc-Koenig,Lecomte, and Aymar, 1994; van Leeuwen et al . , 1995,1996). These studies were motivated primarily by recentexperiments, and will be discussed later in Secs. IV.C,IV.D, and IV.E. Nevertheless, we point out here thatthe construction of an effective reaction matrix Keff hasgreatly helped in analyzing the perturbations of the6dng J56 series by several 4fnf and 4fnh levels (Ay-mar et al . , 1994). This analysis also helped to identifythe 14 doubly excited levels that cause localized pertur-bations of the 7sng 1,3G4 Rydberg series (Luc-Koenig,Lecomte, and Aymar, 1994).We conclude from the two preceding sections on

energy-level positions that Lu-Fano plots permit a uni-fied and intuitive picture of the channel-mixing patterns.They convey the interaction strength in a simple mannerthat brings out aspects of the short-range dynamics com-mon to the global level pattern of an atom. TheR-matrix method in turn can be used to predict channel-mixing parameters, giving reliable energies and classifi-cations even for very complicated perturbed Rydbergseries. In addition, the R-matrix method bypasses diffi-culties encountered in empirical MQDT analyses ofspectra such as the large number of undeterminedchannel-interaction parameters and the strong energydependence of the smooth scattering parameters that isfrequently caused by isolated perturbers. Furthermore,by constructing an effective reaction matrix restricted toclosed channels, one can extend MQDT bound-statetechniques to the energy range of autoionizing levels.We caution, however, against using single-channel quan-tum defects for the classification of such perturbed spec-tra, as is frequently done. This procedure can result inqualitatively incorrect classifications.Other familiar methods have been widely used to cal-

culate singly excited and low-lying doubly excited levelsfor all of the alkaline-earth atoms. These include themulticonfiguration Hartree-Fock studies of singletbound spectra of Ca and Sr (Froese Fischer and Hansen,1981, 1985; Vaeck et al . , 1988), the Spline-Galerkin cal-culations of Brage and Froese Fischer (1994a, 1994b)and the Wigner-Eisenbud-type R-matrix calculations ofOsani et al . (1989a, 1989b, 1991) in Be, Mg, and Ca. Themulticonfiguration Dirac-Fock method was employed tocalculate energy levels of Ba (Rose et al . , 1978; Ko-tochigova and Tupizin, 1987). In contrast to these abinitio calculations just mentioned, numerous calcula-tions have used model potentials to describe the interac-tion between core and valence electrons. Energy levelsof Be and Mg were calculated using a frozen-core ap-proximation in connection with the close-couplingmethod (Norcross and Seaton, 1976; Mendoza, 1981), orwith configuration-interaction techniques (Laughlin andVictor, 1973). Ba was investigated using a model-potential approach including relativistic effects (Hafnerand Schwartz, 1978). The calculations just discussedwere restricted to low-lying levels. The resulting energyvalues have usually agreed poorly with experiment, es-



pecially in Ba. A configuration-interaction proceduredeveloped by Chang (1986), which included higher Ry-dberg states, has been applied to calculate the positionsand widths of 3pns ,nd 1Po autoionizing states of Mg.However, the applicability of this approach to the highlyperturbed and overlapping Rydberg series of the heavieralkaline earths remains to be demonstrated. To date,Chang’s method has never been used to treat systemshaving more than one open channel per symmetry.Calculations of the energy positions of higher-lying

doubly excited levels of alkaline earths are relativelyscarce. Three configuration-interaction calculations(Kim and Greene, 1988; Aymar, 1989; Morita and Su-zuki, 1990) should be mentioned. These calculations,conducted in LS coupling, used a model-potential de-scription of the core-valence interaction. The first studydetermined the n1sn1p

1P Wannier ridge states (Wan-nier, 1953) of Ca up to the 11s11p state (Kim andGreene, 1988). Energy positions of high-lying n1sn1p1P states were obtained by diagonalization of the two-electron Hamiltonian using a set of closed-type two-electron functions that vanish at the boundary r0= 120a.u. The two-electron basis functions were of the sametype as those used in streamlined eigenchannelR-matrix calculations to construct the Hamiltonian Hccintroduced in Sec. III.E. The use of a finite range basishelped to describe the mixing of ridge states with Ryd-berg states and continua. The n1sn1p states were iden-tified in the eigenvalue spectrum by inspecting theeigenvectors. The same procedure was used (Aymar,1989) to calculate the n1s

2 1S Wannier ridge states ofalkaline-earth atoms from Be through Ba. For each ele-ment, the energy positions of the n1s

2 1S states werepredicted up to the (m016)s2 state. A slightly smallerboundary radius r05 100 a.u. was used, the sphere ofradius r0 being found to be large enough to enclose thecharge distribution of the A1 (m017)s ionic state what-ever the element A. Finally, Morita and Suzuki (1990)performed a large-scale configuration-interaction calcu-lation of doubly excited n1sn2s

1S states of Ca to inter-pret their laser spectroscopy observations. This work de-parts from those performed by Kim and Greene (1988)or Aymar (1989) in that Morita and Suzuki (1990) usedtwo-electron basis functions which have zero amplitudeat r→` rather than at finite r0. The diagonalization ofthe two-electron Hamiltonian yielded the energy posi-tions of n1sn2s

1S doubly excited states with6<n1<10 and n1<n2<20. The theoretical energiesagreed well with observed energies, deviations betweentheory and experiment being generally smaller than231024 a.u. Among all the observed states, only twostates were tentatively ascribed to states having n1 =n2, with n1= 9 and 10, respectively. The energy positionsfor the n1s

2 states agreed well with those predicted byAymar (1989).The results obtained by Kim and Greene (1988), Ay-

mar (1989), and Morita and Suzuki (1990) for the Wan-nier ridge states were analyzed in terms of two-electronRydberg formulas. By viewing the pair of electrons as asingle entity attached to the grand-parent ion, different

formulas have been proposed to describe the (core)n1l1n1l2 states as members of a two-electron Rydbergseries converging to the double-ionization limit (seeRead, 1990, and references therein). The total energy ofthese states was expressed in terms of a screening pa-rameter and of a quantum-defect parameter. The two-electron Rydberg formula proposed by Read (1977) is

E52~Z2s8!2

~n12dn1l11m8!2, (4.2)

where Z is the charge of the core (here Z52), dn1l1 isthe quantum defect of the ionic series (core) n1l1, s8 isa screening constant, and the effective quantum defectm8 is a correction to dn1l1 accounting for the change ofthe penetration of the n1l1 electron into the core causedby the presence of the added n1l2 electron. An equiva-lent formula was derived by Rau (1983, 1984):

E524~Z21/42s!2

~n113/22m!2. (4.3)

Algebraically, the expressions (4.2) and (4.3) are essen-tially the same with parameters linearly related as fol-lows:

s5s8/21Z/221/4 (4.4)

and

m53/21dn1l12m8. (4.5)

Note that the m8 parameter of Eq. (4.2) is expected tobe nearly core independent in contrast to the m param-eter involved in Eq. (4.3), which concerns the pair ofelectrons.The applicability of the Rydberg formula to n1l1n1l2

states of alkaline earths was tested by Kim and Greene(1988), Aymar (1989), and Morita and Suzuki (1990)by plotting the predicted energy levels on an effectivequantum-number scale, i.e., n5(22E)21/2, as func-tions of n1 or nn1l1* 5n12dn1l1. This plot should be a

straight line with slope @A2(Z2s8)#21 or@2A2(Z21/42s)#21. The energy levels calculated for1S (Aymar, 1989; Morita and Suzuki, 1990) and 1P(Kim and Greene, 1988) Wannier states showed the ex-pected linear dependence, demonstrating that these lev-els obey the two-electron Rydberg formula of Eqs. (4.2)or (4.3). An interesting result is that the slope obtainedfor 1S Wannier states was almost the same for each el-ement Be to Ba, showing only a slight increase of 3%when going from Be to Ba. Moreover, the slope for Ba isonly 5% larger than that obtained for 1S states of He(Rau, 1984).The study of 1S Wannier states allowed an interpre-

tation of Ba and Ca experiments involving the high-lyingdoubly excited states n1l1n2l2 (l1 and l2 < 2) with largen1 values and n2.n1, i.e., on doubly excited statescalled ‘‘double-Rydberg’’ or ‘‘planetary’’ states (seeBloomfield et al . , 1984; Morita and Suzuki, 1990, andreferences therein). The observed levels were inter-



preted as members of Rydberg series converging toionic A1 n1l1 states. These series are characterized byquantum defects dn1l1 , that are almost constant whenn2@n1. Moreover, the quantum defects increase linearlywith n1 with a slope of about 0.30–0.35 for all partialwaves (l1l2) up through d waves. The independent-particle model predicts a linear dependence on n1, butwith a much smaller slope, in the range 0.20–0.25 (Ay-mar, 1989; Morita and Suzuki, 1990). Starting from theenergy values calculated for the n1s

2 1S states it waspossible to reproduce accurately the experimental re-sults obtained for the quantum defects dn1s of then1sn2s

1S series of Ba (Bloomfield et al . , 1984) and Ca(Morita and Suzuki, 1990). Viewing the n1s

2 state, as thelowest member of the one-electron Rydberg seriesn1sn2s

1S converging to the A1 n1s state, and neglect-ing the variation of dn1s with n2, a linear relationship

between dn1s and nn1s* (or equivalently n1) was found

(Aymar, 1989; Morita and Suzuki, 1990). The slope ofthe linear dependence, depending on the screening pa-rameter s8, was about 0.31–0.35. The n1 dependence ofdn1s was in much better agreement with experimentaldata than the independent-particle model prediction.This suggests that the correlation effects and the polar-ization distortion of the inner orbital induced by thepresence of the outer electron are (not surprisingly) im-portant for highly excited n1sn2s states, even those [email protected] two-electron Rydberg formulas of Eqs. (4.2) and

(4.3) are known to successfully account for the positionof Wannier states of He and H2. It is also worth notingthat generalized two-electron formulas capable of pre-dicting the positions of the other dominant 1P1

o reso-nances (n2>n1) of He and H2 were obtained by com-bining the two-electron formula of Eq. (4.2) with theone-electron Rydberg formula (Sadeghpour andGreene, 1990; Domke et al . , 1991; Sadeghpour, 1991).These generalized two-electron energy formulas ac-count, to a good approximation, for the observed energypositions of the dominant n2

(K5n122,T51)n11 1P1

o

resonances (in Lin’s notation, 1986) of He and H2, withn2>n1. At higher energies, however, different series be-gin to overlap, generating a rich spectrum dominated bymultichannel perturbers. For 1Po symmetry, this newregime occurs above the n1'5 threshold in He andabove the n1'8 threshold in H2.

C. Autoionization widths and branching ratios

A wealth of data on doubly excited levels of alkalineearths has been obtained using multiphoton or multisteplaser experiments. The excitation processes involved inthese experiments are often so complicated that it is dif-ficult to compute reliable theoretical photoionizationspectra comparable with the observed spectra. In par-ticular, as will be documented in Sec. IV.E below, nu-merous spectra have been investigated using multisteplaser experiments based on the so-called isolated-core

excitation technique (Cooke et al ., 1978). The theoreti-cal description of the observed spectra depends on thecorrect description of final-state channel mixing and ofthe state from which the doubly excited levels are ex-cited in the final step. The ability of theoretical multi-channel spectroscopy to extract information (such as en-ergy positions, widths, and classifications of autoionizingresonances) that is independent of the excitation processis particularly helpful in unraveling complex autoioniz-ing spectra. The preceding section (Sec. IV.B) illustratedhow the energy positions and identifications of the reso-nances can be obtained using the theoretical multichan-nel spectroscopy. The present section uses examples todocument the development in Sec. II.D.3. Specifically,quantitative information on the widths and decay prop-erties of autoionizing resonances emerges from a studyof the density-of-states or time-delay matrices.Our first example interprets the ‘‘anomalous’’ auto-

ionization widths of states in the 7sng 1G4 Rydbergseries of Ba that were observed by Wang and Cooke(1993) using the isolated-core excitation method. Theexperiment found that the autoionization widths remainapproximately constant instead of decreasing as n7s

23 . Inaddition, for n>15, the quantum defects increase regu-larly. These anomalies reflect the perturbation of the se-ries by localized perturbers. The even-parity J54 levelslocated between 77 500 cm21 (bottom end of the 7sngseries, i.e., for n7s5l54) and the 7s threshold werestudied using the jj-coupled eigenchannel R-matrixmethod (Luc-Koenig, Lecomte, and Aymar, 1994). TheMQDT treatment used a short-range reaction matrixK0 (see Sec. II.D.3). The energy positions, identifica-tions, and autoionization widths of J54e levels were de-termined by analyzing the energy dependence of theanalytic partial density of states introduced in Eq. (2.71).Fourteen doubly excited levels, low members of Ryd-berg series converging to the 6d and 4f thresholds, werepredicted to lie in the studied energy range and to causelocalized perturbations of the 7sng 1G4 and

3G4 Ryd-berg series. In addition, the 7sng series were affected bysome 4fnp and 7pnf doubly excited levels that werediluted into several high-lying 7sng Rydberg levels. Thecalculated energy positions and widths of 7sng 1G4 Ry-dberg levels agreed with the experimental data (Wangand Cooke, 1993). The anomalous behavior of the7sng 1G4 series for n>15 (almost constant width andgradual increase of the quantum defect) reflects the per-turbation by the 4f8p J54 levels and, in lesser extent,by the 7p5f J54 levels. Those resonances were pre-dicted to be very broad and to lie around the 7s thresh-old. The 7sng 3G4 series was predicted to be perturbedby the 4f5f 3G4 level, which induces a narrowing of the7sng 3G4 widths around n517.Figure 19 shows how the characteristics of the reso-

nances emerge from analysis of the energy dependenceof the densities of states in closed channels. The threecurves of Fig. 19 are associated with the 7sng , 4fnf , and6dnd closed channels in the energy range correspondingto the low-energy end of the 7sng series. Eachdsn1l1l2

o (E) curve was obtained by summing the analytic



partial densities of states [Eq. (2.71)] corresponding tojj-coupled fragmentation channels i5$n1 ,l1 ,j1 ,l2 ,j2%with the same quantum numbers n1 ,l1 ,l2. Theds7sg

0 (E) curve exhibits the 7sng J54 resonances for5<n<9. The extremely broad width for the 7s5g 1G4level at 79 100 cm21 is ascribed to the strong couplingwith the broad 4f7p (or 7p4f) 1G4 resonance. The7s5g 3G4 resonance appears at 79 450 cm21. The ir-regular evolution along the series of the fine-structuresplitting between both 7sng 1,3G4 levels derives fromthe perturbation of the series by numerous doubly ex-cited levels. Three 4f2 resonances (numbers 2, 5, and 6)show up in the curve ds4ff

0 (E) and the three peaks (1, 3,and 4) occurring in the ds6dd

0 (E) profile are associatedwith 6d7d resonances.It was shown, in Sec. IV.B above, that the eigenchan-

nel R-matrix calculations of high-lying 4fnf levels(n>8) with J=4–6 reproduce well the measurements ofde Graaff et al . (1992). The calculations have been ex-tended to the unobserved low-lying levels of the 4fnfseries down to the 4f2 and 4f5f levels. Indeed, becauseof the strong mixing between the 4f5/2nf7/2 and4f7/2nf5/2 Rydberg series, the quantum defect (mod 1) ofhigh-lying 4fnf levels is strongly dependent on the lev-els. The unambiguous determination of the n value of4fnf levels observed by de Graaff et al . (1992) requiresthat the series be followed down to their bottom end.The energy positions, identifications, and widths of 4f2

and 4fnf levels (up to n58) were obtained, as for7sng Rydberg levels, by analyzing the energy depen-dence of the total and partial analytical densities ofstates (Luc-Koenig, Aymar, and Lecomte, 1994). Eigen-channel R-matrix calculations were carried out using ei-ther the LS-coupled approach combined with the jj-LS frame transformation or the jj-coupled method. The4f2 levels were found to be well separated from theother resonances. In contrast, many 4fnf levels(5<n<8) were embedded into high-lying members of7snl (l54,6) and 6dnl (l52,4,6) Rydberg series. Be-

cause of their coupling with Rydberg series, those 4fnflevels give rise to very intricated structures correspond-ing to complex resonance patterns (Giusti-Suzor andLefebvre-Brion, 1984) with an apparent width related tothe spectral range over which the Rydberg series is per-turbed by the 4fnf interloper. A good recipe for deter-mining the unperturbed width or apparent width of dou-bly excited levels embedded into Rydberg series is totreat the corresponding Rydberg series as open channelsin the MQDT calculation and to analyze the energy de-pendence of the density of states associated with thedoubly excited levels. Artificially opening a closed chan-nel in the MQDT calculation removes all the perturbersassociated with that channel. This procedure was used topredict the energy positions and apparent autoionizationwidths of 4fnf levels with J=4–6 and 5<n<8 and thusto determine the n values of high-lying 4fnf levels.Recalling Sec. II.D.3, Smith’s (1960) time-delay ma-

trix [Eq. (2.69)] can be used to analyze resonances. Thedecay properties of some autodetaching Feshbach reso-nances of H2 were obtained by Sadeghpour et al .(1992) with an eigenchannel R-matrix calculation of thepartial photodetachment cross sections, followed by ananalysis of the eigenvectors of the time-delay matrix.The same procedure has also been used by Sadeghpourand Cavagnero (1993) to study the formation and decayof one prominent autoionizing resonance of He whosedecay mode violates radial-correlation propensity rulesatisfied by all other autoionizing resonances observedin He (for references, see Domke et al . , 1991).The positions, autoionization widths, branching ratios,

and identifications of the resonances can also be pre-dicted using the alternative eigenchannel R-matrix ap-proach described in Sec. III.H (Lecomte et al . , 1994).This method is adapted to the analysis of the lowestlevels of Rydberg series described by doubly excitedwave functions Fr completely included within theR-matrix reaction volume; moreover, the doubly excitedlevels are assumed not to interact with other Rydbergseries. Lecomte et al . (1994) introduced a density-of-states matrix

Qp52pxr†xr , (4.6)

expressed in terms of the matrix xr of Eq. (3.52b) whoseelement xai gives the amplitude of the doubly excitedstate Fa in the ith eigensolution of the transformedeigensystem [Eq. (3.52a)] satisfying the outgoing-wavenormalization condition.The resonant structures showing up in the autoioniz-

ing spectra can be analyzed from the energy dependenceof Tr(Qp), which defines a density of states ds(E). Thistotal density of states can be decomposed in partial den-sities of states

dsa~E !52p(ixaixai* , (4.7)

related to a single doubly excited states Fa .Lecomte et al . (1994) have shown that, under the as-

sumption of negligible energy dependence of the non-

FIG. 19. Densities of states (in a.u.) for the even-parity J54states of Ba as functions of the energy relative to the Baground state: full line, ds7sg

0 density of states exhibiting 7sngresonances with 5<n<9; dashed line, ds4ff

0 density of statesshowing the 4f2 3H4 (2), 1G4 (5), and 3F4 (6) resonances;dotted line, ds6dd

0 density of states showing the 6d7d 3G4 (1),3F4 (3), and

1G4 (4) resonances. (From Luc-Koenig, Lecomte,and Aymar, 1994.)



resonant scattering matrix S(0), and thus of the couplingmatrix T [Eq. (3.54)], the density-of-states matrix Qp isidentical to the time-delay matrix Qphys of Smith (1960)defined by Eq. (2.69). (Let us emphasize that, inasmuchas the R-matrix calculation is restricted to open chan-nels, the time-delay matrix Qphys is identical to the short-range time-delay matrix Q .) This allows one to ascribeunambiguously each resonance appearing in the evolu-tion of Tr(Qp) with energy to a particular doubly ex-cited state Fa , and thus to identify every resonance.The 6d2 J=0–4 resonances of Ba, located above the

6p threshold, were analyzed in this way, choosing aR-matrix reaction volume (r0= 40 a.u.) large enough tocontain the 6d2 levels. The doubly excited levels withinthe R-matrix box were identified as eigenstates of theclosed part Hcc of the two-electron Hamiltonian (seeSecs. III.E and III.H). Calculations were carried out inLS and jj couplings. For a given symmetry, the R matrixwas restricted to open channels converging onto the6s , 5d , and 6p thresholds. Figure 20 displaysjj-coupled results for the 6d2 3F4 and

1G4 resonances,which interact with 18 continua. Figure 20(a) shows theenergy dependence of the trace of the time-delay matrixQ and of its eigenvalues q . Among the 18 eigenvalues,only two have non-negligible value. The two peaks vis-ible on the full line near 72 200 cm21 and 73 300 cm21

are associated with the 6d2 3F4 and 1G4 resonances,

which are well isolated from the other resonances. Theirseparation in energy being larger than their widths, theycan be viewed as ‘‘quasi-isolated’’ resonances. The ener-gies of the resonances correspond to the maxima of theTr(Q) profile and the widths can be deduced from thefull widths at half maximum of this profile.Figure 20(b) shows the results obtained with the den-

sity matrix Qp of Eq. (4.6). More precisely, it displaysthe energy dependence of the partial density of statesdsa(E) defined in Eq. (4.7) for each individual 3F4(dashed curve) and 1G4 (dotted curve) levels denoted 1and 2, respectively. Although the dotted curve corre-sponding to the 1G4 peak has a maximum also at theposition of the prominent 3F4 resonance, this secondarypeak has a very small amplitude; this shows that the tworesonances are almost uncoupled. The full curve corre-sponds to the sum ds1(E)1ds2(E) of the individualcontributions of these two 6d2 J54 levels to the trace ofthe matrix Qp defined by Eq. (4.6). The curve associatedwith the full line of Fig. 20(b) differs only slightly fromthe curve of Fig. 20(a) corresponding to the trace of thematrix Q . Part of the differences results from the ap-proximations (energy independence of the nonresonantcontinua and of the matrix T) underlying the identityQ5Qp. However, the differences primarily derive fromthe fact that the contributions to Qp due to doubly ex-cited states other than the 6d2 levels are neglected. Thelargest difference (;20%) occurring at the 1G4 reso-nance peak was related to the neglect of the 7p4f 1G4level whose resonance is extremely diffuse and contrib-utes to the time-delay matrix in a very wide energyrange (see Luc-Koenig, Lecomte, and Aymar, 1994).One major advantage of the resonance analysis in

terms of the partial densities of states rather than withthe time-delay matrix is the possibility of obtainingquantitative information on each individual resonance(including their identification) involved in complexstructures corresponding to overlapping resonances. Fig-ure 21 shows a complicated situation involving the twooverlapping 6d2 3P0 and

1S0 resonances and the close-lying 7s8s 1S0 doubly excited level. Figure 21(a) showsthe energy profiles of the trace and eigenvalues of theQ matrix calculated for J50e. In the higher-energyrange, only one eigenvalue has a nonzero value, indicat-ing that the 7s8s 1S0 level corresponds to an isolatedresonance. In contrast, in the lower energy range, twoeigenvalues have non-negligible values and their profilesdo not correspond to Lorentzian profiles, which meansthat the 6d2 1S0 and 3P0 resonances are interacting.The positions and widths of each individual resonancecannot be extracted from such curves. These character-istics can be obtained by analyzing the energy depen-dence of the partial density of states [Eq. (4.7)] for eachresonance. The jj-coupled results obtained for the 6d23P0 (dashed curve) and 1S0 (dotted curve) resonancesare shown on Fig. 21(b). The total density of state (fullline) obtained by adding the contributions of each levelis identical to that obtained from the time-delay matrixQ , which means that the nonresonant continua and thematrix T are energy independent [Eq. (3.54)]. In con-

FIG. 20. The Ba 6d2 3F4 and 1G4 resonances: (a) sum ofeigenvalues q of the time-delay matrix Q identical to TrQ (fullline) and individual eigenvalues q (18 other lines)—note thatthe curves associated with 16 eigenvalues are almost superim-posed with the horizontal axis q50; (b) trace of the densitymatrix Qp [Eq. (4.6)] (full line) and partial densities of statescalculated with Eq. (4.7) for the 3F4 (dashed line) and 1G4(dotted line) level. Energies are relative to the Ba groundstate. (From Lecomte et al . , 1994.)



trast, differences between the curves associated with theindividual contributions [Fig. 21(b)] and the energy pro-files of the eigenvalues of the matrix Q [Fig. 21(a)] aremore marked than for the J54 case.The probabilities for each 6d2 resonant state to decay

into the different available continua, i.e., the autoioniz-ation branching ratios, were predicted by anLS-coupled R-matrix calculation of the eigenvector ofQ associated with its largest eigenvalue (see Sec. II.D.3).This procedure, valid only for isolated resonances, couldbe applied because for each LS symmetry there is onlyone 6d2 resonance. Branching ratios calculated in thisway are expected to be valid for all isolated and quasi-isolated 6d2 resonances, which are well described inLS coupling.The eigenchannel R-matrix calculations of the 6d2

(Lecomte et al . , 1994), 5d5g (Luc-Koenig et al . , 1995),and 4f7h (van Leeuwen et al . , 1995) levels as well asthose of 6dng ,ni (Aymar et al . , 1994), 7sng (Luc-Koenig, Lecomte, and Aymar, 1994), 4fnf (Luc-Koenig,Aymar, and Lecomte, 1994), and 4fng (van Leeuwenet al . , 1996) Rydberg series accounted for all the rel-evant continua n1l1j1el2j2 (l2<8) built on Ba1 6s , 5d ,6p , . . . . The continua to which each level or series pref-erentially decays were identified using three differentprocedures centered around three different calculatedquantities: the partial photoionization cross sections, the

eigenvectors of the time-delay matrix, or phase-shiftedMQDT parameters. The branching ratios were found todepend strongly on the specific level or series studied.There is no general rule that dictates which multipoleterm will predominate in autoionization caused by theCoulomb interaction. In many cases, resonances au-toionize with large probability into channels associatedwith the lowest (6s and 5d) thresholds; this result is atvariance with the statement (often found in the litera-ture) that resonances autoionize preferentially intochannels attached to the energetically closest ion core.R-matrix calculations that omit some channels built onlow-lying thresholds must be viewed with caution, asthey can give unreliable resonance widths and profiles.Relatively few calculations of autoionization widths in

the alkaline earths exist using theoretical approachesother than the eigenchannel R-matrix method. One isthe configuration-interaction calculation of the positionsand widths of 3pns ,nd 1P resonances of Mg (Chang,1986). Bhalla et al . (1990) and Hahn and Nasser (1993)calculated the positions and widths of several doubly ex-cited states of Ba using the multiconfiguration Hartree-Fock method. Theoretical quantum defects and widthscalculated for n1sn2s

1S and n1sn2d1D Rydberg levels

converging to high-lying Ba1 n1s (n1;7–10) levelswere compared with the experimental data obtained forthese series by Bloomfield et al . (1984). The n1 depen-dence of quantum defects calculated by Bhalla et al .(1990) agrees excellently with experimental results; thecalculated widths, however, disagree strongly with themeasured widths. The agreement between the results ofHahn and Nasser (1993) and the experiment is ratherpoor for both positions and widths.Autoionizing widths for some doubly excited levels of

Ba with a large orbital momentum l2 for the outer elec-tron were calculated by Jones and Gallagher (1988),Poirier (1988, 1994), and Wang et al . (1991) assuming asingle-configuration model for describing the doubly ex-cited levels. Those authors used a Ba1 orbital for theinner electron. Hydrogenic wave functions were used byJones and Gallagher (1988) and Poirier (1988; 1994) forthe outer electron, while Wang et al . (1991) used for thef wave involved in 6pnf levels wave functions that werecalculated in the field of the Ba11 core screened by theinner electron. These calculations reproduce the experi-mental data well for 6pnf levels (Wang et al . , 1991) and6pnl2 levels with l2>4 (Jones and Gallagher, 1988).Calculations show that the dominant decay path for6pnl2 levels with l2>4 corresponds to dipole-allowedautoionization and accurately reproduce the observedrapid decrease of autoionization rates with l2 for theselevels.The narrow autoionization widths of various Ba

5d5g levels located below the 5d3/2 threshold were mea-sured accurately by van Leeuwen et al . (1994). Figure22 shows an example of the 5d5/25g@K# levels, where thesingle-configuration model fails to reproduce the experi-mental widths. These jK-coupled levels, with nonover-lapping valence electrons, decay through the quadrupoleinteraction into the 6sel continua. The single-

FIG. 21. The Ba 6d2 J50 and 7s8s 1S0 resonances: (a) sum ofeigenvalues q of the time-delay matrix Q identical to TrQ (fullline) and individual trace (full line) and individual eigenvaluesq (5 other lines)—note that the curves associated with threeeigenvalues are almost superimposed with the horizontal axisq50; (b) trace of the density matrix Qp [Eq. (4.6)] (full line)and partial densities of states calculated with Eq. (4.7) for the6d2 3P0 (dashed line) and 1S0 (dotted line) levels. Energiesare relative to the Ba ground state. (From Lecomte et al . ,1994.)



configuration model gives autoionization widths system-atically larger than the experimental data (full squares inFig. 22). A detailed investigation of the 5d5g autoioniz-ation mechanisms was performed using the jj-coupledeigenchannel R-matrix approach (Luc-Koenig et al . ,1995). The autoionization widths were derived from thelinewidths of the resonant peaks occurring in the calcu-lated cross section and density-of-states profiles andfrom phase-shifted MQDT parameters. Widths calcu-lated using those two different procedures agree. Twopolarization effects were found to significantly influencethe autoionization dynamics of the 5d5g states. The firsteffect was the direct polarization of the inner valenceelectron by the outer valence electron. This is a dipolarcoupling between the 5dng and the 6pnl channels and,to a lesser extent, between the 5dng channels and the4fnl channels, followed by autoionization through the6pnl and 4fnl channels. This effect was accounted for inthe calculations by including a large number of stronglyclosed channels (the same type included routinely ineigenchannel R-matrix calculations). This direct polar-ization effect significantly narrowed the 5d5g levels withK>7/2, their widths becoming 1.5 to 3 times smallerthan the experimental values (full triangles in Fig. 22).The second significant effect that played a critical role inthe 5d5g autoionization process arose from the interac-

tion of each valence electron with the electric dipolemoment induced in the Ba11 core by the other valenceelectron. This dielectronic polarization correction [Eq.(3.34)] had the same angular dependence as the dipolarinteraction but had an opposite effect; this correctionpartly canceled the polarization of the inner electron di-rectly produced by the outer electron. The widths calcu-lated with both polarization effects included (full circlesin Fig. 22) agreed well with the experiment. This studywas the first one to show that the dielectronic polariza-tion correction can have a significant influence on theautoionization widths of neutral atoms. Its crucial role innegative ions was previously demonstrated by Thummand Norcross (1991, 1992) who investigated the Cs2

negative ion using a fully relativistic Wigner-Eisenbud-type R-matrix calculation.

D. Photoionization from ground and low-lying states

1. Ground-state photoionization in lighter alkaline earths Beand Mg

Various theoretical approaches have been widely usedto investigate ground-state photoionization of Be andMg including double electron excitations. Among them,there are the configuration-interaction technique (Batesand Altick, 1973), the close-coupling method (Dubauand Wells, 1973; Mendoza and Zeippen, 1987), the mul-ticonfiguration Tamm-Dancoff approximation (Radejo-vic and Johnson, 1985), the complex-basis-function tech-nique (Rescigno, 1985), the L2 technique (Moccia andSpizzo, 1989; Chang and Tang, 1992), the multiconfigu-ration Hartree-Fock method (Froese Fischer and Saha,1987), many-body perturbation theory (Altun, 1989), therandom-phase approximation (Chi et al . , 1991; Chi andHuang, 1994), the hyperspherical close-coupling method(Greene, 1981; Zhou and Lin, 1995), the Wigner-Eisenbud-type R-matrix method (Tully et al . , 1990),and the eigenchannel R-matrix method (O’Mahony andGreene, 1985). This last study was the first to demon-strate the ability of the eigenchannel R-matrix methodto handle doubly excited levels of alkaline earths.It is beyond the scope of this review to compare the

results obtained with all these different methods. Wecompare only some unpublished eigenchannelR-matrix results with certain recent predictions andavailable experimental data. O’Mahony and Greene(1985) carried out an LS-coupled three-channel treat-ment (m0snp , m0pns , and m0pnd) of the

1Po spectraof Be and Mg below the m0p threshold. Because theseearly calculations used small sets of two-electron func-tions and a relatively crude unoptimized Hartree-Slaterpotential, we performed new LS-coupled eigenchannelR-matrix calculations of the photoabsorption spectra ofBe and Mg improving both aspects. The differences be-tween old and new results concern mainly the absolutevalues of cross sections, but not the shapes and positionsof m0pns and m0pnd

1P resonances. The new Be ve-locity result @Fig. 23(a)] is to be compared with recenttheoretical predictions of Chi et al . (1991) @Fig. 23(b)].These authors used the multiconfiguration relativistic

FIG. 22. Autoionization widths of 5d5/25g levels of Ba. Thewidths calculated by Luc-Koenig et al . (1995) using differentapproximations (G th) are compared with the experimental val-ues (Gexp) of van Leeuwen et al . (1994): full squares, single-configuration model; full triangles, jj-coupled eigenchannelR-matrix calculation accounting for the direct polarization ofthe inner electron by the outer electron; full circles,jj-coupled eigenchannel R-matrix calculation accounting forthe direct polarization of the inner electron by the outer elec-tron and for the dielectronic polarization term of Eq. (3.34).The ratios G th /Gexpt obtained for the jK-coupled 5d5/25g@K#levels are plotted as functions of K . The l value of the 6selcontinuum toward which each 5d5/25g@K# level autoionizes isindicated below the K values.



random-phase approximation theory (MCRRPA),which is a generalization of the relativistic random-phase approximation including a multiconfigurationwave function for the ground state. Both R-matrix andMCRRPA calculations give a similar description of thebroad 2pns 1Po series, and of the narrow 2pnd 1Po

series, which appear owing to the 2p2-2s2 mixing in theground-state wave function. The positions of resonancesas well as the cross sections at the 2pns peaks, agreewell. Slight differences are visible in the wavelengthrange of the 2p3d and 2p4s resonances, where theR-matrix calculation predicts a stronger interaction be-tween the 2pns and 2pnd channels than the MCRRPAcalculation. However, the one available experiment isnot precise enough to determine which calculation isbetter (Mehlman-Ballofet and Esteva, 1969).Figure 24 compares the new R-matrix result for the

Mg photoabsorption spectrum @Fig. 24(a)] with previoustheoretical and experimental results @Fig. 24(b)]. Onlyvelocity results are shown on @Fig. 24(a)], as the lengthand velocity results for the cross section agree to withinbetter than 2%. In Fig. 24(b), the solid line correspondsto the prediction of Altun (1989), who used the many-body perturbation theory; the dot-dashed and dashedcurves show the experimental results of Ditchburn andMarr (1953) and of Preses et al . (1984), respectively.The relative measurements of Preses et al . (1984) werenormalized to the many-body perturbation theory re-sults at the first resonance 3p4s 1P . Like the homolo-gous Be resonances, the Mg 3pns resonances are broadand the 3pnd narrow. The energy range (7.6–10.6 eV)close to the 3s ionization threshold is very difficult tocalculate accurately owing to the appearance of a Coo-per minimum in the 3s→ep transition. In Be, this Coo-per minimum had occurred in the discrete spectrum.Various theories (Altun, 1989; Chang and Tang, 1992;

Chi and Huang, 1994) predicted quite different crosssection values at the threshold and for the lowest 3p4sresonance. The experimental threshold value of 2.39 60.72 Mb (Yih et al . , 1989) agrees with this newR-matrix result of 2.5 Mb and with the calculation ofChang and Tang (1992), who obtained ; 2.4 Mb. Theseauthors employed a simple configuration-interaction ap-proach using a finite L2-basis set constructed usingB-splines and frozen-core Hartree-Fock orbitals.A key result of the eigenchannel R-matrix and

hyperspherical-coordinate studies of O’Mahony andGreene (1985) and of Greene (1981) is that near them0s threshold the wave functions of Be and Mg statesare analogous to the wave functions describing the2snp62pns states of He and H2 (Cooper et al . , 1963).They demonstrated that the m0pns

1P autoionizingresonances of Be and Mg could be described by wavefunctions that admix the m0sep open channel and the

FIG. 23. Photoabsorption spectrum of Be below the 2pthreshold as a function of the photon wavelength. (a) TheLS-coupled eigenchannel R-matrix result; (b) the MCRRPAprediction of Chi et al . (1991). [Figure 23(b) is from Chi etal . (1991), courtesy of K. T. Cheng.]

FIG. 24. Photoabsorption spectrum of Mg below the 3pthreshold. (a) The LS-coupled eigenchannel R-matrix result;(b) the many-body perturbation theory prediction of Altun(1989) (full line), with experimental results of Ditchburn andMarr (1953) (dot-dashed line) and of Preses et al . (1984)(dashed line). The zero of the cross section near 9.0 eV arisesfrom a Cooper minimum in the 3s→3p transition. [Figure24(b) is from Altun (1989), courtesy of Z. Altun.]



m0pns closed channel with equal amplitude. This ‘‘eq-uipartition’’ of the wave function into the so-called plusand minus eigenmodes is reflected by the large width ofm0pns

1P resonances of Be and Mg, visible in Figs. 23and 24. In fact, these resonances are so broad that thesedoubly excited states decay after only half of a ‘‘Bohr-type orbit’’ is completed. This equipartition, and thestrong similarities between the 1Po spectra of Be andMg, are reflected in the short-range scattering param-eters obtained in a two-channel treatment of them0snp and m0pns channels. As visible on Figs. 1 and 6of O’Mahony and Greene (1985), in Be and Mg the mix-ing angle goes to zero at an energy far below the m0sthreshold and increases to approximately p/4 at higherenergies where the two m0snp and m0pns channels areessentially equally mixed. The variation of the mixingangle between the m0snp and m0pns channels as func-tion of the energy for Be and Mg shows striking similar-ity with the variation of the mixing angle between them0snp and (m021)dnp 1P channels of Ca, Sr, Ba, andRa below the m0s threshold displayed in Fig. 10. Therealso, the mixing is reminiscent of the so-called plus andminus states of H2 and He, in that the two correspond-ing 1P eigenmodes of heavy alkaline earths Ca to Raapparently have similarly strong coupling implied by thestructure (mo21)dnp6mosnp .

2. Ground-state photoionization in heavier alkaline earthsCa to Ra

The heavier alkaline-earth atoms have been experi-mentally investigated far more extensively than have Beand Mg. A wealth of experimental data on them0s

2→J51o photoionization spectra of Ca, Sr, and Bain the region below the m0p threshold has been re-ported during the last three decades. Earlier spectrawere obtained in photoabsorption using light from con-ventional sources or synchrotron radiation (Garton andCodling, 1960, 1968; Garton and Tomkins, 1969b; Brownet al., 1973, 1983; Hudson et al . , 1969, 1970; Gartonet al.., 1974; Brown and Ginter, 1978, 1980; Conneradeet al., 1980). Spectral structures were detected with pho-tographic plates or photoelectric detectors. Recently thequality of the spectra has improved, especially in thespectral resolution, through the use of improved detec-tion techniques (Griesmann et al., 1988, 1992, 1994;Griesmann, 1990; Abutaleb, de Graaff, Ubachs, Hoger-vorst, and Aymar, 1991; Connerade and Farooqi, 1991;Farooqi et al . , 1991, 1992). Many eigenchannelR-matrix calculations have treated photoabsorptionspectra of the heavy alkaline-earth atoms below themop threshold. Earlier calculations carried out for the1Po spectra of Ca (Greene and Kim, 1987) and Sr (Ay-mar, 1987) treated these atoms in LS coupling and in-cluded five LS-coupled fragmentation channels:m0snp , (m021)dnp , (m021)dnf , m0pns , andm0pnd . Then, more elaborated calculations includingspin-orbit effects, either with the jj-LS frame transfor-mation or directly in the variational calculation wereperformed for Ca (Kim and Greene, 1987; Ueda et al.,

1990; Farooqi et al . , 1991), Sr (Aymar et al . , 1987;Greene and Aymar, 1991; Farooqi et al . , 1992), Ba (Ay-mar, 1990; Abutaleb, de Graaff, Ubachs, Hogervorst,and Aymar, 1991; Greene and Aymar, 1991) and Ra(Greene and Aymar, 1991). When combined with theframe transformation, the LS-coupled R-matrix calcula-tions determine three LS-coupled reaction matricesK

1P, K3Pand K

3Dof dimension 5, 5, and 3, respectively.Channels involved for the 1Po and 3Po are the sameand for the 3Do symmetry the LS-coupled channels are:(m021)dnp , (m021)dnf , and m0pnd . The K

LS matri-ces are recoupled into a single jj-coupled reaction ma-trix using the jj-LS frame transformation. The MQDTcalculations performed with the jj-coupled reaction ma-trix obtained from R-matrix calculations in either LS orjj coupling include 13 jj-coupled fragmentation chan-nels: m0snp1/2 , m0snp3/2 , (m021)d3/2np1/2 ,(m021)d3/2np3/2 , (m021)d3/2nf5/2 , (m021)d5/2np3/2 ,(m021)d5/2nf5/2 , (m021)d5/2nf7/2 , m0p1/2ns1/2 ,m0p1/2nd3/2 , m0p3/2ns1/2 , m0p3/2nd3/2 , andm0p3/2nd5/2 .A few theoretical studies of the photoabsorption spec-

tra of Ca, Sr, or Ba were previously carried out usingmany-body perturbation theory (Altun et al . , 1983; Fryeand Kelly, 1987) or the Wigner-Eisenbud-typeR-matrix formulation (Scott et al . , 1983; Bartschat etal . , 1986). Here, we compare some eigenchannelR-matrix results with some of those previous calcula-tions, but we focus mainly on the the comparison withexperimental spectra. Most of the theoretical spectrapresented in this paper correspond to total photoioniza-tion cross sections versus the photon energy or wave-length, without any convolution over the finite experi-mental resolution. Consequently, no meaningfulcomparison can be made between experiment andtheory for the intensity of narrow peaks.Figure 25 compares the theoretical spectrum obtained

between the 4s and 3d thresholds of Ca using thejj-coupled R-matrix method with the experiment of Fa-rooqi et al . (1991). The theoretical curve in Fig. 25(a) isthe geometric mean of the length and velocity curves.The experimental spectrum @Fig. 25(b)] was obtainedusing a tunable coherent vacuum-ultraviolet four-wavemixing source and a thermoionic diode detection. Abso-lute cross sections were obtained by calibrating datawith previous absolute measurements (Parkinson etal . , 1976; McIlrath and Sandeman, 1972). The absorp-tion lines were assigned to 3dnp and 3dnf resonances,members of six series converging to the 3d3/2,5/2 thresh-olds, and perturbed by the 4p5s 1,3P1 resonances(Brown et al . , 1973). The broadest resonances corre-spond to the 3dnp 1P1 levels strongly coupled to the4ses 1P1 continuum. The strong mixing between themosnp and (mo21)dnp 1P channels which dominatesthe bound spectra of all the heavy alkaline-earth atoms(see Sec. IV.A.1) manifests itself also above the mosthreshold. Overall, the agreement between theory andexperiment is remarkably good. The eigenchannelR-matrix calculation closely reproduces the irregularvariation of the widths and intensities along the 3dnp



1P1 series. Some of the corresponding resonances aremarked in Fig. 25(b) by their n values. The 3d6p 1P1resonance at l;177 nm is anomalously narrow and in-tense compared to the highest members of the series.The width irregularity was attributed to the influence ofthe 4p5s 1P1 perturber located in between the 3dnp1P1 resonances with n56 and 7 (Greene and Kim, 1987;Farooqi et al . , 1991; Connerade, 1992). More precisely,destructive interference results from cancellation be-tween the two alternative autoionization paths3dnp→4sep and 3dnp→4pns→4sep (Greene andKim, 1987). Figure 25(a) includes weak features due tothe spin-orbit interaction, though the earliest eigenchan-nel R-matrix calculation of Greene and Kim (1987)showed that an LS-coupled R-matrix treatment gives agood description of the 1P resonances.Two previous calculations were conducted in Ca in

the same energy range, one using the many-body pertur-bation theory (Altun et al . , 1983) and one using theWigner-Eisenbud-type R-matrix formulation (Scott etal . , 1983). Both calculations, which treated Ca in LScoupling and thus dealt with the 1P resonances only,showed distinctly poorer agreement between theory andexperiment for the 1P resonances. The departure of theR-matrix results of Scott et al. (1983) from experimentmay derive from their use of an ab initio all-electrondescription of the Ca11-e interaction as opposed to thesemiempirical potential used in the eigenchannelR-matrix studies (Greene and Kim, 1987; Farooqi etal . , 1991).Many-body perturbation theory was also used by Frye

and Kelly (1987) to calculate the photoabsorption of Srin the analogous energy range, i.e., below the 4d thresh-old. As in Ca, spin-orbit effects were neglected and the

description of the 4dnp and 4dnf 1P resonances is, notsurprisingly, poorer than that obtained with eigenchan-nel R-matrix calculations where fine-structure effectsare included with the jj-LS frame transformation (Ay-mar, 1987).Figure 26 compares the photoabsorption spectrum of

Ca and Sr between the (m021)d5/2 and m0p3/2 thresh-olds. For Ca, the measurements of Griesmann et al.(1988) in Fig. 26(a), are compared with the frame-transformation results of Kim and Greene (1987), shownin Fig. 26(b). For Sr, the measurement of Griesmann(1990) [Fig. 26(c)] is compared with the jj-coupledeigenchannel R-matrix results of Greene and Aymar(1991) [Fig. 26(d)]. Measurements of Griesmann et al .(1988, 1994) and of Griesmann (1990) were obtained us-ing synchrotron radiation and a linear thermoionic diodedetector. In Ca @Fig. 26(a)], as for Fig. 25, absolute cali-bration was achieved using previous absolute measure-ments. In Sr, the experimental data have been normal-ized to achieve agreement with the R-matrix result atthe 4d6p 1P1 resonance which dominates the photoion-ization spectrum between the 5s and 4d3/2 thresholds(see Greene and Aymar, 1991).In both Ca and Sr, the calculations reproduce the

shapes of the observed m0pns and m0pnd J51 reso-nances. In Ca, agreement between experiment andeigenchannel calculations remains superior to that ofcalculations performed by Scott et al . (1983) or by Al-tun et al . (1983).Concerning the measured cross sections, calibrated

as described above, the Ca 4pns peaks are slightlytoo weak compared to R-matrix results, while the Sr5pns resonances are too intense. Recently, Griesmannet al. (1994) have calibrated their data (Griesmann,

FIG. 25. Photoabsorption spectrum of Ca be-tween the 4s and 3d3/2 thresholds. (a) Thejj-coupled R-matrix result; (b) the experi-mental result (wavelength resolution ;0.02nm). Some 3dnp 1P1 resonances are labeledby their n values. (From Farooqi et al . , 1991.)



1990) with the cross section determined by Alexa etal . (1983) at the 5s threshold. In the energy range be-tween the 4d5/2 and 5p3/2 thresholds, the magnitude ofthese normalized cross sections agree well with theR-matrix cross sections. However, the experimentalcross section of the very strong 4d6p 1P1 resonance issmaller than the R-matrix result. Therefore, the normal-ization of the data of Griesmann et al . (1988, 1994) inCa and Sr is somewhat questionable and further abso-lute measurements are desirable. If the experimental

normalization is in fact correct, this discrepancy maysuggest that the energy dependence of calculated dipolematrix element needs to be described more accurately insensitive cases such as these.In both Ca and Sr, the m0pnd resonances are very

broad owing to their strong coupling with the(m021)def continua and are hardly recognizable as in-dividual structures, whereas the members of them0pns series can be clearly identified. Correlation ef-fects are very similar in Ca and Sr and major differencesbetween Ca and Sr primarily reflect the increasing roleof the spin-orbit interaction in Sr. Indeed, anLS-coupled R-matrix calculation correctly describes themain features of the Ca photoabsorption spectrum(Greene and Kim, 1987), but not of Sr. Due to the weak-ness of the spin-orbit interaction in Ca, the relative am-plitude of the two m0pns peaks corresponding to thesame n value differs strongly in Ca [Figs. 26(a) and26(b)]. That is, the Ca 3P1 resonances to the left of theintense peaks in Fig. 26(a) are very weak. The strongerspin-orbit interaction in Sr [Figs. 26(c) and 26(d)] causesthe corresponding resonances to have roughly the sameintensity.Figure 27 compares the high-resolution experimental

Ba result of Brown and Ginter (1978), just below the6p1/2 threshold, shown in Fig. 27(a), with the eigenchan-nel R-matrix results (Greene and Aymar, 1991). Figure27(b) shows the spectrum obtained with R-matrix calcu-lations in LS coupling combined with the jj-LS frametransformation. Figure 27(c) displays the jj-coupled

FIG. 26. Photoabsorption spectra of Ca and Sr between the(m021)d5/2 and m0p3/2 thresholds. (a) Ca measurement ofGriesmann et al. (1988) (wavelength resolution ; 0.05 nm);(b) Ca frame-transformation result of Kim and Greene (1987).(c) Sr measurement of Griesmann (1990) (wavelength resolu-tion ; 0.05 nm); (d) Sr jj-coupled eigenchannel R-matrix re-sult of Greene and Aymar (1991). [Figures for Ca are fromGriesmann et al . (1988) and those for Sr are from Greene andAymar (1991).]

FIG. 27. Photoabsorption spectrum of Ba in the energy rangefrom 60 000 cm21 up to the 6p1/2 threshold: (a) relative mea-surement of Brown and Ginter (1978) (energy uncertainty ;0.11 cm21) that was normalized to give the overall best agree-ment with the R-matrix results; (b) jj-LS velocity (full line)and length results (dashed line); (c) jj velocity (full line) andlength results (dashed line). The vertical bars in (b) and (c)indicate the position of the observed absorption peaks andminima. (From Greene and Aymar, 1991.)



R-matrix result. The results obtained by both eigenchan-nel R-matrix approaches are almost identical and theyreproduce the experimental spectrum accurately. In par-ticular, the periodic enhancement of the cross sectionsdue to low-lying 6p3/2nl levels mixed with 6p1/2n8l Ry-dberg levels is well described. Figures 27(b) and 27(c)display both velocity and length results (shown as solidand dashed lines) which are very close, giving some con-fidence in the convergence of the variational calcula-tions.One important point is that, even in Ba, R-matrix cal-

culations including the spin-orbit terms within the reac-tion volume and those including these terms perturba-tively through the jj-LS frame transformation givealmost identical results. This is true in the energy rangeclose to the 6p threshold but also in the whole energyrange from the 6s threshold up to the 6p3/2 threshold(see Greene and Aymar, 1991). Thus, although no de-tailed comparison between photoabsorption spectra cal-culated with either the fully jj-coupled R-matrix methodor using the jj-LS frame transformation was done forCa and Sr, the conclusions obtained in Ba are expectedto hold for the lightest elements Ca and Sr.R-matrix calculations were also carried out in the

heaviest alkaline-earth atom Ra (Greene and Aymar,1991). Because no experimental spectrum is available inthe autoionizing energy range of Ra we do not repro-duce here the predicted photoionization spectra. Weonly briefly summarize the conclusions obtained previ-ously. As in Ba, spectra predicted with LS-coupledR-matrix calculations combined with the jj-LS frametransformation were compared to those obtained withthe jj-coupled R-matrix method. In Ra, not surprisingly,significant differences between the autoionizing patternpredicted by the two approaches occur. The differencesare much more marked in the low-energy range betweenthe 7s and 6d3/2 thresholds than in the higher-energyrange between the 6d5/2 and 7p3/2 thresholds.Figure 28 compares the photoabsorption spectra for

all the heavier alkaline earths from Ca to Ra betweenthe m0p1/2 and m0p3/2 thresholds. Results in Ca (Kimand Greene, 1987) @Fig. 28(a)], Sr (Aymar, 1987) @Fig.28(b)], and Ba (Aymar, 1990) @Fig. 28(c)] were obtainedwith LS-coupled R-matrix calculations combined withthe jj-LS frame transformation, while those for Ra(Greene and Aymar, 1991) @Fig. 28(d)] were obtainedwith the jj-coupled R-matrix method. In Ca [Fig.28(a)], the theoretical result (thin line) is compared withthe experimental result of Brown and Ginter (1980)(thick line). The quasiperiodic autoionizing structuresformed by asymmetrical absorption profiles are verysimilar in all alkaline-earth atoms from Ca to Ra. Thisgreat similarity throughout the heaviest alkaline-earthatoms reflects the systematic near invariance of the elec-tronic channel mixings. Here, the strongest channel mix-ing corresponds to the mopnd-(mo21)def mixing for1P1 channels.The absolute squares of the off-diagonal elements

uSiju2 of the short-range scattering matrix defined by Eq.(2.43) provide an index of the mixing between channels

i and j . Selected uSiju2 elements obtained for Ca 1Po

(Greene and Kim, 1987) with LS-coupled eigenchannelR-matrix calculations are shown in Fig. 29 as functions

FIG. 28. Photoabsorption spectra of Ca to Ra between them0p1/2 and m0p3/2 thresholds as functions of the energy rela-tive to the m0p1/2 threshold. Results in Ca (a), Sr (b), and Ba(c) were obtained with LS-coupled R-matrix calculations com-bined with the jj-LS frame transformation, while those in Ra(d) correspond to the jj-coupled R-matrix result. In Ca, thetheoretical result (thin line) is compared with the experimentalresult of Brown and Ginter (1980) (thick line). (From Aymar,1987, 1990; Kim and Greene, 1987; and Greene and Aymar,1991.)



of the photoelectron energy between the 4s and 4pthresholds. The very large widths of 4pnd 1P levels,visible in Figs. 26 and 28(a), reflect the extremely strongmixing between the 4pnd and 3dnf channels. Such am0pnd-(m021)dnf channel mixing is shared by all theheavier alkaline earths, as visible on Fig. 28. Kim andGreene (1988) showed that this exceptionally strongchannel mixing continues to dominate the photoabsorp-tion spectrum of Ca at higher energies. The marginallyweaker, but nevertheless strong 3dnp-4snp mixing gen-

erates the broad autoionization widths of 3dnp 1P reso-nances of Ca for n Þ 6 (see Fig. 25). In this case also, thepair of channels 3dnp-4snp remains strongly coupled athigh energy. The (m021)dnp-m0snp mixing is weakerin Sr, Ba, and Ra than it is in Ca, at least below them0p threshold, but it is still far from negligible.A final example of photoabsorption spectra in the al-

kaline earths considers the energy range well above them0p threshold, where almost no experimental data areavailable to date. Figure 30 shows the Ca photoabsorp-tion spectra predicted by Kim and Greene (1988) fromthe 4p threshold up to nearly the 6s threshold. Calcula-tions were carried out with the LS-coupled eigenchan-nel R-matrix method. On Figs. 30(a) and 30(c), the up-per curve is the velocity result and the lower one thelength result. This study was the first one performedwith the eigenchannel R-matrix method at an energyhigher than the m0p threshold. For this reason this studyshed new light on fundamental aspects of R-matrix andMQDT technologies. The investigated energy range in-cluded the 5s , 4d , 5p , 4f , and 5d thresholds; the num-ber of open and weakly closed channels (up to sixteen)included in the calculation depended on the energyrange under study. The reaction volume radius was 31a.u. The resonances pertaining to the 5snp , 4dnp ,4dnf , 5pns , 5pnd , 4fnd , 4fng , and 6snp series, as wellas the lowest resonances associated with the 5dnp and5dnf series, were identified. The 5s5p and 6s6p 1P au-toionizing states, which involve two electrons with thesame degree of excitation, are the analog of the so-called plus states that dominate the photoabsorption

FIG. 29. Short-range scattering matrix elements for the 1Po

symmetry of Ca. Absolute squares of selected matrix elementsobtained with LS-coupled eigenchannel R-matrix calculationare shown as functions of the photoelectron energy betweenthe 4s and 4p thresholds. (From Greene and Kim, 1987.)

FIG. 30. Photoabsorption spec-tra of Ca predicted below the6s ionization limit withLS-coupled eigenchannelR-matrix calculations: (a) en-ergy for E between 20.32 a.u.and 20.22 a.u.; (b) for E be-tween 20.22 a.u. and 20.17a.u.; (c) for E between 20.17a.u. and 20.12 a.u.. The uppercurve is the velocity result andthe lower one the length result.Energies in a.u. are referred tothe double-ionization limit.(From Kim and Greene, 1988.)



spectrum of He (Cooper et al . , 1963). In contrast [Figs.30(a) or 30(c)], the 5s5p and 6s6p resonances are sur-prisingly weak.For ground-state photoionization, only total cross sec-

tions were measured experimentally. However, someother observables more sensitive to channel mixing andspin-orbit effects were predicted using the eigenchannelR-matrix method. In particular, Kim and Greene (1987)calculated for ground-state photoionization of Ca, in ad-dition to the total cross section, the partial cross sec-tions, the angular distributions of emitted electrons, andthe alignment of the Ca1 fragments. The energy rangeconsidered in that work was from the 4s threshold up tothe 4p3/2 threshold. Not surprisingly, the weak spin-orbitinteraction, accounted for through the jj-LS frametransformation, was seen to affect the photoelectronasymmetry parameters and the alignment parametersmuch more strongly than the total cross section.

3. Photoionization from low-lying excited states

The earliest experimental investigations on autoioniz-ing states of alkaline-earth atoms mainly dealt with theodd-parity J51 levels, which can be observed in absorp-tion from the ground state. Now, owing to the develop-ment of laser spectroscopy, a wealth of data has beenobtained for odd- and even-parity autoionizing levelswith various J values. This section concerns photoioniza-tion from low-lying excited levels only. Photoionizationspectra investigated using multistep laser experimentsbased on the the so-called isolated-core excitation (ICE)technique (Cooke et al . , 1978) will be treated in Sec.IV.E. As explained in Sec. III.F , the procedure used tocalculate photoionization spectra from low-lying states isidentical to the procedure employed for calculating pho-toabsorption spectra. Photoionization cross sections arecalculated using short-range dipole matrix elements,which only include contributions within the reaction vol-ume. This volume must be chosen to be large enough toenclose the full initial-state wave function. The use oftoo large a boundary radius r0 complicates theR-matrix calculation of final-state channel mixing. Cal-culations of photoionization spectra from low-lyingstates of alkaline earths performed to date have beenrestricted to r0<50. a.u. They dealt with various photo-ionization spectra from excited states of Ca (Aymar andTelmini, 1991; Assimopoulos et al . , 1994; Luc-Koenig,Bolovinos, et al., 1994), Sr (Kompitsas et al . , 1990, 1991;Goutis et al . , 1992), and Ba ( Greene and Theodosiou,1990; Gounand et al . , 1991; Luc-Koenig and Aymar,1991; Armstrong, Wood, and Greene, 1993; Bartschatand Greene, 1993; Telmini et al., 1993; Wood et al., 1993;Carre et al., 1994; Aymar and Luc-Koenig, 1995;Lecomte et al., 1995; Luc-Koenig et al., 1995; Lagadecet al., 1996). For illustration, we concentrate on photo-ionization from low-lying excited states of Ba. Resultson total photoionization cross sections and on other ob-servables will be discussed.

a. Total photoionization cross sections

Figure 31 deals with the 6pnp and 6pnf J51 levels ofBa, below the 6p1/2 threshold. The excitation spectrumrecorded by de Graaff et al . (1990) from the 5d6p3P0 excited state @Fig. 31(a)] is compared with the theo-retical spectrum @Fig. 31(b)] obtained with the frame-transformation treatment (Luc-Koenig and Aymar,1991). The calculated spectrum reproduces the energypositions, widths, and shapes of the observed autoioniz-ing resonances. The very narrow resonances correspond-ing to the Rydberg series 6p1/2np1/2 and 6p1/2np3/2 andto the 6p3/211p1/2 and 6p3/211p3/2 J51 levels are super-imposed on broad structures, the 6p3/2nf J51 reso-nances with n57 and 8. The calculation reproduces ac-curately the large differences between the widths of6pnp and 6pnf resonances. The extremely large widthsof 6p3/2nf J51 levels reflect the very strong coupling of6pnf channels with 5deg continua. All Ba 6pnf stateshave similar anomalously large autoionization rates,with decay times of approximately half of the Bohr orbitperiod.

FIG. 31. Photoionization spectra for the 6pnp , nf J51 levelsof Ba below the 6p1/2 threshold: (a) experimental spectrumrecorded by de Graaff et al . (1990) from the 5d6p 3P0 excitedstate; (b), (c) theoretical spectra obtained with LS-coupledeigenchannel R-matrix calculations and the jj-LS frame trans-formation corresponding to the 5d6p 3P0 and 6s6p

3P0 initialstates, respectively. The vertical bars in (b) and (c) indicate thepositions of the 6pnp J51 resonances calculated from an ef-fective short-range reaction matrix referring to the 6pnpclosed channels (see Sec. IV.B). Energies are relative to the Baground state. (From Luc-Koenig and Aymar, 1991.)



The autoionizing pattern depends on the initial statefrom which the resonances are excited. Figure 31(c) dis-plays the calculated excitation spectrum of 6pnp ,nfJ51 levels from the 6s6p 3P0 initial level. The calcu-lated cross sections differ greatly for the two initialstates. The 6pnf resonances have vanishing cross sec-tions in excitation from the 6s6p 3P0 initial level. In-deed, the direct 6s6p→6pnf excitation is not allowed,in contrast to the 5d6p→6pnf excitation.Similar dependence of the profiles of autoionizing

lines on the choice of intermediate states used in theexcitation process has been observed by Keller et al .(1991). Several 5dnd autoionizing resonances of Bawere observed in a multiphoton resonant ionization ex-periment involving either the 6snp 1P1 or the 5dnp1P1 level as intermediate level. The shape of the ob-served 5dnd resonances was seen to drastically changewith the intermediate state from which they were ex-cited.A major test case for theoretical and experimental

methods has proven to be the photoionization of theexcited Ba 6s6p 1P1 level. A large number of experi-mental ( Burkhardt et al . , 1988; Kallenbach et al . , 1988;He et al . , 1991, 1995; Keller et al . , 1991; Willke andKock, 1991, 1993; Lange, Eichmann, and Sandner, 1991)and theoretical (Bartschat and McLaughlin, 1990;Greene and Theodosiou, 1990; Bartschat et al . , 1991;Greene and Aymar, 1991; Bartschat and Greene, 1993;Wood et al . , 1993) papers have been devoted to thisexcited-state photoionization process.Figure 32 compares the jj-coupled eigenchannel

R-matrix results of Wood et al . (1993) (lower curves)with the experiment of Lange, Eichmann, and Sandner(1991) (upper curves) in the wavelength region between417 nm (6s threshold) and 355 nm. Only relative photo-ionization cross sections were measured and in Fig. 32,the experimental cross sections were normalized by op-timizing the agreement between theory and experimentin the region of the broad resonances near 372 nm. Inaddition, the theoretical spectra have been convolvedwith the experimental resolution of 0.2 cm21. Twopulsed lasers were used in the experiment. The first laserwas used to excite the 6s6p 1P1 level and the secondlaser photoionized this level. Both lasers were linearlypolarized with the polarization vectors being either par-allel [Fig. 32(a)] or perpendicular [Fig. 32(b)]. In theparallel case, the allowed final states were Jf50,2,MJf

50, while in the perpendicular case they wereJf51,2, uMJf

u51. Breakdown of these selection rulescan lead in excited-state photoionization spectra to un-expected resonances that do not belong to the final-statesymmetries selected by a given experiment. A centralpoint of the treatment of Wood et al . (1993) was theinclusion of hyperfine effects in the odd isotopes of neu-tral barium to explain the presence of unexpected fea-tures in the photoionization spectra recorded by Lange,Eichmann, and Sandner (1991). Wood et al . (1993) haveshown that the resonances marked by the arrows in Figs.32(a) and 32(b) are only present in the calculations

when hyperfine effects are taken into account. ‘‘Elec-tronically forbidden’’ resonances with Jf51 arise in Fig.32(a), while those with Jf50 arise in Fig. 32(b). Asshown by Wood et al . (1993), the cross section forphotoionization of a Je51 state from a J050 groundstate in which both exciting and photoionizing lasers arelinearly polarized has the general form

s~J0→Je→Jf!5s~ iso!~1→0 !@112gav~2 !P2~cosu!#

1s~ iso!~1→1 !@12gav~2 !P2~cosu!#

1s~ iso!~1→2 !@11 15 gav

~2 !P2~cosu!# .(4.8)

In Eq. (4.8), the factor gav(2) represents averages over

FIG. 32. Photoionization cross section of Ba 6s6p 1P1 laser-excited level as a function of wavelength of the photoionizinglaser: (a) for parallel polarization of the two lasers and (b) forperpendicular polarization of the two lasers. The baseline ofthe normalized experimental cross section (upper curves) hasbeen set to 500 Mb. Experimental resonances, which are over-lapped with theory, are labeled with horizontal bars. Thejj-coupled R-matrix cross sections include the effect of hyper-fine depolarization of the 6s6p 1P1 level. Numbers in paren-theses are peak intensities of theoretical cross sections. Theresonances marked with the arrows are forbidden by electronicselection rules and are only present in the calculation whenhyperfine effects are taken into account. (From Wood et al .1993.)



time and over the relative isotopic abundances of natu-ral Ba of an expression for the time dependence of theeffects of the quadrupolar hyperfine interaction on theJe excited state. It is a reasonable approximation to as-sume that complete depolarization of the excited stateoccurs for the Ba isotope with I53/2; in this limit onehas gav

(2)50.864. In Eq. (4.8), u is the angle between thelinear polarization vectors and P2 is a Legendre polyno-mial. The ‘‘isotropic’’ cross section s(iso)(Je→Jf) is thetotal photoionization cross section for the transitionJe→Jf that would be calculated if the excited levels wererandomly oriented, i.e., with equal population in allmagnetic sublevels MJe

. The calculation successfully re-produces the experimental data, including the forbiddenresonances. The positions and widths of the resonancesare generally very well reproduced, while some persis-tent discrepancies in the peak intensities and the reso-nance line shapes are still not resolved. As describedelsewhere (Aymar et al . , 1982, 1983; Camus et al . , 1983;Greene and Aymar, 1991) the resonances are ascribedto 5dns , nd levels and to the 6p2 1S0 level which formswith neighboring 5dnd J50 levels a complex resonancenear 372 nm. The eigenchannel R-matrix calculations(Greene and Theodosiou, 1990; Greene and Aymar,1991; Wood et al . , 1993) confirmed the identificationsdeduced from empirical multichannel quantum-defecttheory, in particular that of the 6p2 1S0 (Aymar et al . ,1982), which has been the subject of some controversy.Note finally that the isotropic eigenchannel R-matrixcross section @s(iso)(1→0)1s(iso)(1→1)1s(iso)(1 →2)]at the 6s threshold (417 mm) agrees reasonably well withthe absolute measurements of He et al . (1991) and ofWillke and Kock (1993).Apparent effects of the hyperfine interaction on

excited-state photoionization cross sections were alsofound by Armstrong, Wood, and Greene (1993). Thoseauthors studied photoionization of the 5d6p 3D1 levelboth experimentally and theoretically. There also, ex-perimental spectra were compared with jj-coupledeigenchannel R-matrix calculations and incorporation ofhyperfine effects was found to be necessary for a com-plete description of the measured cross sections. The im-portance of hyperfine interaction in resonant three-photon ionization of Ba via excited levels was alsoappreciated by Mullins, Chien, Hunter III, Keller, andBerry (1985) and by Hunter et al . (1986); hyperfine-interaction effects were incorporated in the theoreticaldescription in these papers.Isotropic photoionization cross sections of the Ba

6s6p 1P1 level were also calculated using the Wigner-Eisenbud-type R-matrix formulation (Bartschat andMcLaughlin, 1990; Bartschat et al . , 1991). One-electronspin-orbit, mass-correction, and Darwin terms of theBreit-Pauli Hamiltonian were included in the variationalcalculation. The calculation of Bartschat and co-workersshowed distinctly poorer agreement with the experimen-tal spectra (He et al . , 1991, 1995; Lange, Eichmann, andSandner, 1991; Willke and Kock, 1991, 1993) than theeigenchannel R-matrix results. The most striking dis-

crepancy concerned the 6p2 1S0 resonance, which failedto appear in the spectrum calculated by Bartschat etal . (1991). This discrepancy between the eigenchanneland Wigner-Eisenbud-type R-matrix calculations wasresolved by Bartschat and Greene (1993), who per-formed a set of test calculations with both R-matrix ap-proaches. To make the comparison meaningful, all testcalculations used the same empirical core potential ofEq. (3.26) and incorporated relativistic effects throughthe spin-orbit interaction operator only. These calcula-tions showed that the photoionization cross section nearthe 6p2 1S0 resonance cannot be obtained by a standardclose-coupling expansion involving only physical boundstates; instead, the correct description of short-rangecorrelation effects and of the relaxation of the 6p orbitalrequires the inclusion of doubly excited configurationsof the continuum-continuum type. It should be notedthat the eigenchannel R-matrix calculations routinely in-clude doubly excited configurations confined within theR-matrix box and among them those of continuum-continuum type. In other words, short-range correlationand polarization effects are accounted for by usingstrongly closed channels described by two-electron basisfunctions constructed from closed-type orbitals. Conver-gence of the 6p2 1S0 resonance using a purely ionic ba-sis set in the eigenchannel R-matrix calculation requires324 closed-type basis functions. One reason for this sur-prisingly slow convergence for the Ba 6p2 1S0 reso-nance, and also for the 7p2 1S0 (Lecomte et al . , 1994),Ca 4p2 1S0 (Aymar and Telmini, 1991; Assimopouloset al . , 1994), and Sr 5p2 1S0 (Kompitsas et al . , 1991)resonances is that these levels exhibit strong angularelectron correlations. More precisely, all these levelswere found to correspond to a strong admixture of them08p

2 and (m0821)d2 1S0 configurations (m085m0 orm011). Similarly strong electron correlations werefound for the analogous 1D2 resonances and for higher-lying n1sn1s

1S Wannier ridge states of alkaline earths(Aymar, 1989).The work of Bartschat and Greene (1993) also indi-

cated the need for caution regarding the apparent con-vergence of the close-coupling results in calculationsperformed with a limited number of short-range corre-lation functions. Recently, Mende et al . (1995) encoun-tered the same difficulties in describing with Wigner-Eisenbud-type R-matrix calculation the Sr 5p2 1D2resonance observed in the photoionization cross sectionof the Sr 5s5p 1P1

o excited level. To achieve reasonableagreement between theory and experiment, they simu-lated in a semiempirical way the short-range correlationand relaxation effects that are not fully described by thestandard close-coupling expansion.Various photoionization spectra from low-lying ex-

cited levels of Ba calculated with the eigenchannelR-matrix approach in either LS coupling (plus frametransformation) or jj coupling have been compared (see,for example, Greene and Aymar, 1991; Telmini et al . ,1993). Spectra are generally very similar except in somecases in which the low-lying level is not well described inLS coupling. Rather different results were found for the



excitation spectra of 6p1/2nf J51–3 series of Ba excitedfrom the 5d6p 3F2 level in the energy range where the6p3/28f J=1–3 levels are located. These levels were notexcited in the experimental spectrum recorded by Ab-utaleb, de Graaff, Ubachs, and Hogervorst (1991), norin the theoretical spectrum calculated in jj coupling,while they are visible in the spectrum obtained in LScoupling (see Fig. 7 of Telmini et al . , 1993). Recall thatthe frame-transformation treatment incorporates fine-structure effects in the final states only but assumes thatthe initial level is well described in LS coupling. The5d6p 3F2 level (LS weight of ;80%) corresponds tothe pure jj-coupled 5d3/26p1/2 J52 level (97%). This ex-plains why the LS calculations poorly describe photo-ionization from this level. Ground states of alkalineearths as well as the Ba 5d6p 3P0 level involved in thephotoionization process considered in Fig. 31 are welldescribed in LS coupling. However, this is not the casefor various excited states of Sr and mainly of Ba. Thus, aprerequisite of any calculation of photoionization spec-trum in LS coupling (and frame transformation) mustbe to check whether the initial state is correctly de-scribed in that coupling. This reflects a limitation of theframe-transformation calculations.

b. Partial photoionization cross sections and photoelectronangular distributions

Even-parity doubly excited levels of Ba have recentlybeen investigated in a two-step laser experiment thatcovered a large energy range from the 6p3/2 threshold to(nearly) the 7s threshold. Excitation of the Ba groundstate via 5d7p 3P1 and

3D1 levels produced photoelec-trons that were analyzed in two groups of energy-resolved continua 6s , 5d3/2,5/2el (‘‘fast’’ electrons) and6p1/2,3/2el (‘‘slow’’ electrons). Laser beams were polar-ized and several combinations of the laser polarizationswere used. The experimental partial cross sections werecompared with results obtained with the alternativeeigenchannel R-matrix formulation (Lecomte et al.,1994) described in Sec. III.H. The R-matrix calculationswere carried out in jj coupling using an R-matrix box ofradius r0=50 a.u. Both experiment and calculations werecarried out in this study (Lecomte et al . , 1995). Figure33 compares the experimental and theoretical partialphotoionization cross sections in the 71 400–72 400cm21 energy range. The electron yields were recordedvia the 5d7p 3D1 level, with both lasers having the samecircular (s1) polarization. Only J52 final states werepopulated. All 17 J52 open channels were included inthe R-matrix calculation; no additional closed channelswere introduced, except within the reaction volumethrough ‘‘closed-type’’ basis functions. The resonancesresulted from the coupling between the open channelsand the 6d2 doubly excited states that fit completelywithin the reaction volume. The positions and widths of6d2 states were predicted by Lecomte et al . (1994). Ex-perimental and theoretical spectra were in good agree-ment. Only the calculated and measured ‘‘fast’’ electronyields [Fig. 33, curves (a)] were adjusted, showing that

the branching ratio of the experimental partial cross sec-tions was reproduced by the calculation. The peak inFig. 33 [curves (a) and (b)] is ascribed to the 6d2 3F2state; the 6d2 3P2 state, predicted to lie in the sameenergy range, is more weakly excited (the correspondingdipole moments are in a ratio of ; 3). The higher-lying4f7p 3F2 state at 78 760 cm21 was found to have astrong influence on the profiles. This is evident in thetheoretical slow electron yield [Fig. 33, curves (b)],where the energy position and profile (long-dashedcurve) are strongly modified when the direct excitationof the 4fnp channel is disregarded (dot-dashed curve).The 4f7p 3F2 resonance, strongly excited from the in-termediate level, has a very large autoionization width(G= 1600 cm21). Its coupling with the continua is largeenough to induce a strong mixing with the 6d2 3F2 state.The energy difference between the maxima of both ex-perimental cross sections was found to result from inter-ference effects between the direct excitation of the 6d23F2 state and the indirect excitation via the 4f7p 3F2state. The fact that the calculated spectrum [Fig. 33,curves (b)] was more asymmetric than the experimentalspectrum may indicate that the calculation overesti-mated the excitation of the 4f7p 3F2 state. The eigen-channel R-matrix calculations were also able to success-fully account for observations in the higher-energy rangefrom 76 000 to 84 000 cm21 (Lecomte et al . , 1995).There, resonances correspond either to doubly excitedstates, such as the 7p2 3P2 level, enclosed within thereaction volume, or to Rydberg states such as 7snd ,which are associated with closed channels. The proce-dures developed by Lecomte et al . (1994) were ex-tended to handle such situations involving doubly ex-cited levels confined within the reaction volume thatinteract with levels pertaining to Rydberg series. Special

FIG. 33. Measured and calculated partial photoionizationcross sections of the Ba 5d7p 3D1 state as functions of theenergy relative to the Ba ground state: (a) in unresolved 6s ,5d3/2,5/2el continua and (b) in unresolved 6p1/2,3/2el continua.The experimental spectra (full curves) are compared with thecalculated spectra (long-dashed curves). Cross sections calcu-lated by disregarding the excitation of the 4f7p 3F2 state arealso shown (dot-dashed curves). Only the calculated and mea-sured partial cross sections corresponding to electrons ejectedinto the 6s , 5d3/2,5/2el continua (a) were adjusted. (FromLecomte et al . , 1995).



attention has been given to the identification of eachindividual resonance in complex autoionizing structures.All resonances were found to exhibit strong electroncorrelations.We mention here some experiments on angular distri-

butions of photoelectrons obtained by resonance-enhanced multiphoton ionization through excited levelsof alkaline earths. A review of experimental and theo-retical studies carried out up to 1987 was given by Smithand Leuchs (1987). Resonant multiphoton ionization viavarious intermediate levels of Ca, Sr, and Ba has beenused to study configuration mixing in these intermediatelevels (Matthias et al., 1983; Leuchs and Smith, 1985;Mullins, Chien, Hunter III, Jordan, and Berry, 1985;Mullins, Chien, Hunter III, Keller, and Berry, 1985;Mullins, Hunter III, Keller, and Berry, 1985; Hunter IIIet al., 1986; Keller et al., 1991). The angular distributionsof photoelectrons ejected into channels attached to dif-ferent ion cores were measured.The photoelectron angular distributions observed by

Mullins and co-workers and by Hunter III et al . (1986)were fitted to an appropriate parametric expression ac-counting for hyperfine interaction occurring from the co-herent excitation of hyperfine states. These hyperfine ef-fects are strong in some cases (Mullins, Chien, HunterIII, Keller, and Berry, 1985; Hunter III et al., 1986).Data obtained by Matthias et al . (1983) and Leuchs

and Smith (1985) via 6snd 1,3D2 and 6sns1S0 Rydberg

levels, respectively, which interact with certain 5d7ddoubly excited states, have been successfully interpretedby parametrizing the angular distributions in terms ofionization amplitudes and closed-channel coefficientspreviously obtained with semiempirical MQDT models.From the data they obtained in Ba via low-lying levels,Hunter III et al . (1986) suggested that semiempiricalMQDT analysis of the J52e and J50e bound spectra(Aymar et al., 1978; Aymar and Robaux, 1979) misla-beled some levels. Recent R-matrix calculations(Greene and Aymar, 1991) confirm the original labelsgiven in Moore’s table (1958) to J52e levels, and thusdo not support the reassignment obtained by semiempir-ical multichannel quantum-defect theory. In contrast,R-matrix calculations support the 6s8s 1S0 label (Ay-mar et al., 1978) given to the level at 34 731 cm21 (6p21S0 in Moore’s table), in contradiction with the sugges-tion of Hunter III et al . (1986). As was detailed in theprevious subsection, several experimental and theoreti-cal studies have demonstrated that the 6p2 1S0 levelcorresponds to a broad autoionizing resonance around44 800 cm21.The various experiments quoted just above deal with

ionization into a structureless continuum, i.e., with final-state energies far from any autoionization resonance. Incontrast, the two-photon experiment of Keller et al .(1991), via the 5d6p or 6s6p 1P1 level, measured theangular distributions of photoelectrons across 5dnd au-toionizing resonances. Like the profiles of the autoion-izing resonances, the angular distributions were found todrastically change with the intermediate level used inthe excitation process.

We discuss below, in Sec. IV.E, several experiments inalkaline earths in which the angular distributions of pho-toelectrons were measured using multistep laser excita-tion based on the isolated-core excitation scheme; in thelast step, the autoionizing levels were excited from a Ry-dberg level. Several experiments were performed over alarge energy range across autoionizing resonances. Incontrast, angular distributions of photoelectrons ob-tained by multistep ionization through low-lying excitedlevels were measured at fixed energy or over a restrictedenergy range.An indirect determination of the angular distributions

of photoelectrons ejected from Ba 6pns , nd J51 au-toionizing levels into 6sep and 5d3/2,5/2ep , f continuawas recently reported by Lagadec et al . (1996). Mea-surements covered a 1000 cm21 energy range. Autoion-izing levels were populated from the Ba ground stateusing the three-step laser excitation scheme 6s2

→6s6p1P1→6p2 3P0→6pns , ndJ51. Linearly polar-ized laser beams were used. The variation of the asym-metry parameters characterizing the differential partialcross sections corresponding to electrons ejected intocontinua attached to the 6s or 5d3/2,5/2 ion cores weredetermined by measuring, for different combinations ofthe orientations of the laser polarization vectors, the dif-ferential partial cross sections for a fixed solid angle ofphotoelectron ejection. The experimental measurementswere successfully accounted for by eigenchanneljj-coupled R-matrix calculations (Lagadec et al., 1996).

E. Photoionization from Rydberg states

An impressive number of experimental investigationson autoionizing states of the alkaline earths Mg to Bahas been carried out using multistep laser experimentsbased on the isolated-core excitation (ICE) experimen-tal technique. Earlier investigations focused first on Sr(Cooke et al . , 1978; Xu et al., 1986, 1987; Zhu et al.,1987) and Ba (Gounand et al., 1983; Bloomfield et al. ,1984; Tran et al., 1984; Kachru et al., 1985; Bente andHogervorst, 1990; Hieronymus et al., 1990; Jones et al.,1991a, 1991b; Lange, Aymar, et al., 1991; de Graaffet al., 1992, and references therein), then on Ca (Langeet al., 1989; Morita and Suzuki, 1990), and more recentlyon Mg (Dai et al., 1990; Schinn et al., 1991; Lindsay, Cai,et al., 1992; Lindsay, Dai, et al., 1992). Finally, noticethat an increasing number of laser experiments beingperformed in the alkaline earths now focus on doublyexcited states with both electrons in a highly excited or-bit, or on states with high angular momentum l2 for theouter electron, or on states having both qualities (Camuset al., 1989, 1992; Eichmann et al., 1989, 1992; Jones andGallagher, 1990; Roussel et al., 1990; Jones et al., 1991a;Camus, Cohen, et al., 1993; Camus, Mahon, and Pruvost,1993; Wang and Cooke, 1993; van Leeuwen et al., 1995,1996; Seng et al., 1995, and references therein). Formore details, the reader is referred to the book that Gal-lagher (1994) wrote recently on Rydberg atoms.



At the present time, the eigenchannel R-matrix tech-nologies have not been extended to handle very high-lying levels. The highest levels at which experimentaldata have been compared with R-matrix calculations arelocated near the 6f threshold of Sr, the 8s threshold ofBa (Wood and Greene, 1994), and the n56 thresholdsin H2 and Li2 (Pan et al . , 1994).In the ICE experiments performed on autoionizing

states, multistep laser excitation reaches a particularbound (or long-lived autoionizing) Rydberg level. Then,starting from this Rydberg level, autoionizing reso-nances are reached by ‘‘isolated-core excitation’’ of theinner valence electron, the Rydberg electron being keptoutside as a spectator. In the ICE approximation, thedipole operator acts only on the core electron becausethe outer electron spends too little time near the nucleusto absorb a visible or ultraviolet photon. In such an ex-citation process, only particular autoionizing levels areexcited and the direct continuum excitation is usuallynegligible. Experiments using one- and two-photon ex-citation of the core electron were performed but we willconsider mainly processes involving one-photon transi-tions, which can be compared with eigenchannelR-matrix results more easily.When the outer electron is at distances larger than

r0, the wave function of the Rydberg initial level to bephotoionized can be written as

c05rCF0~v!W~n0 ,l0 ,r !n023/2 , (4.9)

where F0(v) is the initial-state channel function, n0 isthe effective quantum number, and l0 the orbital mo-mentum of the outer electron. In Eq. (4.9), W(n0 ,l0 ,r)is the energy-normalized Coulomb function—introduced in Eq. (2.53)—that decays exponentially atinfinity. Such an approximation for the initial wave func-tion neglects exchange and all correlation effects for theinitial state. The initial state is generally assumed to bedescribed in a particular LS , jj , or jK coupling scheme.Calculation of photoionization cross sections [Eqs.(2.59) and (2.60)] requires the determination of dipolematrix elements d(2) [Eq. (2.58)] connecting the initial-state wave function [Eq. (4.9)] to the N0 energy-normalized physical solutions C i8

(2) defined in Eq. (2.55),which obey incoming-wave boundary conditions (N0 isthe number of open channels).Because the isolated-core approximation relies on

negligible continuum excitation one has

di8~2 !

5^c0uDuC i8&5(iPc

diZii8, (4.10)

with the component di of the dipole matrix element as-sociated with the closed channel i , characterized by theeffective quantum number n i and the orbital momentuml i for the outer electron, given by

di5n023/2^F0uDuF i&^W0~n0 ,l0 ,r !uWi~n i ,l i ,r !& iPc .

(4.11)

In Eq. (4.11) the one-photon transition moment implic-itly contains all the angular coefficients related to the

decoupling of the individual angular momenta and spinsfrom the total angular momentum, as well as those re-lated to the initial-state coupling. It also incorporatesangular factors related to the polarizations of the lasersinvolved in the multistep laser excitation process.As shown by Bhatti et al . (1981), for Coulomb func-

tions normalized per unit energy range, the overlap in-tegral can be approximately expressed as

^W0~n0 ,l0 ,r !uWi~n i ,l i ,r !&

52sinp~n i2n0!

p~n i22n0

2!n02n i

2d~ l0 ,l i!. (4.12)

The Kronecker delta function defines a subset i0 ofclosed channels with l i5l0 that are directly accessible inthe isolated-core excitation.Experimental ICE results have been obtained not

only for excitation spectra and for the positions andwidths of levels, but also for the branching ratios andangular distributions of the ejected electrons. These lat-ter observables are known to be highly sensitive to in-terseries interactions and to provide a much more strin-gent test of theory.

1. Excitation spectra

In both ordinary photoabsorption spectra and ICEspectra, a large part of the energy dependence of thephotoionization cross sections is contained in theZii8(E) closed-channel coefficients [Eq. (2.54b)] in-volved in the closed part of the physical solutions of Eq.(2.55). However, there are two main differences be-tween the ICE spectra and ordinary photoabsorptionspectra. First, in the ICE spectra, only certain closedchannels are excited and thus these spectra, do not dis-play Beutler-Fano (Fano, 1961) profiles such as thoseoccurring in the photoabsorption spectra, where struc-tures are complicated by interferences between thediscrete-channel and continuum-channel excitation. Thesecond difference comes from the fact that a large partof the energy dependence of the ICE profiles is con-tained in the overlap integral of Eq. (4.12). The squaredoverlap integral is maximum for n i5n0, is zero for n i =n01p , where p is a nonzero integer, and has subsidiarymaxima for n i =n01p11/2.The marked differences between isolated-core excita-

tion and ground-state photoionization spectra are illus-trated in Fig. 34. Figures 34(a) and 34(b) show part ofthe photoabsorption spectrum of Sr, where the 5pns ,5pnd J51 autoionizing levels are excited from the 5s2

ground state. Figures 34(c) to 34(f) display ICE spectrawhere, in the final transition, the 5pns J51 levels areexcited from a particular 5sns 1S0 Rydberg level. Theenergy range involved in either Figs. 34(c) and 34(d) orFigs. 34(e) and 34(f) corresponds to a small part of theenergy range covered by Figs. 34(a) and 34(b), aroundthe 5p3/214s peak, denoted (*) in Fig. 34(b). Theoreticalspectra @Figs. 34(b), 34(c), or 34(e)] obtained withLS-coupled R-matrix calculations combined with the



jj-LS frame transformation are compared with corre-sponding experimental spectra @Figs. 34(a), 34(d), or34(f)].As we have already described various photoabsorp-

tion spectra in detail, we comment only briefly on Figs.34(a) and 34(b). The photoabsorption spectrum calcu-lated by Aymar (1987) @Fig. 34(b)] reproduces ex-tremely complex structures occurring in the the spec-trum recorded by Brown et al . (1983) @Fig. 34(a)]. Thesharp structures correspond to the 5p1/2ns and5p3/2n8s resonances. The 5pnd resonances are toobroad to be identifiable.Using the experimental ICE technique, Xu et al .

(1986) and Zhu et al . (1987) reported a complete set of

measurements for the 5p1/2ns and 5p3/2ns J51 autoion-izing levels of Sr. These levels were investigated usingthree-step laser excitation via the 5s5p 1P1 and 5sns1S0 Rydberg levels with n>10. Experimental data onisolated or structured resonances concern the excitationspectra as well as the energy and angular distributions ofejected electrons. The isolated resonances have approxi-mately Lorentzian profiles because of the negligible con-tinuum excitation. We consider here only the excitationspectra of the more complex structured resonanceswhose features reflect the interactions between the5p1/2ns ,nd and 5p3/2ns ,nd series. Two different types ofspectrum reflecting the interseries interactions are

FIG. 34. Photoionization spectra for the odd-parity 5pns , nd J51 levels of Sr near the 5p1/2 threshold. Photoabsorption spectrafrom the 5s2 ground state [(a) and (b)] are compared with isolated-core excitation spectra from 5sns 1S0 Rydberg states [(c to f)].Experimental data are compared with results obtained with LS-coupled R-matrix calculations combined with the jj-LS frametransformation. (a) The photoabsorption spectrum recorded by Brown et al . (1983); (b) the R-matrix result; the vertical barsindicate the position of the observed absorption peaks and minima and the 5p3/214s level is denoted by an asterisk (*). Theexperimental ICE 5s14s → 5p3/214s spectrum recorded by Xu et al . (1986) in the vicinity of the 5s→5p3/2 ionic transition (d) iscompared with the R-matrix result (c). The shake-up satellite spectrum of the ICE 5s20s → 5p1/220s transition recorded by Xuet al . (1986) far from the 5s→5p1/2 ionic transition (f) is compared with the R-matrix result (e). The dashed curves in (d) and (f)are the MQDT empirical fits of Xu et al . (1986). [Figures 34(a) and 34(b) are from Aymar (1987) while Figs. 34(c) to 34(f) arefrom Aymar and Lecomte (1989).]



shown in Figs. 34(c) and 34(d) and in Figs. 34(e) and34(f).We first comment on Figs. 34(c) and 34(d), which

compare theoretical and experimental results for the5s14s → 5p3/214s spectrum. Figure 34(d) shows thespectrum recorded by Xu et al . (1986), who obtainedtheir results by tuning the wavelength of the last stepexcitation laser in the vicinity of the 5s→5p3/2 ionic line.The frame-transformation result @Fig. 34(c)] obtainedby Aymar and Lecomte (1989) closely reproduced therecorded spectrum, in particular the central features in-volving three strongly interacting levels 5p3/214s ,5p1/228s , and 5p1/227d . The shape of the spectrum iscontrolled mainly by the energy variation of theZii8(E) coefficients, because the spectrum is within thecentral lobe of the overlap integral. The spectrum exhib-its complex structures associated with the mixing of the5p3/214s level and the nearby 5p1/2ns and 5p1/2nd levels.Although only one di0 dipole matrix element associatedwith the 5p3/2ns channel has a non-negligible amplitude,the 5p1/2ns and 5p1/2nd resonances appear in the spec-trum because of their coupling with the 5p3/214s level.The positions and shapes of the 5p1/2ns and 5p1/2ndresonances agree with the results of the experiment,while the relative magnitudes of these secondary peaksare slightly underestimated by the calculation. TheR-matrix calculation reproduces the experimental ob-servations, as did the MQDT simulation performed byXu et al . (1986) using a simplified semiempirical six-channel MQDT model based on the phase-shiftedMQDT formalism developed by Cooke and Cromer(1985) @dashed curve on Fig. 34(d)]. The broad5p1/2nd resonances are described distinctly better by theR-matrix calculation.Figures 34(e) and 34(f) correspond to another kind of

ICE spectrum in which the autoionizing levels were ex-cited by tuning the wavelength of the excitation laser farfrom any ionic transition. More precisely, the spectrumof Fig. 34(f), recorded by Xu et al . (1986) from the5s20s 1S 0 level, was obtained by scanning the wave-length of the third laser to the blue side of the5s→5p1/2 ionic transition. This shake-up spectrum ofthe 5s20s→5p1/220s transition compares well with theframe transformation R-matrix results of Aymar andLecomte (1989) @Fig. 34(e)]. The precise form of thesatellite peaks depends now on the Zii8(E) coefficients,on both di0 dipole moments associated with the5p1/2ns and 5p3/2ns channels, and on the variation of theoverlap integral. The minima between each 5p1/2nspeak correspond to the zeros of the overlap integral,while the maxima correspond approximately to those ofthe Zii8(E) coefficients. The strong asymmetry of thepeaks results from the distortion due to the overlap in-tegral, whose zeros are very close to the maxima of theZii8(E) coefficients. The irregular behavior nearn=27–30 reflects the presence of the 5p3/214s perturber.The calculated cross section perfectly reproduces all thesubtle features occurring in the recorded spectrum. Thequality of the R-matrix calculation is comparable to that

of the MQDT simulation performed by Xu et al. (1986)@dashed curve on Fig. 34(f)]. One last remark concernsthe features on the left side of the ns peaks (particularlyvisible near E=69 383 cm21), which cannot be repro-duced by the calculation since, as explained by Xu etal . (1986), they are due to the impurity of the initial5s20s level, which contains a small amount of nearby5snd levels.Complex structures, such as those visible in Figs. 34(c)

and 34(d), have been observed in several ICE spectra ofSr (Xu et al . , 1987) and Ba (see, for example, Gounandet al . , 1983, Bente and Hogervorst, 1990, and Hierony-mus et al . , 1990). They correspond to m0p3/2n8l statesdegenerate with m0p1/2nl levels having large n values.Shake-up or shake-down spectra of a given transition,such those shown on Figs. 34(e) and 34(f), were alsofrequently recorded in Ba. A common procedure used inthe ICE experimental method is to saturate the centralpeak using high laser power in order to observe manysatellite peaks. In this way, it is possible to obtain thepositions and widths of doubly excited states that, forexperimental reasons, cannot be easily excited directly(see, for example, Tran et al . , 1984, and Bente and Hog-ervorst, 1990).The excitation spectra discussed above dealt with low-

lying autoionizing levels located around the m0pjthresholds. They involve n1l1n2l2 doubly excited stateswith a small value of the orbital momentum l2 (l2<2) ofthe outer electron. Approximating the initial and finalwave functions as products of one-electron wave func-tions that span different regions of space neglects over-lap and exchange between the two valence electrons.For low-l2 levels, the nonoverlapping property occursonly when n2@n1, and thus the wave-function exten-sions of the Rydberg initial levels are too large to becontained within a reaction volume of moderate size,and the explicit calculation of the dipole matrix ele-ments becomes inefficient. In this regime, the ICE ap-proximation determines the dipole matrix elements farmore efficiently. Eigenchannel R-matrix calculationsthat used the ICE approximation were carried out byAymar and Lecomte (1989), Lange, Aymar, et al .(1991), Dai et al . (1990), and by Schinn et al . (1991);these were used to interpret experimental ICE spectra.Some spectra have been measured that deal with

higher-lying autoionizing levels with larger values of theorbital momentum l2. One example can be found in theexcitation spectra of the Ba 4f5g J53 levels from5d5g J52 levels observed by Jones et al . (1991a) in athree-step laser experiment based on an isolated-coreexcitation scheme. All three lasers were circularly polar-ized in the same sense, ensuring that the final states were4f5g levels with J53 only. The 4f5g levels are of par-ticular interest because the orbital momentum of eachelectron is equal to its maximum allowable value in thisenergy range: l1=n121, l2=n221. The classical orbits ofthe two electrons would therefore be circular in the ab-sence of interactions; in this sense, the observation of



Jones et al . (1991a) was the first one involving autoion-izing levels for which both electrons are ‘‘in circular or-bits.’’Photoionization cross sections from the jK-coupled

5d5/25g@5/2# and 5d3/25g@5/2# J52 levels were calcu-lated using the jj-coupled eigenchannel R-matrixmethod (Aymar and Luc-Koenig, 1995). For the firsttime, excitation spectra calculated using the ICE ap-proximation were successfully reproduced by a separatecalculation of dipole matrix elements restricted to theinterior of the R-matrix reaction volume; in this case itsradius was r0=50 a.u. The theoretical excitation spectraagreed with the 5dj5g@5/2# J52 →4f j85g J53 spectraobserved by Jones et al . (1991a). The ICE approxima-tion was found to adequately describe the excitation ofthe 4f5g levels. The validity of the ICE approximationrelies mainly on the absence of spatial overlap betweenthe inner- and outer-electron orbitals. The nonoverlap-ping property occurring for the 4f5g levels, which in-volve two electrons having almost the same degree ofexcitation, is peculiar to the double-circular states. An-other key result of the theoretical study of the double-circular 4f5g J53 levels (Aymar and Luc-Koenig, 1995)is the large influence of electron correlations on theproperties of these levels. Despite the negligible spatialoverlap between the wave functions of the two valenceelectrons, correlation effects have been shown to affectthe widths and excitation cross sections of the 4f5g lev-els very strongly. The low autoionization rates observedfor the 4f5g levels cannot be reproduced by neglectingthe mixing of 4f5g levels with several Rydberg series,most importantly the 6dnh series. The excitation spec-trum from the 5d5/25g [5/2] level exhibits, in addition tothe peaks associated with the four 4f5g J53 levels,structures that were assigned to 6d8h levels. These lev-els are not directly excited in the ICE process but ap-pear in the excitation spectrum owing to their mixingwith the 4f5g levels.More recently, experimental excitation spectra of

higher-lying 4fng J=1–3 levels of Ba, recorded in athree-step ICE experiment, were successfully accountedfor by eigenchannel R-matrix calculations (van Leeuwenet al . , 1996).Another example, depicted in Fig. 35, deals with the

4f5/27h J8=4–6 levels of Ba that lie in the energy rangebetween the 6d3/2 and 6d5/2 thresholds. These levelswere excited from the jK-coupled 5d3/27h@11/2# J=5level in a resonant two-step ICE experiment with pulsedlasers of van Leeuwen et al. (1995). The excitation spec-trum of Fig. 35(a) was recorded by detectingBa11 ions produced by photoionization of the Ba1

6d3/2 ion. The experimental spectrum was comparedwith the partial photoionization cross sections of the5d3/27h@11/2# J=5 →4f5/27h J8 processes (summed overJ8=4, 5, and 6) corresponding to electron ejection intothe 6d3/2el8 continua only [Fig. 35(b)]. The theoreticalspectrum in Fig. 35(b) obtained with jj-coupledR-matrix calculations has been convolved with an in-strumental linewidth of 0.2 cm21 (van Leeuwen et al . ,1995).

The 4f5/27h levels interact with 6d5/2ni levels andcause a complex interference pattern. All resonances fallwithin the central lobe of the overlap integral of Eq.(4.12), which peaks at the position of the 5d3/2→4f5/2ionic transition [see Fig. 35(a)]. The 4f5/27h J8=4–6states correspond to the broad central features in Figs.35(a) and 35(b). The narrow resonances were identifiedas members of 6d5/2ni Rydberg series converging to the6d5/2 ionization limit. These resonances, which are notdirectly accessible in the ICE process, appear because oftheir mixing with the 4f7h levels. The 6d5/2ni levels onthe low-energy side of the broad 4f5/27h J8=4–6 reso-nances are characterized by negative values of the line-shape parameter q (Fano, 1961) and those on the rightside by positive values of q . The widths of 6d5/2ni levelswere found to vary strongly with n . The 6d5/223i level,located below the 6d3/2 ionization limit, as well as the6d5/224i level, are extremely narrow. The predictedwidths for these levels are respectively 0.08 and 0.15cm21, which are below the instrumental linewidth of 0.2cm21. This spectrum is complex, as it includes finalstates with three different values of J8, each of whichrequires the correct description of interactions among alarge number of channels (up to 47). In view of thiscomplexity, the agreement between the experimentaland theoretical spectra in Fig. 35 is excellent. TheR-matrix approach successfully reproduces the compli-cated interference pattern involving q reversal and line-narrowing effects in the 6d5/2ni series interacting withthe 4f5/27h levels.It was found that the final-state angular momentum

J855 dominates in the complex interference pattern.Above the Ba1 6d3/2 threshold, the J855 spectrum in-volves 20 closed channels interacting with 24 open chan-nels. Examination of the phase-shifted MQDT param-eters deduced from R-matrix calculations (see Sec.II.D.3) have shown that, to a good approximation, the44-channel J855 problem can be reduced to a three-channel problem. In fact, the 4f5/27h@11/2# level ismainly coupled to the 6d5/2ni@11/2# Rydberg series andboth the 4f7h level and the 6dni Rydberg levels decaymainly into the same 6d3/2ei@11/2# continuum. A three-channel MQDT treatment of the J855 spectrum hasbeen performed (van Leeuwen et al . , 1995) using a setof five phase-shifted MQDT parameters, whose valueswere fixed at the values obtained from the completetreatment. The three-channel MQDT model clearlyshowed that the interfence pattern does not result frominterferences in the excitation process but from interfer-ences in the spectral density of autoionizing states. Thedramatic changes of the widths and the q reversal alongthe 6d5/2ni autoionizing Rydberg series result from in-terference between direct autoionization and indirectautoionization via the 4f5/27h broad interloper of theRydberg series.Eigenchannel R-matrix calculations were conducted

by Wood and Greene (1994) in Ba and Sr, in an energyrange higher than the m0p threshold, which was probedby the experiments of Camus, Cohen, et al . (1993) andof Eichmann et al . (1992). We discuss here the results



obtained in Ba near the 8s threshold. Asymmetric8snl levels were excited from 6snl levels that werepopulated by a Stark-switching technique (Cooke etal . , 1978). Photoionization cross sections have beenmeasured using a two-photon isolated-core excitation(Camus, Cohen, et al . , 1993). Figure 36 compares therelative experimental and theoretical photoionizationcross sections for the 6snl initial state with n514 andl56 over an energy range of ; 300 cm21, i.e., fromn8s516 to n8s512. Eigenchannel R-matrix calculationswere performed in jK coupling with a reaction volumeradius of 100 a.u. Calculations included open channelsconverging to the 7s , 6dj , 4f j , and 7pj thresholds onlyand closed channels converging to the 5f j , 8s , 7dj ,8pj , and 5gj thresholds. Because the experimental ICEprocess corresponds to the absorption of two photons, it

is no longer possible to express easily the transition am-plitudes in terms of the analytical form [Eq. (4.12)] ofthe overlap integral of the outer-electron wave func-tions. Thus, Wood and Greene (1994) calculated ap-proximate two-photon transition amplitudes. The reso-nances in Fig. 36 are due to the 5f5/2nl levels (5f5/2threshold at -0.01061 a.u.) overlapping with the 5f7/2nl8levels (5f7/2 threshold at -0.0105 a.u.). Given the ex-treme complexity of the experiment, the calculation ac-counts for the measurement of Camus, Cohen, et al.(1993) reasonably well, except for the intensities of the5f jnl8 resonances. A possible explanation for the dis-crepancy concerning the intensities of the 5f jnl8 reso-nances is that the approximations underlying the calcu-lation of the dipole 2moments were overly simplified. Asecond possible reason for the discrepancy is the pres-

FIG. 35. Isolated-core excitation spectra 5d3/27h@11/2# J55 → 4f5/27h J85426 of Ba as functions of the energy relative to theBa ground state: (a) experimental spectrum: the Ba11 signal is divided by the Ba1 signal; the narrow resonances were identifiedas 6d5/2ni levels. The 6d5/227i J8=4 level at 88 037 cm

21 is denoted by an asterisk (*). (b) Theoretical spectrum calculated with thejj-coupled eigenchannel R-matrix method. (From van Leeuwen et al . , 1995.)



ence of perturbers in the energy range of Fig. 36. Woodand Greene (1994) found that 5gjnd and 7djng perturb-ers lie in the that energy range. Moreover, they foundthat the 7djnl9 channels play a crucial role in determin-ing the intensities of the 5f jnl8 resonances. A slight er-ror in the position of the 7djnl9 perturbers could beresponsible for the too large intensities of the calculated5f jnl8 resonances.The intensities of 5f jnl8 resonances are surprisingly

large in view of the fact that the direct mixing of 8snlchannels with the 5f jnl8 channels requires a change ofthree units of angular momentum. Wood and Greene(1994) analyzed the mechanism responsible for the in-teraction between the 8snl56 and the 5f jnl8 channelsby calculating the short-range scattering matrix elementsuSiju2 connecting the 8snl56 channel to the 5f jnl8 chan-nels. The largest coupling was found for the 5f7/2nl857channel. However, this scattering probability is dramati-cally reduced when the 7djnl9 channels are consideredas open in the MQDT treatment, i.e. when all perturbersassociated with these channels are removed (see Sec.IV.C). This provides strong evidence that the interactionbetween 8snl56 and the 5f jnl8 channels is being medi-ated through an interaction with the 7djnl9 channels.Camus, Cohen et al . (1993) speculated that the inter-

action between the 8snl and the 5f jnl8 channels couldbe due to a direct octopole coupling. This suggestionwas not supported by the calculations of Wood andGreene (1994), who repeated both calculations of theshort-range scattering matrix elements by omitting allmultipoles higher than k52 in the expansion of the1/r12 interaction. The calculations gave results very closeto those obtained from the calculations including allmultipoles.

2. Branching ratios and angular distributions of electronsejected from autoionizing levels

Several experimental investigations on angular distri-butions of electrons ejected from odd-parity doubly ex-

cited states m0pns J51 have been carried out in Mg(Lindsay, Cai, et al . , 1992), Ca (Lange et al . , 1989), Sr(Zhu et al . , 1987), and Ba (Hieronymus et al . , 1990;Lange, Aymar, et al . , 1991, and references therein). Inaddition, angular distributions of electrons ejected fromthe more complex 3pnd states of Mg have been alsomeasured (Lindsay, Cai, et al . , 1992). In all these stud-ies, the autoionizing Rydberg states were reached usingthree-step laser excitations via the m0sm0p

1P1 and aRydberg level m0sns

1S0 or m0snd1D2. The three la-

sers were linearly polarized in the same direction.We first address the simplest case in which the third

laser excites the m0pns J51 states from a sphericallysymmetric m0sns

1S0 bound Rydberg state. The differ-ential cross section for photoionization to a given ionicstate i can be expressed (Yang, 1948) as

ds i

dV~u!5

s i

4p@11b iP2~cosu!# , (4.13)

where s i is the partial cross section in channel i inte-grated over V, u the angle between the third laser polar-ization and the momentum of the electron ejected in thesolid angle dV , and P2(cosu) is the second-order Leg-endre polynomial. The asymmetry parameter b i charac-terizes the angular distribution of electrons ejected inchannel i . In Mg, electrons are ejected into continuabuilt on the 3s core while in Ca, Sr, and Ba, the elec-trons can be ejected into continua built on cores m0s ,(m021)d3/2 , (m021)d5/2 , and even m0p1/2 for levelslocated above the m0p1/2 limit. The measured observ-ables that were compared to calculations include the b iparameters, and in Ca, Sr, and Ba the branching ratiosri5s i /(s i that determine the relative probabilities ofautoionization to the various ionic-core states.The general angular properties of photoelectron an-

gular distributions have been worked out by Dill andFano (1972), by Fano and Dill (1972), and by Dill (1973)using the concept of angular momentum transfer. Theangular distribution of the ejected electrons is expressedas a sum of incoherent contributions corresponding todifferent magnitudes of the angular momentum j t trans-ferred between the unobserved photofragments

jW t5 jW i1sW2JW 0 , (4.14)

where jW i is the total angular momentum of the residualion core, JW 0 the total orbital momentum of initial state,and sW the spin of the ejected electron. There are parity-favored and parity-unfavored transfers. The differentialpartial cross section [Eq. (4.13)], or equivalently the b iparameters, can be expressed in terms of reduced dipolematrix elements djisJ5^J0uuDuu@(j is)j is ,l#J2&, where theminus sign indicates that the final state is normalizedaccording to the incoming-wave boundary condition [seeEq. (2.55)]. The final total angular momentum isJW5 jW is1 lW and lW is the orbital momentum of the ejectedelectron. As was mentioned in Sec. II.E, matrix ele-ments are easier to evaluate in j ij is coupling (denoted byJcJcs in Sec. II.E) than in jj coupling, and the short-

FIG. 36. Relative photoionization cross sections for the two-photon excitation spectrum of Ba from the 6snl initial statewith n514 and l56 as functions of the energy relative to thedouble-ionization limit. The experimental spectrum (top) re-corded by Camus, Cohen, et al . (1993) is compared with thejK-coupled eigenchannel R-matrix result of Wood and Greene(1994) shown as a mirror image (bottom). The position of theBa1 ionic line is indicated for reference. (From Wood andGreene, 1994.)



range reaction matrices KJ in the j ij is coupling can beobtained from the LS- or jj-coupled matrices K throughsimple angular recoupling transformation.Details on the computation of the b i parameters are

given elsewhere (Zhu et al . , 1987; Aymar and Lecomte,1989; Hieronymus et al . , 1990; Lindsay, Cai, et al . ,1992; Lindsay, Dai, et al., 1992) and are not reportedhere. We restrict our present comments to the bs param-eter corresponding to autoionization into a m0s core inphotoionization from a m0sns

1S0 level. There, the an-gular momentum transfer is simply the total spin of thevalence electron pair which is either 0 (parity favored)or 1 (parity unfavored), the final state being identical toa m0sep

1P1 or3P1 state. The bs parameter may have a

value between -1 and 2. Deviation of bs from 2 can re-flect the strength of the parity-unfavored term, whichoriginates in the spin-orbit interaction or final-state cor-relation effects.Some results on the energy dependence of the asym-

metry and branching-ratio parameters have been ob-tained in the heavier alkaline earths. Figure 37 comparesthe experimental energy dependence of cross sections,asymmetry parameters and branching ratios obtained inCa and Ba with the eigenchannel R-matrix results. Dataon the 4p3/214s level of Ca @Figs. 37(a)–37(d)] were ob-tained by Lange et al . (1989) starting from the 4s14s1S0 Rydberg level and scanning the third laser in thevicinity of the 4s→4p3/2 ionic resonance line. Data onthe 6p3/220s state of Ba @Figs. 37(e)–37(k)] were ob-tained by Lange, Aymar, et al . (1991) in a similar way.The Ca 4p3/214s level is located below the 4p1/2 thresh-old. It is almost degenerate with the 4p1/216s—denotedas s3 on the top of Fig. 37(a). The positions of the neigh-boring 4p1/214d and 4p1/215d are denoted by d2 andd3, respectively. In contrast, the Ba 6p3/220s level isabove the 6p1/2 threshold. Ba data are plotted as func-tions of the effective quantum number n6p3/2 associatedwith the 6p3/2 limit. The Ba 6p3/220s level(n6p3/2;15.7) lies in between the 6p3/218d and 6p3/219dlevels, which are around n6p3/2;15.3 and n6p3/2;16.3, re-spectively. The branching-ratio and asymmetry param-eters corresponding to decay into A1 m0s are denotedrs and bs ; those corresponding to decay into the unre-solved A1 (m021)d3/2 and (m021)d5/2 , rd and bd . InBa, autoionization into the 6p1/2 core is characterized byrp and bp . On each figure, 37(a)–37(k), the experimen-tal data (dots) are compared with the results obtainedwith the LS-coupled eigenchannel R matrix combinedwith the jj-LS frame transformation (full lines). In Ca,theoretical values were computed using the short-rangereaction matrix K determined by Kim and Greene(1987). In Ba, experiment and calculations were con-ducted simultaneously (Lange, Aymar, et al . , 1991).The positions, widths, and shapes of the resonances inCa @Fig. 37(a)] and Ba @Fig. 37(e)] are nicely repro-duced by the calculations (note that the heights of ex-perimental resonance profiles are the only adjustable pa-rameters in the comparison between theory andexperiment). The Ca 4p3/214s profile is slightly asym-

metric because this level interacts with nearby4p1/2ns ,nd levels. The Ba 6p3/220s level has an approxi-mately Lorentzian line profile, as is typical for the iso-lated resonances observed using the ICE experimentalmethod. The calculations also closely reproduce the en-ergy variations of the various branching-ratio param-eters @Figs. 37(b) and 37(i)–37(k)], in particular the lo-calized variations on either side of the Ba 6p3/220sresonance, which occur in the vicinity of the neighboring6p3/2nd (n518,19) levels. In Ba and mainly in Ca, alarge fraction of autoionizing electrons populate the(m021)d3/2,5/2 ionic levels. In Ca, this results in a popu-lation inversion of the Ca+ ion produced by photoioniza-tion.We turn now to the angular distribution parameters

@Figs. 37(c), 37(d), and 37(f)–37(h)]. In Ba, the calcula-tion perfectly reproduces the plateau value of bs;1.8around the 6p3/220s resonance where departure frombs52 is due mainly to spin-orbit effects. Marked dipsoccur in the bs spectra of Ba (on either side of the reso-nance) and Ca (at the resonance center). These dips lo-cated at the positions of the Ba 6p3/2nd (n518,19) per-turbers and of the Ca 4p1/216s perturber reflect final-state channel mixing. These dips are not perfectlydescribed by the theory. The general structure of thepronounced energy variation of bd in Ba is well de-scribed by theory, but poorer agreement between theoryand experiment is obtained for bd in Ca and for bp inBa.Finally, it is interesting to note that the electron

branching ratios and angular distributions of various6p1/2ns and 6p3/2ns levels of Ba investigated by Lange,Aymar, et al . (1991) were also calculated using the Kmatrix obtained with the jj-coupled R-matrix method.Although results are not always completely identical tothose provided by the LS-coupled R-matrix method(and frame transformation), the overall agreement withexperiment of Lange, Aymar, et al . (1991) is of thesame quality.A second example deals with 6pns J51 levels of Ba

located just below the 6p1/2 threshold in the energyrange in which the 6p3/212s level is degenerate withhigh-lying 6p1/2ns ,nd levels with n>22. These levelswere investigated by Hieronymus et al . (1990), startingfrom the 6s12s 1S0 level and inducing the core transi-tion 6s→6p3/2 . Data concern the ion yields as well asthe electron branching ratios and angular distributionsas functions of the energy of the laser driving the coretransition. Experimental data on electron angular distri-butions are compared with R-matrix results in Fig. 38.Experimental and theoretical asymmetry parameters areplotted as functions of the effective quantum numbern6p1/2 associated with the 6p1/2 limit. The broad (G;85cm21) 6p3/212s resonance is centered at ;62 150cm21, i.e., at n6p1/2;27.5. Figures 38(a), 38(c), and 38(e)show the experimental results that were analyzed by Hi-eronymus et al . (1990) using semiempirical MQDTmodels, whose results are shown on Figs. 38(b), 38(d),and 38(f). Three different seven-channel MQDT models



FIG. 37. Electron branching ratios and angular distributions of Ca 4pns and Ba 6pns autoionizing Rydberg states. The experi-mental data (solid points) for the 4p3/214s J51 resonance of Ca [(a) to (d)] excited from the 4s14s 1S0 Rydberg level and for the6p3/220s J51 resonance of Ba [(e) to (k)] excited from the 6s20s 1S0 Rydberg level are compared with results (solid lines)obtained with LS-coupled R-matrix calculations combined with the jj-LS frame transformation. Data in Ca are plotted asfunctions of the energy relative to the Ca ground state and those for Ba as functions of the effective quantum number n6p3/2associated with the 6p3/2 limit. The Ca 4p3/214s J51 level located below the 4p3/2 threshold can autoionize to the m0sel and(m021)del continua (denoted s and d, respectively), whereas in Ba the 6p3/220s J51 level autoionizes in addition to them0p1/2el continua (denoted p). The curves show: (a) and (e), resonance profiles; (b) and (i)–(k), branching ratios; (c), (d) and(f)–(h), asymmetry parameters b . [Ca figures are from Lange et al . (1989), courtesy of W. Sandner and Ba figures are from Lange,Aymar, et al. (1991).]



(one for each final ionic state) were adjusted by Hiero-nymus et al . (1990) on the ion-yield and asymmetry pa-rameters. Figures 38(g)–38(i) show the eigenchannelR-matrix results.We first comment briefly on the experimental results

analyzed in greater detail by Hieronymus et al . (1990).The variations of the asymmetry parameters bs , bd3/2

,and bd5/2

associated with the 6s , 5d3/2 , and 5d5/2 finalionic states show pronounced modulations, almost peri-odic in n6p1/2, the effective quantum number associatedwith the 6p1/2 limit. As explained by Hieronymus etal . (1990), the positions of the resonances are stronglycorrelated with the positions of the 6p1/2ns ,nd Rydberglevels, which are not excited directly but only throughtheir coupling with the 6p3/212s resonance. However thedata are not strictly periodic in n6p1/2, changes in the

various shapes occurring with increasing energy. Thesechanges, particularly visible in Fig. 38(e), are due to the6p3/210d (at n6p1/2;18) and 6p3/211d (above the 6p1/2limit) levels located on both sides of the 6p3/212s reso-nance.We turn now to the description of eigenchannel

R-matrix results obtained from two different short-range reaction matrices of dimension 13. The first one,obtained by combining LS-coupled R-matrix calcula-tions with the jj-LS frame transformation, is the onethat successfully reproduced the photoabsorption spec-trum (Aymar, 1990) and the ICE data of Lange, Aymar,et al . (1991). The second matrix K was obtained withthe jj-coupled R-matrix method. Both K matrices givealmost identical results for bs and bd3/2

and, for the sakeof clarity, only LS results are shown in Figs. 38(g) and

FIG. 38. Angular distributions of Ba 6p1/2ns , nd J=1 levels degenerate with the 6p3/212s level and excited from the 6s12s 1S0level by driving the 6s→6p3/2 core transition. Data are plotted as functions of the effective quantum number n6p1/2 associated withthe 6p1/2 limit. Experimental results of Hieronymus et al . (1990) for the asymmetry parameters bs , bd3/2

, and bd5/2, associated with

the final 6s , 5d3/2, and 5d5/2 ionic states are shown (a), (c), and (e), respectively. The curves (b), (d), and (f) show the results ofthe MQDT simulations carried out by the same authors. At the right, (g)–(j) show the R-matrix results: (g) LS result for bs ; (h)jj-LS result for bd3/2

; (i) jj-LS result for bd5/2; (j) jj result for bd5/2

. [Figs. 38(a) to 38(f) are from Hieronymus et al . (1990),courtesy of H. Hieronymus.]



38(h). In contrast, significant differences exist for bd5/2between the frame-transformation and jj results shownon Figs. 38(i) and 38(j), respectively. The calculationscorrectly give the periodic modulations as well as thechanges with increasing energy observed experimentallyfor the asymmetry parameters bs and bd3/2

. Comparisonof Figs. 38(i) and 38(j) with Fig. 38(e) clearly shows thatthe marked decrease of the resonance amplitudes ob-served for bd5/2

is better reproduced by the fullyjj-coupled R-matrix calculation. In addition, the bd5/2spectrum is described slightly better by this R-matrixcalculation than by the MQDT simulation of Hierony-mus et al . (1990), whereas the fitted and calculated re-sults are very close for bs and bd3/2

.Although not shown here, the R-matrix calculations

also trace out the modulations observed in the branch-ing ratios rs , rd3/2 and rd5/2; here no significant differ-ences between the LS results (and frame transforma-tion) and the jj results are found. The detailed shapes ofthe resonances showing up in the total ion yield are alsovery well reproduced by the two different R-matrix cal-culations, while some discrepancies exist for the relativeheights of the successive peaks. Interestingly, this showsthat resonance profiles (usually considered to be lesssensitive to channel mixing than angular distributions)can in some cases give a more sensitive test of theory.This conclusion was also reached by Lange, Aymar, etal . (1991).In Mg, the angular distributions of ejected electrons

from autoionizing 3pns J51 levels of Mg were mea-sured by Lindsay, Cai, et al . (1992) as functions of theenergy of the third laser, which drives the isolated-coreexcitation 3sns→3pns . The experimental bs spectrum(full line) obtained for the 3s14s→3p14s excitation iscompared with the R-matrix calculation (dotted line) inFig. 39. The spectrum covers an energy range of ; 800cm21 centered on the Mg1 ion transitions 3s→3p1/2and 3s→3p3/2 marked by vertical bars at the top of thefigure. The calculations were carried out using a seven-channel reaction matrix determined by Greene (1990b)with LS-coupled R-matrix calculations combined withthe jj-LS frame transformation. The LS-coupled chan-nels were: (3snp , 3pns , 3pnd) 1P1, (3snp , 3pns ,3pnd) 3P1, and 3pnd 3D1. The same K-matrix wasalso successfully employed to reproduce the line shapesof 3pnd J51 autoionizing levels (Schinn et al . , 1991).Figure 39 shows how the calculations reproduce the ex-perimental data almost perfectly. The only discrepanciesthat cannot be accounted for by experimental errors oc-cur very close to the Mg1 resonance lines.Examination of Fig. 39 reveals that b;2, except for

dips, with minima up to b;21. A broad dip appearsbetween the ionic lines and narrower dips are located atthe positions of the 3p1/2,3/2ns ,nd autoionizing levels.The b;2 value corresponds to the excitation of a pure3pns 1P1 channel, which corresponds to a coherent su-perposition of 3p1/2ns and 3p3/2ns channels, followed bydirect scattering into the 3sep 1P1 continuum. Theb52 value is observed far from the Mg1 resonance

lines and away from the 3pns ,nd resonances. Indeed,far from the Mg1 lines, both 3p1/2ns and 3p3/2ns chan-nels are excited and starting from a 3sns 1S0 level it isreasonable, in Mg where spin-orbit effects are veryweak, to assume that a 3pns 1P1 level is populated anddecays immediately. Departures from b;2 occur at thepositions of the 3pns ,nd resonances that are not exciteddirectly. Similar dips are exhibited by the the bm0s

spec-tra in Ca and Ba (see Figs. 37 and 38), but the dips aredeeper in Mg than in Ca and Ba. Departures fromb;2 between the ionic lines are caused by large differ-ences between the 3p1/2ns and 3p3/2ns excitation ampli-tudes in that energy range.A far more complicated type of angular distributions

arises when electrons are ejected from the autoionizing3p12d J51 and J53 levels of Mg, which is the situationtreated in Fig. 40. These levels were reached using athree-step excitation scheme via the 3s12d 1D2 level(Lindsay, Dai, et al . , 1992). The three lasers were lin-early polarized in the same direction, which ensured thatonly J51 and J53 levels were excited, and which alsopreserved cylindrical symmetry of the resulting angulardistribution. The differential photoionization cross sec-tion for electrons ejected from an aligned 3snd 1D2 Ry-dberg level has the form:

ds

dV~u!5

s

4p@11bP2~cosu!1gP4~cosu!

1eP6~cosu!# , (4.15)

where s is the total cross section integrated over thesolid angle V , u the angle between the third laser polar-ization and the direction of the ejected electron, and

FIG. 39. Angular distributions of ejected electrons from theautoionizing 3p14s J51 state of Mg. The experimental resultsfor the asymmetry parameter b for the transition 3s14s →3p14s (full line), as functions of the photon energy, are com-pared with results (dotted line) obtained with LS-coupledR-matrix calculations combined with the jj-LS frame transfor-mation. The vertical bars at the top indicate the positions ofthe 3s → 3p1/2 and 3s → 3p3/2 ionic transitions. (From Lind-say, Cai, et al . , 1992, courtesy of T. F. Gallagher.)



Pk(cosu) is the kth Legendre polynomial. The three an-gular parameters b , g , and e as functions of the energyconstitute the experimental data that were measuredand compared to R-matrix calculations. The experimen-tal b , g , and e parameters obtained over a range of 400cm21 centered on the peaks of the 3s12d→3p12d tran-sition correspond to the jagged solid line, and the theo-retical results obtained with two different calculationsare shown as smooth solid and dotted lines, respectively.Two different R-matrix determinations of the

LS-coupled reaction matrices KJ51 and KJ53 were car-ried out by Greene (1990b). The first set, successfullyused to interpret the excitation spectra of 3pnd J51and J53 states of Mg (Dai et al . , 1990; Schinn et al . ,1991) was obtained using a reaction volume of radiusr0512 a.u. The 3png J53 channels were ignored andonly five LS-coupled channels were introduced: (3snf ,3pnd) 1F3, (3snf , 3pnd)

3F3, and 3pnd 3D3. Corre-sponding results are shown on Fig. 40 as dotted lines.The second treatment, whose results are marked bysolid lines, used a larger radius r0520 a.u. and includedin addition the 3png 1F3,

3F3, and3G3 channels. Both

calculations included fine-structure effects through thejj-LS frame transformation. Figure 40 shows that theangular distribution parameter spectra are complex andvary irregularly with energy. They appear drastically dif-ferent from the b spectra for the 3pns states (Fig. 39).This is not surprising because the 3pnd states are morecomplex and two angular momenta J51 and J53 areexcited. In contrast with the situation for the 3pns bparameter, there is no obvious correlation between thefluctuations of the b , g , and e parameters and the posi-

tions of the 3pns ,nd J51 and J53 resonances. Theresults obtained with the two different calculations arealmost the same and, in light of the experimental uncer-tainties, it is clear that theory agrees well with the ex-periment. Among the matrix elements involved in thecalculation of the b and g parameters, those corre-sponding to pure J51 and J53 terms can be distin-guished from those associated with mixed-J terms. Itwas found that the structures of the b and g spectrawere essentially formed by the mixed-J terms and alsothat the 3pnd resonances autoionized preferentially intothe 3sef continua, favoring the ejection of the photo-electrons along the polarization axis.We conclude from this section on observables re-

corded with the ICE experimental method that theeigenchannel R-matrix method, when combined withmultichannel quantum-defect theory, provides a power-ful tool for theoretical multichannel spectroscopy. Thismethod can describe many subtle features of the excita-tion spectra, branching ratios and angular distributionsof the emitted electrons. In all the cases studied, agree-ment ranging from excellent to satisfactory was achievedbetween high-precision experiments and theory. To ourknowledge, no other theoretical method has been usedto calculate such observables in alkaline earths to date.Of course, various sets of short-range scattering param-eters were fitted to agree with particular sets of experi-mental data, in semiempirical MQDT analyses. For ex-ample, Hieronymus et al . (1990) fitted six differentmodels to the data they obtained for 6pns J51 levels ofBa, below or above the 6p1/2 ionization limit. However,as Aymar and Lecomte (1989) analyzed in detail for aspecific example in Sr, the fitted parameters differ con-siderably from those calculated with the eigenchannelR-matrix method. This implies that in some complicatedmultichannel systems such semiempirical fits do not rep-resent the short-range interactions alone, but rather rep-resent an aggregate influence of all forces that are notconsidered explicitly.

V. OPEN-SHELL ATOMS

The eigenchannel R-matrix approach in combinationwith multichannel quantum-defect theory has been suc-cessfully applied to the calculation of photoionizationand bound-state properties of several open p-subshellatoms and of two open d-subshell atoms, scandium andtitanium. The first eigenchannel R-matrix study carriedout in an atom with more than two valence electronsconcerned Al, which has three electrons in the n=3 shell(O’Mahony, 1985), followed by Si, which has four va-lence electrons in the same shell (Greene and Kim,1988). This calculation was the first application of thestreamlined eigenchannel R-matrix approach to de-scribe atomic systems with more than two valence elec-trons. The efficiency of the streamlined eigenchannelR-matrix approach (Greene and Kim, 1988) led to theadoption of this reformulation in almost all subsequenteigenchannel R-matrix calculations. Calculations havesince been carried out for the halogen atoms (F to I)

FIG. 40. Angular distributions of electrons ejected from theautoionizing 3p12d J51 and J53 levels of Mg. The threeexperimental angular parameters b , g , and e , as functions ofthe photon energy (jagged lines), are compared with twoR-matrix calculations (full and dotted lines)—see the text. Thehorizontal lines indicate the zeros, sJ is the relative photoion-ization cross section (in arbitrary units) contributed by theJ51 (solid line) and J53 (dot-dashed curve) final states. Thetop of the figure shows the effective quantum number scalesrelative to the 3p1/2 and 3p3/2 thresholds in unit steps. (FromLindsay, Dai, et al . , 1992, courtesy of T. F. Gallagher.)



(Robicheaux and Greene, 1992, 1993a), for the carbon-group atoms (C to Sn) (Robicheaux and Greene,1993b), and more recently for the oxygen-group atoms(O to Te) (Chen and Robicheaux, 1994). Another theo-retical study of this type by Miecznik et al . (1995)treated Al ground-state photoionization. These series ofcalculations were motivated largely by a desire to ex-tract similarities and contrasts in the Rydberg-channeldynamics of chemically similar elements, and by a desireto isolate and categorize the most important physicaleffects. They also test the accuracy of the theoreticaldescription of complex atoms that can be achievedthrough small-scale calculations.The exploration of channel-interaction dynamics in

complex open-shell atoms (Armstrong and Robicheaux,1993; Robicheaux and Greene, 1993c, 1993d) began inearnest with calculations for scandium, the simplesttransition-metal atom. A second complex atom with anopen d subshell was studied subsequently, in calcula-tions performed for titanium (Miecznik and Greene,1996). Results for atoms with an open p subshell and foratoms with an open d subshell are summarized below.

A. Atoms with an open p subshell

1. Aluminum

We begin by describing a calculation carried out byO’Mahony (1985) in Al. The even-parity 2D symmetryof Al involves the 3s2nd Rydberg series perturbed bythe 3s3p2 configuration, as was discerned from an earlysemiempirical MQDT analysis carried out by Lin (1974).O’Mahony’s calculation provided a better understandingof the strong 3s2nd–3s3p2 interaction. This mixingserves as a prototype for the 3s23pqnd-3s3pq12 inter-action, which is known to dominate the valence electrondynamics of third-row metalloide atoms and ions. Thismixing was analyzed in parallel with the 3snd-3p2 mix-ing in Mg 1De. The 3s2nd-3s3p2 channel mixing in Al2De and the 3snd23p2 channel mixing in Mg 1De nearthe lowest threshold were found to be very strong. Inthis regard, these systems are similar to the m0snp-m0pns

1Po mixing in Be and Mg, and to the m0snp-(m021)dnp 1Po mixing in the heavier alkaline earths(see Sec. IV).The early eigenchannel R-matrix calculation in Al

(O’Mahony, 1985) was conducted in LS coupling. More-over, it used a relatively crude unoptimized Hartree-Slater potential. In contrast, the most recent calculationof the photoionization spectrum of Al (Miecznik etal . , 1995) used the improved eigenchannel R-matrixtechniques developed for open-shell atoms (see Sec.III.D). Moreover fine-structure effects were includedthrough the jj-LS frame transformation. One major mo-tivation of this study was to analyze the influenceof spin-orbit effects on the photoionization spectrumof the 3s23p 2Po ground state below the 3s3p 1Po

ionization threshold. There are ten channelsand three LS symmetries that can be excited by onephoton from the ground state: [(3s2 1Se)ns 2Se],

[(3s2 1Se)nd 2De], [(3s3p 3,1Po)np 2Se, 2Pe, 2De],[(3s3p 3,1Po)nf 2De]. In LS coupling, the channels cor-responding to different LS symmetries do not interact,since all non-Coulombic interactions, including fine-structure effects, are omitted from the Hamiltonian. Thedecay of (3s3p 3,1Po)np 2Pe resonances is forbiddenbelow the 3Po threshold within this LS-coupling ap-proximation. The spin-orbit interaction couples thechannels mentioned above to each other and to a largenumber of other channels associated with five additionalLS symmetries: 4Se, 4Pe, 4De, and 2,4Fe. EigenchannelR-matrix calculations were carried out using a box of 15a.u. Configuration-interaction wave functions were de-termined for the three target states of Al1, namely3s2 1Se, 3s3p 3Po, and 3s3p 1Po. Basis functions usedto describe all eight final LS symmetries were con-structed by attaching s , p , d , f , and g orbitals to thecomponents of the expansion of the ionic states. Addi-tional correlation-type functions such as, for instance,3s3p2 were also included. Length and velocity photo-ionization cross sections from the 3s23p 2P1/2,3/2

o groundstate were calculated by assuming a statistical mixture ofthe two initial J=1/2 and J=3/2 levels. Length and veloc-ity results agreed to 1% accuracy typically, except atenergies where they were small. Theoretical results werecompared with the experimental measurements of Roig(1975) and with the Wigner-Eisenbud-type R-matrix cal-culation of Tayal and Burke (1987), which was con-ducted in LS coupling.The cross section at the 3s2 threshold has an unusu-

ally large value that derives from the presence of the3s3p2 2De level, which is diluted into the 3s2nd 2De

bound states and the 3s2ed 2De continuum (Lin, 1974;O’Mahony, 1985; Komninos et al . , 1995). The calcu-lated threshold value of 61 Mb agreed well with the ex-perimental value of 65 67 Mb (Roig, 1975) while Tayaland Burke (1987) found a slightly smaller value of 55Mb. Aside from spin-orbit effects, the photoionizationcross section below the 3s3p 3P2

o threshold agreed withthe spectrum calculated by Tayal and Burke (1987). Thelowest autoionizing resonance is the 3s3p2 2Se level.The next broad resonances are members of the(3s3p 3Po)np 2De series, upon which sharp peaks aresuperimposed. These narrow peaks were ascribed to theLS-allowed (3s3p 3Po)np 2Se and (3s3p 3Po)nf 2De

series and to the LS-forbidden (3s3p 3Po)np 2Pe and(3s3p 3Po)np 4De series, which appear in the spectrumowing to their fine-structure-induced mixing with chan-nels of the 2De symmetry. Due to the occurrence ofnumerous sharp and overlapping resonances that con-verge to the closely spaced 3s3p 3PJ

o thresholds, thespectrum including spin-orbit effects takes on a compli-cated appearance. Conspicuous spin-orbit effectsarise just below the 3s3p 3P2

o threshold, including dra-matic examples of enhancement or reduction of theresonance widths. For instance, the widths of thehigh-n LS-forbidden (3s3p 3Po)np 4D3/2,5/2

e and(3s3p 3Po)np 2P1/2,3/2

e resonances are larger or of thesame order of magnitude as the widths of the



LS-allowed (3s3p 3Po)np 2Se resonances. Indeed,since the widths for the 2De series are about 10 3 timeslarger than for 2Se, a very small admixture of 2De char-acter may cause the LS-forbidden states to autoionizefaster than the 2Se states. Due to the n-dependence ofthe mixing between the (3s3p 3Po)np 2P3/2

e and(3s3p 3Po)np 2D3/2

e series, the widths exhibit an irregu-lar evolution as a function of energy along variousLS-forbidden series. The quantum defects of severalRydberg levels were found to agree with the availableexperimental data (Roig, 1975). In addition, Mieczniket al . (1995) analyzed the transition from LS to jj cou-pling for high-lying Rydberg levels below the 3s3p 3Pin the context of statistical distributions. While the spec-trum exhibits a type of ‘‘chaotic’’ behavior according tothese distributions, striking regularity remains in thepattern of energy levels and widths. One specific effectthat is noteworthy is the periodic occurrence of verylong-lived resonances.Despite the strong influence of spin-orbit effects on

the photoionization spectrum below and among thespin-orbit-split 3s3p 3P0,1,2

o thresholds, these effects re-main almost negligible between the 3s3p 3P2

o and3s3p 1P1

o thresholds since the 1P1o threshold has no

such splitting.

2. Halogens

Photoionization cross sections of the m0s2m0p

5 2Po

ground state have been calculated in F (m0=2), Cl(m0=3), Br (m0=4), and I (m0=5) near the m0s

2m0p4

thresholds (Robicheaux and Greene, 1992), and in Cland Br near the higher-lying m0sm0p

5 thresholds (Ro-bicheaux and Greene, 1993a). In the series of R-matrixcalculations conducted for F through I, a reaction vol-ume of radius r05 9 a.u. was used. The same choice ofbasis functions was adopted for the different atoms, al-though the radial orbitals differed from one atom to an-other. The basis function sets were chosen to obtain con-verged results for I, which made necessary the inclusionof basis functions unnecessary for lighter halogens. Them0s

2m0p5 2Po ground state was described with ;300

basis functions. The target states in order of increasingenergy were the m0s

2m0p4 3Pe, 1De, and 1Se ionic

states. When combined with the jj-LS frame transfor-mation, there were seven different final LS symmetries(2Se, 2,4Pe, 2,4De, 2,4Fe) and 15 final-state channels thatentered the R-matrix calculations. These channels wereconstructed by attaching s and d waves to the threeconfiguration-interaction target states. The basis set de-pended on the LS symmetry and the largest basis setsize was ;700 for the 2De symmetry. The jj-LS frametransformation in its simplest and most common formcouples only closed channels converging to the same2Sc11Lc threshold. (The index c refers to the ionic core.)For all atoms, except those with a single electron or holein the target valence shell, the spin-orbit interaction notonly splits the target states having the same Lc ,Sc butdifferent Jc , but also couples the states with the sameJc but different Lc ,Sc . In the halogens the m0s

2m0p4

3P0e state mixes with the m0s

2m0p4 1S0

e state, and them0s

2m0p4 3P2

e state mixes with the m0s2m0p

4 1D2e

state. These mixings were accounted for by anintermediate-coupling frame transformation obtainedempirically from the experimental ionization energies. Itis worth noting that some states would not autoionizewithout this frame transformation. Note that thisintermediate-coupling frame transformation has notbeen utilized in eigenchannel R-matrix calculations car-ried out for other open-shell atoms.Figure 41 shows the photoionization cross sections for

Cl, Br, and I between the 3Pe and 1De thresholds. Themost striking feature of this figure is the remarkablesimilarity between Cl @Figs. 41(a), 41(b)], Br @Figs.41(c), 41(d)], and I @Figs. 41(e), 41(f)]. The broad reso-nances are superposition of two broad autoionizing se-ries, (1De)nd 2Pe and (1De)nd 2De, while the sharpresonances are (1De)ns 2De and (1De)nd 2Se. Interest-ingly, (1De)nd 2Se resonances would not be able to au-toionize in this theoretical description, except throughthe operation of the intermediate-coupling frame trans-formation. The photoionization cross section for F in thesame energy range (not shown here) strongly differsfrom those displayed in Fig. 41 for the heavier halogens.In F, the (1De)nd resonances are sharper due to thelack of overlap between a d electron and the innerm0p electron. Figure 41 shows that the calculated photo-ionization spectra reproduce most features in the experi-mental data (Berkowitz et al . , 1981; Ruscic andBerkowitz, 1983; Ruscic et al . , 1984b). The agreementwas found to be the best in Cl. For Br, the broad(1De)nd 2Pe and (1De)nd 2De resonances were shiftedslightly too high in energy, the error in the calculatedquantum defects amounting to ;0.05. Similarly goodagreement between theoretical and experimental (Rus-cic et al . , 1984a) spectra was observed for F, althoughan error of ;0.01 in the quantum defect of the(1De)3s 2Se level caused a reversal in the order of thelowest autoionizing lines compared to the experiment.For the photoionization spectra between the 1De and1Se thresholds, experimental data exist only for Cl and I(Ruscic and Berkowitz, 1983; Berkowitz et al . , 1981).Good agreement between theory and experiment wasfound for Cl. The poorer agreement observed for Iprobably stems from the fact that the much larger spin-orbit splittings in I are too strong to be fully accountedfor by frame-transformation procedures.The photoionization cross sections of some halogen

atoms have been calculated using different theoreticalapproaches. Each halogen atom from F through I wasstudied by Manson et al . (1979) using both the Hartree-Fock and the central-potential-model approximations.Cl was investigated by Starace and Armstrong (1976)using the random-phase approximation with exchange,by Fielder and Armstrong (1983) using the multiconfigu-ration Hartree-Fock method, by Shahabi et al . (1984)with an open-shell transition matrix method, by Lam-oureux and Combet Farnoux (1979) with the Wigner-



Eisenbud-type R-matrix method, and by Brown et al .(1980) using many-body perturbation theory. All thesecalculations were performed in strict LS coupling. Reso-nance structures were calculated only in the last twostudies mentioned. In addition, I photoionization wascalculated by Combet Farnoux and Ben Amar (1986)using a K-matrix method. The resonance structures ob-tained by Robicheaux and Greene (1992) compared wellwith those predicted by Brown et al . (1980), whereasonly a moderate agreement with the predictions ofCombet Farnoux and Ben Amar (1986) was found. An-other point of comparison of the eigenchannelR-matrix results of Robicheaux and Greene (1992) withprevious calculations is the magnitude of the total crosssection at the 1Se threshold. In Cl, the theoretical valueof 3862 Mb is smaller than the experimental value of43.663.5 Mb obtained by Samson et al . (1986). Thethreshold cross sections predicted for Cl by other meth-ods range from 25 Mb to 60 Mb. Values obtained by theeigenchannel R-matrix calculations are smaller than thevalues obtained by Manson et al . (1979) except for F,where they agree. The threshold value of the I crosssection obtained by Robicheaux and Greene (1992) wassmaller than the value predicted by Combet Farnouxand Ben Amar (1986).Robicheaux and Greene (1992) further analyzed the

similarities among the halogens by comparing the short-range scattering parameters obtained for every halogenatom as functions of energy. The striking similarity be-tween the heavier halogens that is apparent in Fig. 41, aswell as the major differences from F, emerge clearlyfrom this comparison. Moreover, this analysis of channelmixing helps to explain why all the ns resonances arenarrow, while the nd resonances are very broad in allhalogen atoms except F.A second eigenchannel R-matrix study for the halo-

gens determined partial and differential photoionizationcross sections of Cl and Br near the higher-lyingm0sm0p

5 3,1Po thresholds (Robicheaux and Greene,1993a). The eigenchannel R-matrix calculations weresimilar to those performed to calculate photoionizationcross sections at lower energy. However, the(m0sm0p

5 3Po,1Po)np channels were added in orderto describe the m0sm0p

5np autoionizing resonances.Observables calculated included partial photoionizationcross sections and asymmetry parameters characterizingthe energy and angular distributions of photoelectronsejected, leaving the ionic core in the states m0s

2m0p4

3Pe, 1Se, and 1De. Results calculated for Cl below the3s3p5 3Po threshold compared well with the experimen-tal data obtained by van der Meulen, Krause, et al .(1992). The agreement deteriorated at energies near the1Po threshold. Most features in the Br experimentalpartial cross sections below the 4s4p5 3Po threshold(van der Meulen, Krause, and Lange, 1992) were alsoreproduced by the R-matrix calculation. The calcula-tions accounted for singly excited state resonances only,but omitted doubly excited states such as the Cl3s23p33d4p level. This omission appears to be respon-

FIG. 41. Photoionization of the m0s2m0p

5 2P3/2o ground state

of halogens Cl to I between the m0s2m0p

4 3P and 1D thresh-olds. The calculated length (solid line) and velocity (dottedline) cross sections [(a), (c), and (e)] have been convolved withthe experimental resolution of 0.28 Å. The theoretical resultsin Cl (a), Br (c), and I (e) are compared with the photoabsorp-tion spectra (b), (d), and (f) recorded by Ruscic et al . (1983),Ruscic et al . (1984b), and Berkowitz et al . (1981), respec-tively. (From Robicheaux and Greene, 1992).



sible for most of the discrepancies between the calcula-tions and the experimental data.Robicheaux and Greene (1993a) compared the values

they obtained in Cl for partial cross sections and asym-metry parameters at a fixed energy far from resonancesto the values derived by Shahabi et al . (1984) with anopen-shell transition matrix method that neglected theresonance structures. Both sets of partial cross sectionsvalues agreed well with the experimental data of van derMeulen, Krause, et al. (1992), while Robicheaux andGreene’s asymmetry parameters agreed better with theexperimental results than did the values obtained byShahabi et al. (1984). Robicheaux and Greene (1993a)also compared the resonance structures they obtained inCl with those predicted by Brown et al . (1980) usingmany-body perturbation theory. Because these latter au-thors carried out their calculations in LS coupling, thecomparison dealt only with LS results. Both sets oftheoretical results agreed well below the 3Po threshold,while differences appeared near the 1Po threshold.

3. Carbon-group atoms

The carbon-group atoms are characterized by am0s

2m0p2 3Pe ground state with m0=2 for C, m0=3 for

Si, m0=4 for Ge, and m0=5 for Sn. The first applicationof the streamlined eigenchannel R-matrix approach(Greene and Kim, 1988) dealt with the J=0 even-paritybound spectrum of Si, which involves the two3s23p1/2np1/2 and 3s23p3/2np3/2 Rydberg series. It wasfound that LS-coupled R-matrix calculations combinedwith the jj-LS frame transformation accounted well forthe experimental energy positions (Martin and Zalubas,1983).A series of small-scale calculations carried out by Ro-

bicheaux and Greene (1993b) in the carbon group con-sidered the odd-parity bound Rydberg spectra withJ=0–3 and the photoabsorption spectra. TheseLS-coupled eigenchannel R-matrix calculations fol-lowed by the frame transformation were focused on theregion below and between the m0s

2m0p2P1/2

o and2P3/2

o thresholds. The R-matrix volume radius wasr0515 a.u. As for the halogens, the basis-function setsused to describe initial and final states were the same forthe different atoms; the ground state, the final states,and the m0s

2m0p2Po target state were all computed

using configuration interaction. The m0s2m0p

2 3Pe

ground state was described by ;150 basis functions.Study of the odd-parity spectra with J=0–3 required theR-matrix calculations to be performed for six differentLS symmetries: 1,3Po, 1,3Do, 1,3Fo. The largest basis-setsize was ;275 for 1,3Po symmetries. Basis functions forthe final-state symmetries were constructed by attachinga s or d wave to the odd-parity m0s

2m0p2P

configuration-interaction target state. In addition tothese close-coupling-type variational basis functions,correlation-type basis functions were introduced as well.Basis functions associated with strongly closed channelswere constructed by attaching a p or f wave to theconfiguration-interaction even-parity target states

m0sm0p2, m0s

2m0d , and m0s2(m011)s . These latter

channels are relevant to the study of the odd-parityspectra with J=0–3 because the (m0s

2m0p2Po)ns and

(m0s2m0p

2Po)nd Rydberg series converging to thetwo ionic thresholds m0s

2m0p2PJc

o (Jc=1/2 and 3/2) are

perturbed by m0sm0p3 states.

Oscillator strengths for transitions from them0s

2m0p2 3P0

e ground state to odd-parity bound levelswith J=1 were calculated for all elements. For C, whereprevious experimental and theoretical values were avail-able for some lines (Nussbaumer and Storey, 1984;Goldbach and Nollez, 1987; Goldbach et al . , 1989),theoretical results compared well with the data. Ro-bicheaux and Greene (1993b) compared the photoab-sorption spectra between the m0s

2m0p2PJc

o thresholdsfor the different atoms (see their Fig. 8). The cross sec-tions exhibited one (m0s

2m0p2P3/2

o )ns autoionizingRydberg series and two (m0s

2m0p2P3/2

o )nd autoioniz-ing series converging to the 2P3/2

o threshold. Thes-wave resonances were sharper than the d-wave reso-nances except in C, where all resonances had smallwidths because, as in F, the d waves did not interactstrongly with the core. The Sn and Ge autoionizationspectra resembled each other while the Si and mainlythe C, the spectra looked very different. The only ex-perimental spectrum available for comparison was adensitometer trace constructed by Brown et al . (1977a)for Ge. The agreement between experiment and theorywas very good for the positions of the resonances butsomewhat poorer for the line shapes.We consider now the results obtained for the energy-

level positions of odd-parity bound Rydberg levels withJ=0–3. Extensive experiments conducted on the carbon-group atoms were interpreted using approximate Lu-Fano plots drawn through experimental quantum de-fects, or through the use of semiempirical MQDT fits(Brown et al . , 1975; Feldman et al . , 1976; Brown etal . , 1977a, 1977b; Ginter et al . , 1986; Ginter and Ginter,1986). The theoretical energy values provided by eigen-channel R-matrix calculations were compared with ex-perimental data using Lu-Fano plots. Very good agree-ment was found for all symmetries in C and Si and in theJ=0 symmetry of Ge and Sn. For J51–3, the agreementbetween the experimental and theoretical results ispoorer for Ge and Sn than was obtained for the lighteratoms. The main reason for the discrepancies is thestrong energy dependence of several short-range scatter-ing parameters due to the presence of the m0sm0p

3

states. In Ge and Sn, these states cause rapid energydependence of the quantum defects in 1,3Po and 1,3Do

symmetries, with the strongest effects occurring nearthreshold in the 3Do symmetry. This large energy de-pendence makes the application of the jj-LS frametransformation somewhat problematic. Indeed, depend-ing upon whether the scattering parameters were re-ferred to theoretical or experimental thresholds, notice-ably different results were found. Importantly, them0sm0p

3 states cause an energy dependence of the scat-tering parameters in all atoms. However, the carbon pa-



rameters are energy dependent only at low energy, whilein Si, Ge, and Sn, the energy dependence is strong overa much larger range. Carbon and silicon have muchsmaller spin-orbit splittings than heavier carbon-groupelements, whereby the jj-LS frame-transformation ap-proximation remains valid in C and Si.Note finally that the calculation of Robicheaux and

Greene (1993b) provided the first nearly ab initio de-scription of these spectra for the carbon-group atoms.The calculations described the energy dependence of dy-namical scattering parameters better than semiempiricalfits (Brown et al . , 1977a, 1977b; Ginter et al . , 1986;Ginter and Ginter, 1986). Indeed, the parameters intro-duced in the fits were, in most cases, assumed to be en-ergy independent.

4. Chalcogens

Chen and Robicheaux (1994) calculated the photoion-ization cross sections of the m0s

2m0p4 3P2

e groundstates of O (m052), S (m053), Se (m054), and Te(m055) using LS-coupled eigenchannel R-matrix cal-culations combined with the jj-LS frame transforma-tion. As was described in Sec. III.D, calculations werecarried out with the Hamiltonian referring to the fullatom [Eq. (3.41)]. The dipole polarizability of the corewas accounted for in the Se and Te calculations only.The calculations were conducted using the Wigner-Eisenbud-type reformulation of the eigenchannel ap-proach (Schneider, 1975; Robicheaux, 1991). Calcula-tions were restricted to photoionization of the outermostm0p subshell, the relevant target states in LS couplingbeing m0s

2mop3 4So, 2Do, and 2Po, in order of increas-

ing energy. In LS coupling, there are nine channels andthree LS symmetries that can be excited by one photonfrom the ground state: [(4So)ns 3So, nd 3Do],[(2Do)nd 3So, nd 3Po, ns , nd 3Do], [(2Po)ns ,nd 3Po, nd 3Do]. When spin-orbit effects are neglected,channels corresponding to different LS symmetries donot interact; in particular the (2Do)nd 3Po states arenot coupled to any continuum, which means that thedecay of (2Do)nd 3Po resonances is forbidden in LScoupling below the 2Do threshold. The spin-orbit inter-action can couple the channels mentioned above to eachother and to a large number of other channels such asthe (4So)ns 5So, nd 5Do, and similar channels associ-ated with 2Do and 2Po thresholds. Effects of the spin-orbit interaction were incorporated approximately,through the jj-LS frame transformation. A reaction vol-ume of radius of 11 a.u. was used for O and one of 13 a.ufor the heavier chalcogens. The rare-gas-core electronswere frozen and no correlation or excitation of theinner-core electrons was included. Configuration-interaction wave functions for the target states weredetermined variationally as a superposition of onlyfive configurations: m0s

2m0p3, m0s

2m0pm0d2,

m0s2m0p

24f , m0sm0p3m0d , and m0p

5. For a given con-figuration, all possible intermediate angular couplingswere introduced into the many-electron basis set. Thesame list of configurations was used for the different

atoms. The differences between the atoms were the ra-dial orbitals of all electronic states involved.Photoionization cross sections were calculated at

final-state energies between the 4So and 2Do thresholdsand between the 2Do and 2Po thresholds. Theoreticalcross sections were convolved with the experimentalwidth using a preconvolution technique (Robicheaux,1993). Chen and Robicheaux (1994) focused on theheavier chalcogens S, Se, and Te, which have been stud-ied far less than oxygen (for references see Seaton,1987b). The spectrum of O differs from the spectra ofthe other chalcogens since it consists of sharp reso-nances only. As in C (Robicheaux and Greene, 1993b)and F (Robicheaux and Greene, 1992) in the secondrow, the absence of broad features in O is a consequenceof the small overlap between the 2p and nd orbitals. Werestrict our discussion to the heavier chalcogens, wherecorrelations are stronger than in O.The calculated photoionization cross section of S be-

low the 2Do threshold reproduced the experimentalspectrum of Gibson et al . (1986a). The positions andwidths of the resonances pertaining to the four seriesattached to the 2Do threshold were well described. Inparticular, as observed, the LS forbidden (2Do)nd3Po appeared as very intense and narrow resonances.However, the calculation did not reproduce the relativeintensities of those resonances perfectly. There are three(2Po)3d 3Do, 3Po and (2Po)5s 3Po perturbers attachedto the 2Po threshold which fall below the 2Do threshold;the description of those perturbing resonances wasslightly poorer. Indeed, the error in energy of the per-turber is DE (a.u.)5dm/n3, which can be large when theeffective quantum number n in the perturbing channel issmall. The intensities and shapes of nearby resonancesare very sensitive to the exact position of the perturbers.The resonances associated with the cross section be-tween the 2Do and 2Po thresholds are dominated by the(2Po)nd 3Do and 3Po resonances, which are broad andoverlapping. Agreement with experiment (Gibson etal . , 1986a) was quite satisfactory.Photoionization cross sections of S have been calcu-

lated using different theoretical approaches (Mendozaand Zeippen, 1988; Tayal, 1988; Altun, 1992, and refer-ences therein). Tayal (1988) used the Wigner-Eisenbud-type R-matrix approach, Mendoza and Zeippen (1988)adopted the close-coupling approximation, while Altun(1992) implemented many-body perturbation theory.Because all of those calculations were performed in astrict LS-coupling approximation, they failed to showthe LS-forbidden (2Po)nd 3Po resonances below the2Do threshold. Chen and Robicheaux (1994) comparedtheir positions and classifications, for states in the au-toionizing series converging to the 2Do and 2Po thresh-olds, to those of Gibson et al . (1986a) and of other cal-culations. Below the 2Do threshold, the calculations ofTayal (1988) and Altun (1992) gave global agreementwith the experiment, but the errors in the quantum de-fects of some autoionizing levels were larger ( up to 0.2)than those calculated with the eigenchannel R-matrixmethod (error within 0.02 for most resonances). The



eigenchannel R-matrix calculations also resolved a con-troversy existing in the literature on the assignment ofthe (2Do)nd 3So and (2Do)ns 3Do resonances; the ex-perimental classifications of those series should be inter-changed. All previous calculations gave good agreementwith experiment in the energy range between the 2Do

and 2Po thresholds.The origin of spectral features in S was analyzed in

some detail by Chen and Robicheaux (1994). The originof the broad and sharp features of the S spectrum as wellas the differences of the quantum defects of the(2Do)nd 3So, (2Po)nd 3Po, and 3Do series could beunderstood through an examination of the matrix ele-ments that couple those channels. Chen and Robicheaux(1994) also studied the convergence of the wave-function expansion, and have shown the importance ofpolarization-type basis functions for the description ofatomic dynamics. Moreover, they compared the resultsof their full electron (16) calculation with those they ob-tained using a model potential to account for the effectsof closed shells and six valence electrons. Both calcula-tions were conducted with the same configuration list.For all levels, the quantum defects of both calculationswere found to differ by less than 0.01; negligible differ-ences were also found for dipole moments. This showedthat the Ne-like core could be well approximated with alocal potential.Figure 42 compares the photoionization cross sections

of S and Se calculated in LS coupling. This figure clearlyshows the similarities between the two atoms; it is obvi-ous that the two atoms have nearly identical electro-static interactions between the valence and Rydbergelectrons. Figure 43 shows Se cross section calculatedusing the jj-LS frame transformation to incorporatefine-structure effects. The calculated results gave global,but not detailed, agreement with the experiment of Gib-son et al . (1986b). As in S, most of the discrepancies arenear the perturbers at ; 1120 Å attached to the 2Po

threshold. In this energy range there are two peaks as-signed by the experiment as 4s4p4(4P5/2

e )4p states.

Chen and Robicheaux (1994) found that the 4s4p5

states should not be assigned to any of the experimentalresonances. They ascribed the perturbers to the(2Po)4d and (2Po)6s 3Po states, which are analogous tothe perturbers of the (2Do)ns ,nd series of S.The quality of the Te calculation of Chen and Ro-

bicheaux (1994) has noticeably deteriorated, comparedto Se. This might indicate some limitation of the frame-transformation treatment for including spin-orbit ef-fects. Another possibility is that wave function exhibitspoorer convergence. Although the resonances are morestrongly mixed in Te than in Se, Chen and Robicheaux(1994) managed to identify some of them. As in the caseof the Se 4s4p5 levels, they concluded that the 5s5p5

classification given in the experiment (Berkowitz etal . , 1981) to some of the peaks is inappropriate. Noother theoretical calculation is available for comparisonin either Se or Te.

B. Open d-subshell atoms

1. Scandium

An extensive study of the Sc atom was performed byRobicheaux and Greene (1993c, 1993d) and Armstrongand Robicheaux (1993) using the LS-coupled eigen-channel R-matrix method combined with the jj-LSframe transformation. The main purpose of these inves-tigations was to test whether the eigenchannelR-matrix method could reproduce the extremely com-plicated spectra of a transition-metal atom. The goodagreement between theory and experiment obtained forphotoionization spectra from the Sc ground state3d4s2 2D3/2,5/2

e and from the Sc 3d3 2D3/2e excited state

demonstrated, for the first time, that the eigenchannelR-matrix techniques could achieve the accuracy neededto describe the complex spectra of Sc. The studies per-

FIG. 42. Photoabsorption cross sections of S (lower) and Se(upper + 1500 Mb) between the m0s

2m0p3 4So and 2Do

thresholds. LS results are drawn vs a dimensionless quantity35(v2E4S)/(E2D2E4S). (From Chen and Robicheaux,1994).

FIG. 43. The photoabsorption cross section of Se obtained bycombining LS-coupled R-matrix calculations with the jj-LSframe transformation (lower curve) is compared with the ex-perimental result (in arbitrary units) of Gibson et al . (1986b).The zero of the experimental curve corresponds to the hori-zontal line. The vertical scale of the theoretical curve has beenadjusted to reproduce the experimental mean background.(From Chen and Robicheaux, 1994, courtesy of F. Ro-bicheaux.)



formed in Sc did not aim solely at the calculation ofaccurate spectra. They also probed the reasons for thesuccess of the calculations, as well as the limitations, andthey explored the underlying atomic dynamics of Sc.The theoretical description of Sc is complicated by the

large number of closely spaced thresholds. The firstfourteen states of Sc1 have even parity, with the con-figurations 3d4s , 3d2, and 4s2. The first thirteen thresh-olds (3d4s 3D1

e , 3d4s 3D2e , 3d4s 3D3

e , 3d4s 1D2e ,

3d2 3F2e , 3d2 3F3

e , 3d2 3F4e , 3d2 1D2

e , 4s2 1S0e , 3d2

3P0e , 3d2 3P1

e , 3d2 3P2e , 3d2 1G4

e , in order of increas-ing energy) have a spread of only 14 000 cm21, while the3d2 1S0

e threshold is at 25 955 cm21 above the 3d4s3D1

e ionic ground state. The small spread of even-paritythresholds is partly due to the near degeneracy of the4s and 3d orbitals. The Rydberg series attached to theclosely spaced thresholds interact with each other, pro-ducing complicated spectra. Almost every Rydberg se-ries is perturbed and the calculated short-range scatter-ing parameters need to be very accurate to place theperturbers in the correct position. As was emphasizedabove, the theoretical spectrum may bear no resem-blance to the experimental spectrum if a perturber haseven a small error in its quantum defect.The eigenchannel R-matrix techniques used for the Sc

calculations are similar to those employed for the openp-subshell systems. However, the complexity of Sc com-pared to open p-subshell atoms led Robicheaux andGreene (1993c) to carefully analyze the factors that af-fect the convergence of the calculation. Here we de-scribe the most vital features of the method developedfor Sc. The R-matrix volume radius was r0=21 a.u. As inthe open p-subshell atoms, the parameters a l

i and rc in-troduced in the one-electron Hamiltonian Hv of Eq.(3.38) were fitted to optimize agreement between thecalculated energy levels of Hv and the experimental lev-els of Sc11. However, in Sc, a large weight was put inthe optimization on fitting the spin-orbit splitting of theSc11 3d levels, which ensured a better shape of theSc11 nd orbitals.As in previous work, the calculation began by the

choice and construction of the target wave functions.Convergence tests showed that the lower odd-parityionic states (from 26 000 cm21 to 39 000 cm21 above theSc1 ground state) play a large role in Sc dynamics. In-deed, perturbers such as 3d4p5s and 4s4p4d fall in theenergy range close to the higher even-parity thresholds;it is crucial to obtain a reasonably good description ofthese perturbers in order to obtain quantum defects thatare reasonably well converged. Moreover, the introduc-tion of these odd-parity target states was found to becrucial in order to describe the polarizabilities of thelow-lying even-parity states of Sc1.The variational basis functions were constructed with

natural orbitals, as in the calculations for p-subshell at-oms. In Sc, the natural orbitals were chosen to give thebest overall convergence for the lowest even- and odd-parity ionic target states. The initial states, the finalstates, and the target states were computed using

configuration-interaction methods. The basis functionsused to construct the even-parity 3d4s2 2D3/2,5/2

e and3d3 2D3/2

e initial states were obtained by attachings-wave, d-wave, and g-wave orbitals to the even-paritytarget states and p-wave and f-wave orbitals onto theodd-parity target states. A large number of LS- andjj-coupled channels entered the calculations of photo-ionization cross sections from the ground-state fine-structure levels 3d4s2 2D3/2,5/2

e . Construction of thejj-coupled short-range reaction matrices for final Jf val-ues ranging from Jf=1/2 to Jf=7/2 required LS-coupledeigenchannel R-matrix calculations to be carried out foreleven LS symmetries: 2,4So, 2,4Po, 2,4Do, 2,4Fo, 2,4Go,and 4Ho. For a given final LS symmetry, between twoand thirteen channels entered the calculation. For agiven Jf , anywhere from 23 to 43 channels entered thecalculation in the jj-coupled representation. Open andweakly closed channels were constructed by attachingp-wave and f-wave orbitals onto the even-parity targetstates. In addition to these channels, a large number ofstrongly closed channels were included in the calcula-tions. They were constructed by attaching s-wave,d-wave, and g-wave orbitals to the odd-parity targetstates.Two papers (Robicheaux and Greene, 1993c, 1993d)

treated the photoabsorption spectrum out of theground-state fine-structure levels 3d4s2 2DJg

e

(Jg=3/2,5/2). This spectrum was studied experimentallyby Garton et al . (1973). Robicheaux and Greene(1993c) calculated length and velocity cross sections forthe six different 2DJg

e →Jf transitions: 3/2→1/2,3/2→3/2, 3/2→5/2, 5/2→3/2, 5/2→5/2, and 5/2→7/2.Very good agreement was found between length andvelocity results. Calculations covered the wavelengthrange from the lowest 3d4s 3D1

e threshold up to 3d21G4

e threshold. The spectrophotographic plates of Gar-ton et al . (1973) were compared with theoretical calcu-lations performed for a statistical mixture of Jg=3/2 andJg=5/2 ground-state fine-structure levels; this reflectedthe level of thermal excitation of the Sc vapor in theexperiment.Robicheaux and Greene (1993c) constructed simu-

lated ‘‘theoretical plates’’ by translating photoabsorp-tion intensity into a grayscale image. To mimic satura-tion effects, cross sections less than 3 Mb were displayedas white, while cross sections greater than 18 Mb wereshown as black. All major experimental features werereproduced by the calculation. The numerous perturbingstates complicated the appearance of the spectrum. Ad-ditional absorption lines associated with the substantialpopulation in both of the ground-state fine-structure lev-els added further complexity as well.Global aspects of the Sc dynamics were analyzed by

Robicheaux and Greene (1993c), who studied the scat-tering probability matrix Sij that characterizes the mix-ing strengths of different channels. The photoionizationpropensities provided additional insight: The photoab-sorption spectrum was expected to display pairs of au-toionizing lines with the ground-state splitting. For ex-



ample a 2Do state should appear in both 2D5/2e →

2D3/2o and 2D5/2

e → 2D5/2o . Although the lines tend to ap-

pear in both Jf symmetries that are possible, usually oneline has much more oscillator strength. By analyzing theangular part of the reduced dipole matrix elements, Ro-bicheaux and Greene (1993c) showed that the favoredtransitions were simply those for which DJ5DL . Thismeans, for example, that the cross section for 2D5/2

e →2D3/2

o is fourteen times smaller than the 2D5/2e → 2D5/2

o

cross section.The exploration of the dynamics of Sc on a large scale

permitted a qualitative understanding of what deter-mines the positions and shapes of various autoionizinglines and the origin of several marked features of the Scspectrum. Robicheaux and Greene (1993c) successfullylabeled the more prominent Rydberg series for excita-tion from the 3d4s2 2D3/2

e ground state. However, a bet-ter understanding of Sc (and a more complete classifica-tion of the autoionizing lines) can be obtained only bystudying the details on a fine scale. Robicheaux andGreene (1993d) classified the lines measured by Gartonet al . (1973) and compared the calculated lines withtheoretical spectra. The classification was performed intwo steps. First, the experimental and theoretical lineswere paired visually to give the best agreement betweentheoretical and experimental quantum defects and oscil-lator strengths. Then the classification was performedwith the procedure described in Sec. IV.B, i.e., by throw-ing away all the open channels in the MQDT calculationthat converts the problem to a search for bound levels.A crucial aspect of the classification was the derivationof an estimation of the level of acceptable error in thepositions of the perturbers. Another key aspect was as-certaining from the scattering probability matrix howstrongly a perturber interacted with various Rydberg se-ries. The theoretical classification agreed for the mostpart with the experimental classification, but there weresome discrepancies. While several lines could not beclassified, owing to uncertainties in the calculation, Ro-bicheaux and Greene (1993d) were able to complete theclassification of numerous ‘‘partially classified’’ lines ofGarton et al . (1973) and to classify most of their linesthat had remained totally unclassified. All the linescould be classified in either the LS or jj coupling andthe classification was presented in the coupling schemefor which the state was the purest.The comparison of theoretical and experimental lines

permitted an evaluation of the accuracy of the calcula-tion. At energies far from perturbers, the errors in quan-tum defects were less than 0.03, the (typically smaller)quantum defects for the nf series being more accuratethan those for the np series. Indeed, the f-wave channelshardly interact with other channels because the f-waveelectrons do not penetrate into the complicated core re-gion of Sc, while the p-wave channels display strongerinteractions. The oscillator strengths are highly suscep-tible to cancellation effects and are expected to be lessaccurate than the quantum defects. In the vicinity ofperturbing states, the quantum-defect errors are largerdue to the relatively large errors in the positions of the

low-n perturbers. The high accuracy achieved for theshort-range scattering parameters was crucial for theclassification. It probably would not have been possibleto classify most of the lines if the calculated quantum-defect errors had been greater than ;0.06.The fact that, in the experiment of Garton et al .

(1973), the two lowest bound levels 3d4s2 2D3/2e and

3d4s2 2D5/2e were simultaneously photoionized intro-

duced difficulty in the identification of all the experi-mental lines. To overcome this difficulty, and to test fur-ther the eigenchannel R-matrix method developed forSc ground-state photoionization, a new measurement ofthe relative photoionization cross section for the excitedSc 3d3 2D3/2

e level was carried out by Armstrong andRobicheaux (1993). An initial state having the sameSg , Lg , and parity as the ground state was used in orderto probe the same final states as those observed by Gar-ton et al . (1973). The relative measurement of photo-ionization cross section was performed by resonant two-step excitation via the 4s24p 2P1/2

o and subsequentphotoionization of the 3d3 2D3/2

e level. The photoioniza-tion cross section was measured over the energy rangefrom the Sc1 3d4s 3D1

e lowest threshold to just abovethe 3d2 3F4

e threshold. The experimental spectrum wascompared with a photoionization spectrum calculatedusing the same reaction matrices with Jf51/2–5/2 asthose used to describe final states excited from theground state. The only difference between these two cal-culations concerned the description of the initial stateand thus the values of dipole matrix elements and tran-sition frequencies. Figure 44 compares the unconvolvedtheoretical cross section and the measured Sc1 ion sig-nal in the wavelength range from the 3d4s 3D1

e thresh-old at 600.8 nm up to 540 nm, i.e., below the 3d4s1D2

e ionization threshold at 521.2 nm. Most of theprominent sharp lines were ascribed by Armstrong andRobicheaux (1993) to (3d4s 1D2

e)nf 2P1/2 , (3d4s1D2

e)nf 2D3/2 , and (3d4s1D2

e)nf 2F5/2 Rydberg levels.In addition, in this wavelength range are the lowestmembers of the (3d2 3FJc

e )nf Jf Rydberg series corre-sponding to n=5 (588–585 nm) and to n=6 (545–542

FIG. 44. Photoionization cross section of the Sc 3d3 2D3/2e ex-

cited level as a function of the wavelength of the photoionizinglaser. The wavelength range goes from the Sc1 3d4s 3D1

e

threshold at 600.8 nm to 540 nm, below the 3d4s 1D2e thresh-

old at 521.2 nm. The length theoretical cross section (uncon-volved) is compared with the experimental ion signal, shown asa mirror image. (From Armstrong and Robicheaux, 1993,courtesy of F. Robicheaux.)



nm), as well as the (3d2 3Pe)5p 4Po and (3d2 3F2e)7p

perturbing states near 582 nm. These perturbers arebroader than the (3d4s 1D2

e)nf resonances. The mea-sured and calculated cross sections agreed quite welloverall. It should be noted that the theoretical cross sec-tion was obtained from short-range scattering param-eters and dipole matrix elements that were calculatedbefore the experiment was completed. The most markeddisagreement in resonance location between experimentand theory visible on Fig. 44 appears near l= 573.6 nm.However, the corresponding (3d4s 1D2

e)8f 2F5/2o level

has an error in its quantum defect of only ;0.01–0.02,relative to the experiment.Similar good agreement was found at shorter wave-

lengths for the series converging to the 3d4s 1D2e

threshold. Above the 3d4s 1D2e threshold, the positions

of the resonances that belong to the various (3d23FJc

e )nf Jf interacting Rydberg series were also in verygood agreement. The most pronounced disagreementwas found for the (3d2 1D2

e)4f broad perturber, whichcorresponds to a low-n level, whose position is very sen-sitive to the accuracy of the calculation. In Fig. 44 thecalculated spectrum was unconvolved. In a wavelengthrange involving high-n resonances, a better comparisonbetween theory and experiment was achieved by convo-lution of the theoretical spectrum with the laser linewidth. Figure 45 shows a small portion of the cross sec-tion where the spectrum has been convolved with aGaussian line profile of 0.4 cm21. This figure displaysthe interacting Rydberg series attached to the three3d2 3Fe thresholds. The (3F2

e)nf (n;44 near the middleof the figure) series is perturbed by the (3F3

e) nf stateswith n=27 to 30 and by the (3F4

e) nf states with n=21and 22. The positions and shapes of the peaks are accu-rately reproduced by the calculation.

Armstrong and Robicheaux (1993) also compared thephotoionization cross sections from the 3d4s2 2D3/2

e

ground level and from the 3d3 2D3/2e excited level in the

range from the lowest 3d4s 3D1e threshold to just above

the 3d2 3Fe thresholds. The photoionization cross sec-tion differs greatly for the two initial states. In particu-lar, the cross section from the 3d3 2D3/2

e state is domi-nated by f-wave resonances; this is expected owing todominance of the one-electron dipole matrix elementfor the 3d→nf excitation over the 3d→np and4s→np excitations. Consequently, the cross sectionfrom the 3d3 2D3/2

e level consists of mostly symmetricsharp lines except for some of the broad perturbers,while the cross section from the ground state displaysmany asymmetric Beutler-Fano profiles (Fano, 1961).

2. Titanium

The progress achieved by eigenchannel R-matrix cal-culations for Sc (Armstrong and Robicheaux, 1993; Ro-bicheaux and Greene, 1993c, 1993d) led Miecznik andGreene (1996) to tackle the theoretical description oftitanium, the second transition-metal atom, whoseground state is 3d24s2 3F2

e . Photoabsorption and photo-ionization spectra were calculated from the four excitedlevels 3d2(3Fe)4s4p(3Po) 3D2

o , 3d2(3Fe)4s4p(3Po)3G3,4

o , and 3d3(4Fe)4p 5F5o , whose cross sections were

measured (Page and Gudeman, 1990; Sohl et al . , 1990).(Below, those initial levels will be referred to as the3d24s4p 3D2

o , 3d24s4p 3G3,4o , and 3d34p 5F5

o levels.)The theoretical description of Ti, like Sc, is complicatedby the large number of closely-spaced ionic target states.These cause irregularities in the pattern of interactingRydberg series that converge to those target states asthe principal quantum number increases. The first 15LS-coupled states of Ti1 have even parity and lie within25 000 cm21 above the Ti1 ground state. All thesestates correspond to different LS terms of either the3d24s , 3d3, or 3d4s2 configuration. Fine-structure ef-fects were incorporated through the jj-LS frame trans-formation and the use of experimental (fine-structure-split) ionization thresholds. When fine-structure effectswere taken into account, there were 34 even-parity ion-ization thresholds associated with the 14 LS-coupledtarget states relevant to the study, i.e., more than twiceas many as in Sc1. The lowest odd-parity 3d24p thresh-olds lie above 25 000 cm21. Additional complicationsresult from the presence of four electrons in the nearlydegenerate 3d and 4s subshells. One major motivationof the Ti project was to investigate whether the eigen-channel R-matrix could overcome those additional diffi-culties and achieve an accuracy comparable to thatfound for Sc. Comparisons of the theoretical cross sec-tions with the spectra measured by Sohl et al . (1990)and by Page and Gudeman (1990) were used to test theaccuracy of the calculations. As discussed above in Sec.III.D, Miecznik and Greene (1996) adopted an all-electron (21) Hamiltonian to describe Ti1, in order toaccount for the crucial complications of the Ti atom.They also developed a new method to generate radial

FIG. 45. Photoionization cross section of the Sc 3d3 2D3/2e ex-

cited level as a function of the wavelength of the photoionizinglaser. The figure covers a small wavelength range below theSc1 3d2 3F2

e threshold at 466.2 nm, where the Rydberg seriesattached to the 3d2 3F2

e , 3F3e , and 3F4

e thresholds are interact-ing. The length theoretical cross section convolved with aGaussian function of 0.4 cm21 full width at half maximum iscompared with the experimental ion signal, shown as a mirrorimage. (From Armstrong and Robicheaux, 1993, courtesy of F.Robicheaux.)



orbitals that incorporated the mutual screening and cor-relation of all electrons in the targets. In contrast withmost of the earlier eigenchannel R-matrix calculations,radial orbitals were obtained in a multiconfigurationHartree-Fock approximation (Froese Fischer, 1977) thatdirectly generated natural orbitals for the target states.Eigenchannel R-matrix calculations were performed

in LS coupling using a box of radius r0516 a.u., whichwas large enough to enclose the charge distribution ofthe four odd-parity initial levels. Calculations were car-ried out for even-parity final states from -500 cm21 to8000 cm21 relative to the first ionization threshold3d24s 4F3/2

e . The four LS-coupled thresholds lying inthis energy range are: 3d24s 4Fe, 3d3 4Fe, 3d24s 2Fe,and 3d24s 2De, in order of increasing energy. As for allopen-shell atoms, the calculation began with the con-struction of target-state wave functions. As in Sc, thelower odd-parity ionic states were found to play a largerole in Ti dynamics, primarily through their influence onthe polarizabilities of low-lying even-parity levels ofTi1. Target states were constructed from multiconfigu-ration Hartree-Fock orbitals that gave the fastest con-vergence for all even-parity Ti1 states below 25 000cm21 and for some lowest-lying 3d24p odd-parity Ti1

states. The even- and odd-parity target states, the odd-parity initial states, and the even-parity final states werecomputed using configuration interaction. The3d24s(2Fe)4p and the 3d24s(4Fe)4p configurationswere found to contribute strongly to the wave functionsof the 3d24s4p 3Do and 3d24s4p 3Go initial states. Onthe other hand, the 3d24s4p components were negli-gible in the 3d34p 5F5

o initial state.Construction of the jj-coupled short-range reaction

matrices for final Jf values ranging from Jf=1 to Jf=6required LS-coupled eigenchannel R-matrix calcula-tions to be carried out for 19 LS symmetries. Open andweakly closed channels for final even-parity symmetrieswere constructed by attaching s-wave and d-wave orbit-als to the even-parity configuration-interaction targetstates. Besides these channels, a large number ofstrongly-closed channels were included in the calcula-tions. They were constructed by attaching p-wave andf-wave orbitals onto the odd-parity configuration-interaction target states. Additional four-electron corre-lation functions were also included in the final-statewave-function expansions.The photoabsorption and photoionization cross sec-

tions from the 3d24s4p 3D2o , 3d24s4p 3G3

o , and3d24s4p 3G4

o levels were compared with the experi-ment of Sohl et al . (1990). This experiment probed theenergies both above and below the first ionizationthreshold 3d24s 4F3/2

e because of the presence of a weakexternal field that lowered the ionization barrier. Thefinal states were in the energy range of the fine-structure-split 3d24s 4Fj

e and 3d3 4Fje thresholds.

Highly-excited Rydberg levels with principal quantumnumbers n> 19 converging onto the 3d24s 4Fj

e thresh-olds were recorded. Figure 46 compares the calculatedspectrum for photoabsorption from the initial 3d24s4p

3G4o state with the experimental spectrum recorded by

of Sohl et al . (1990). The theoretical cross section waspreconvolved (Robicheaux, 1993) with the experimentalresolution of 0.6 cm21. Figure 46 deals with the energyrange just below the 3d24s 4F7/2

e threshold, where thedominant Rydberg series are the (3d24s 4F7/2

e )nd and(3d24s 4F9/2

e )nd series. Good agreement betweentheory and experiment is apparent over the whole en-ergy range. The differences between the measured andcalculated quantum defects of the dominant (3d24s4F7/2

e )nd resonances are about 0.03. The theoreticalcross section in Fig. 46 is a 15-channel calculation thatincludes closed channels converging to the 3d24s 4FJc

e ,

3d3 4FJee , and 3d24s 2FJc

e thresholds. This cross section

was compared with that obtained in a six-channel calcu-lation that included only channels associated with the3d24s 4FJc

e targets. The widths, line-shape parameters

q (Fano, 1961) and strengths of the resonances werevery different in the two calculations. Indeed, Miecznikand Greene (1996) found that a broad perturber at;55 259 cm21, ascribed to the (3d24s2Fe)6d 1H5

e level,strongly modified the resonances over a very wide en-ergy range, including the range displayed in Fig. 46.Miecznik and Greene (1996) also compared the calcu-

lated photoionization cross section of the 3d34p 5F5o

level with the experimental data of Page and Gudeman(1990), at energies below the 3d3 4F9/2

e threshold. The

FIG. 46. Photoionization cross section of the Ti 3d24s4p3G4

o excited level as a function of the energy relative to the Tiground level 3d24s2 3F2

e . The experimental measurement ofSohl et al . (1990) (upper curve) is compared with the jj-LSresult, shown as a mirror image; the theoretical cross sectionhas been convolved with the experimental resolution of 0.6cm21. The two theoretical lines are length and velocity results.(From Miecznik and Greene, 1996).



calculation reproduced the positions of the major ex-perimental peaks, the (3d3 4F9/2

e )nd resonances, but theagreement was poorer for intensities and widths.The experimental excited-level cross sections of Sohl

et al . (1990) and Page and Gudeman (1990) coveredenergies only slightly (1215 cm21) above the Ti1

ground state. Miecznik and Greene (1996) predictedphotoionization cross sections of the same excited levelsat higher energies up to 8000 cm21 above that lowestionization threshold. This calculation extended nearlyup to the energy of the fourth LS-coupled 3d24s 2De

threshold at 8505 cm21. The dominant Rydberg seriesoccurring in photoionization spectra of the four differentinitial levels were classified by Miecznik and Greene(1996). The similarities and differences among the spec-tra from different initial levels were interpreted usingsimple arguments based on the selection rules for thedipole operator and on the configuration-interaction ex-pansions of the initial states. The photoionization crosssections of the 3d24s4p 3D2

o and 3d24s4p 3G3,4o excited

levels are dominated by the (3d24s 4Fe)nl and(3d24s 2Fe)nl resonances. The (3d3 4Fe)nl resonances,which are very weakly excited from the 3d24s4p 3D2

o

and 3d24s4p 3G3,4o excited levels, dominate the photo-

ionization cross sections of the 3d34p 5F5o level. The

cross sections of the 3d24s4p 3D2o and 3d24s4p 3G3,4

o

excited levels exhibit similar features at most final-stateenergies. This results from the fact that all those initialstates have the common configuration3d2(3F)4s4p(3P) and differ only in the LS-term value.Moreover, the comparable strength of the (3d24s4Fe)nd resonances and the (3d24s 2Fe)nd resonancescould be understood as resulting from the near equalityof contributions from the (3d24s 4Fe)4p and (3d24s2Fe)4p components in the initial-state configuration-interaction expansions. Miecznik and Greene (1996)found that below the fourth threshold, the autoionizingresonances converging to the four LS-coupled lowest-lying thresholds were not the only ones present. Otherlow-lying members (n<6) of resonant Rydberg seriesattached to higher-lying thresholds were intense and hadlarge widths. Additional short-range 3d24p2 perturbersalso arose. A major resonance peak in the cross sectionfrom the 3d34p 5F5

o level, for instance, was ascribed tothe 3d2(3Pe)4p2(3Pe) 5De level. The presence of allthose perturbers complicates the spectra tremendouslyat higher-energies, causing extremely complex interfer-ence patterns. Miecznik and Greene (1996) estimatedthe error in quantum defects for Rydberg levels to be atmost 0.03, which translates as an error of ;100–200cm21 for the low-n perturbers. Nevertheless, additionalexperimental tests of the predictions remain desirable.The Sc and Ti results are prototype demonstrations

that show the capabilities of LS-coupled eigenchannelR-matrix calculations combined with frame-transformation methods and multichannel quantum-defect theory. The fact that the complicated electron dy-namics of Sc and Ti could be unraveled bodes well forfuture efforts in the other transition-metal elements,

which are still more complicated. It is important tomaintain or improve the level of accuracy, i.e., to limitthe error in quantum defects to ; 0.03. Otherwise it willbe difficult to obtain a sensible theoretical descriptionfor open-shell atoms with more d electrons. This willpose a difficult challenge for future calculations.

C. Concluding remarks

Theoretical multichannel spectroscopy can describethe complicated dynamics of open p-subshell atoms withZ < 53 near the lower ionizations thresholds. Calcula-tions have traced out systematic trends in the Rydbergdynamics of chemically similar elements. The approachwas also successful in unraveling the extremely compli-cated dynamics of the simplest transition metals Sc andTi. The jj-LS frame-transformation treatment has beensurprisingly successful at the description of a class ofspin-orbit effects that are important at energies nearfine-structure-split thresholds. In short, these resultsshow that the eigenchannel R-matrix method can beused to obtain reliable photoionization spectra and clas-sifications of the resonances even for extremely compli-cated spectra such as those of Sc and Ti. Despite itsnotable successes, the frame-transformation treatmentdoes have some limitations, however. Examples visiblefrom the calculations in open p-subshell atoms includethe bound-level spectra of Ge and Sn (see Sec. V.A.3).The study of the halogen atoms showed the need for anintermediate-coupling frame transformation when thetarget states do not have definite values of the total or-bital L and spin S angular momenta (see Sec. V.A.2). Inaddition, once the spin-orbit splittings of core levels ex-ceed about 1 eV, as in the iodine calculations, a moresatisfactory description of spin-orbit effects begins to be-come imperative. A ubiquitous phenomenon, whichplagues theoretical multichannel spectroscopy of open-shell atoms and limits its ultimate capability to describehighly complex spectra, is the presence of numerouslow-n perturbers located near the bottom end of Ryd-berg series converging to higher thresholds. An accuratedescription of these perturbing levels is needed becauseeven a small error in quantum defect can eliminate allresemblance between the theoretical spectrum and ex-periment. As in alkaline earths, one way to improveupon the jj-LS frame-transformation approximation,and to achieve a better description of low-n perturbers,is to include the spin-orbit interaction within the reac-tion volume and to perform the variational calculationsin jj coupling. The LS-coupled eigenchannel R-matrixcalculations carried out for open-shell atoms were rela-tively modest and were carried out on desktop worksta-tions. Calculations in jj coupling would probably bemore appropriate for a high-speed supercomputer, espe-cially for a transition-metal or lanthanide atom.A number of calculations for open-shell atoms have

been conducted with other theoretical approaches. Themost widely used has been the Wigner-Eisenbud-typeR-matrix approach that was developed by the Belfastgroup (Burke and Robb, 1975). For references, the



reader is referred to the comprehensive R-matrix com-pilation volume that was written recently by Burke andBerrington (1993). Most of the calculations have so fartreated atoms and ions with atomic number below 20. Inparticular, we cite the extensive calculations of accurateradiation absorption and emission properties carried outin the framework of the worldwide ‘‘Opacity Project’’ ofSeaton and co-workers (Seaton, 1987a; Berrington etal . , 1987, and subsequent papers referred to as ‘‘atomicdata for opacity calculations’’). In that framework, theR-matrix method provided the energies and wave func-tions of bound states, oscillator strengths, photoioniza-tion cross sections, and parameters for line broadeningby electron impact. Calculations were obtained on alarge scale for neutral atoms as heavy as neon, and fortheir isoelectronic sequences. Calculations for ions ofcosmically abundant elements containing more than 10electrons are planned. Some have already been carriedout, such as the investigation by Le Dourneuf et al .(1993) of Fe1 photoionization. Theoretical calculationsof the ‘‘Opacity Project’’ are being used to obtain im-proved values for the opacity of stellar envelopes; theseshould be of interest across a range of problems in phys-ics and astronomy (Zeippen, 1995).Apparently, the number of theoretical photoioniza-

tion calculations performed for the open d-subshell ele-ments has been relatively small. In addition to the cal-culation of Fe+ photoionization (Le Dourneuf et al . ,1993), electron impact ionization of Cr and electron im-pact excitation of Fe+ have been investigated by Reidet al. (1992) and Pradhan and Berrington (1993), respec-tively. Moreover, photoionization of Zn and Hg werestudied by Bartschat and co-workers (Bartschat andScott, 1985a, 1985b; Bartschat, 1987). In these last twocalculations mentioned, the 3d subshell in Zn or the5d subshell in Hg is closed for the ground state only, butnot in some target states included in the calculations.The large number of channels has limited most

Wigner-Eisenbud-type R-matrix calculations to a strictlynonrelativistic approximation, i.e., neglecting all fine-structure effects. However, Scott and Burke (1980) de-veloped a Breit-Pauli R-matrix formulation and appliedit to the study of electron impact excitation of Fe+

(Pradhan and Berrington, 1993). Photoionization of Znand Hg was also investigated with this approach (Bar-tschat and Scott, 1985a, 1985b; Bartschat, 1987).

VI. CONCLUSIONS AND PERSPECTIVES

Theory has developed a new capability in recentyears: the capability to calculate and interpret atomicspectra of unprecedented complexity. The set of tech-niques used is termed multichannel spectroscopy. Theterm is intended to convey the notion that many of thetools of multichannel collision theory, such as the scat-tering matrix, remain applicable even in the presence ofclosed channels, provided the tools are suitably general-ized.Some of these tools were initially introduced from a

phenomenological or semiempirical viewpoint, but

theory has progressed to the point that key quantitiessuch as R matrix or the S matrix can now be calculatedfrom first principles. One technique that is particularlywell suited to carry out such nearly ab initio calculationsis the R-matrix method; it has been discussed at lengthin this review. While our examples have been drawn pri-marily from the eigenchannel version of R-matrixtheory, this is a ‘‘detail’’ that remains largely a matter ofpreference. Indeed, the most important details are notwhether the eigenchannel or Wigner-Eisenbud variantsof R-matrix theory are used; these two variants have infact been shown to be exactly equivalent if identical ba-sis sets are used in both calculations. Other theoreticalmethods, such as the Schwinger variational principle(Schwinger, 1947) or the complex Kohn variational prin-ciple (Kohn, 1948) can also describe complicated spec-tra, in principle. Applications of these methods to mul-tichannel Rydberg spectra can be expected to berelatively inefficient, however, if they are not extendedto take advantage of scattering theory ideas in closedRydberg channels; this is a huge simplification exploitedby multichannel quantum-defect theory. Aside from theflexibility of the theoretical formulation adopted, themost important detail that controls the likely success ofany given calculation is the choice of the type (and num-ber) of variational basis functions, and on the actualphysics included in the real or model Hamiltonian of thesystem.Multichannel spectroscopy is a theoretical description

that not only determines complex spectra, but whichalso obtains the amplitudes and phases for ejection of anelectron (or other particle) into the various accessiblechannels that are open or weakly closed. The informa-tion contained in this set of intermediate quantities,namely the smooth short-range scattering parameters Sand d used by multichannel quantum-defect theory, con-siderably enhances the predictive power of multichannelspectroscopy. Knowledge of the ‘‘short-range’’ scatter-ing matrix implies a far more detailed understanding ofthe final-state dynamics than can be extracted from asingle total or partial cross section spectrum.Inclusion of the relativistic spin-orbit interaction at

different levels of approximation enhances the capabil-ity of multichannel spectroscopy to describe high-lyingRydberg levels near fine-structure-split ionizationthresholds. Impressive agreement between theory andexperiment has been obtained for extremely compli-cated spectra, including the perturbed pattern ofalkaline-earth atom autoionizing states at high energieswithin 0.02 a.u of the double-escape threshold. Even thecomplex autoionizing spectra of some transition-metalelements have been calculated to spectroscopically use-ful accuracy. In Sc and Ti, for instance, errors in thecalculated scattering parameters were found to be smallat energies far from perturbers: errors in the calculatedquantum defects were typically less than ; 0.03. Near aperturber, however, quantum-defect errors can be largerbecause the low value of the perturber principal quan-tum number n magnifies the error in the perturber bind-ing energy according to DE'Dm/n3. This problem ap-pears to be ubiquitous in the heavy open-shell atoms,



because the large density of ionization thresholds gener-ates many perturbing levels in any given energy range.Accurate results have been obtained for other observ-

ables besides energy-level positions and total photoion-ization cross sections. These include anisotropic observ-ables more sensitive to electron correlations and spin-orbit effects, such as the angular distributions ofelectrons ejected from alkaline-earth atom autoionizinglevels. Multichannel spectroscopy has capabilities thatextend beyond the accurate reproduction of experimen-tal spectra. It also permits classification of observedresonance structures and a detailed analysis of themechanisms (such as the dominant channel couplingsand autoionization pathways) that govern the Rydberg-electron dynamics. It also provides a powerful way toelucidate global trends and systematics in the electroncorrelations and channel interactions. Despite the inher-ent complexity of multichannel spectra, the underlyingshort-range dynamics of these spectra often exhibit re-markable simplicity, and are usually controlled by asmall number of physical parameters. This review high-lights the extent to which spectral regularities andchannel-interaction invariances are revealed despite theseemingly intractable complexity of the cross sections.Numerous spectral regularities occur along the series ofalkaline-earth atoms and along the series of chemicallysimilar open p-shell atoms. Likewise, the complicatedelectron dynamics in scandium and titanium present vis-ible similarities.The numerous examples in this review have concen-

trated on the alkaline earths and on some open-shellatoms. These examples share one thing in common: anattractive Coulombic long-range potential is experi-enced in each channel by the lone escaping electron.The examples have also dealt with field-free photoion-ization spectra excited by one photon from the groundstate or an excited state. To convey a glimpse of thegenerality of these methods, we turn now to a few appli-cations of theoretical multichannel spectroscopy to abroader class of systems and processes. Some results ob-tained by other powerful approaches will also be men-tioned. Possible extensions that are suggested by thesuccess of some of these calculations are commentedupon. We also draw attention to limitations of thesemethods.We begin with systems for which the long-range

electron-core interaction differs from an attractive Cou-lombic (plus centrifugal) potential; this category in-cludes He, H2, and other negative ions. In H2, an es-caping electron experiences a 1/r2 dipole potential atlarge r . This unusual dipolar character results from thepermanent electric dipole moment that is established inthe excited hydrogen fragment H(n) by the electric fieldof the outer electron. This dipole moment is essentially‘‘permanent’’ in this context rather than ‘‘induced,’’ ow-ing to the approximate degeneracy of the excited H(n)thresholds. Moreover, the effective dipole interaction isattractive in some channels, which generates an infinitenumber of levels converging to the correspondingthreshold. In other channels the dipole moment repels

the outermost electron. In He, the long-range potentialin each channel has a combined Coulomb and dipolecharacter. For negative ions other than H2, the escapingelectron usually experiences a long-range polarizationpotential whose asymptotic form is -a/2r4, with a theatomic polarizability when the core is isotropic. For an-isotropic core states a channel-dependent polarizabilityemerges that depends on both the scalar and tensor po-larizabilities of the corresponding atomic states. Multi-channel spectroscopy accounts for long-range multipoleinteractions beyond the reaction volume in H2, He, andnegative ions, either through the use of quantum-defecttheory, generalized to account for the relevant long-range potential (Greene et al . , 1979, 1982; Watanabeand Greene, 1980), or else by direct integration of theclose-coupling equations without exchange (Pan et al.,1994).The first eigenchannel R-matrix calculation for nega-

tive ions was carried out for the alkaline-earth negativeions (Kim and Greene, 1989). Three-electronLS-coupled eigenchannel R-matrix calculations werecombined with a generalized one-channel quantum-defect treatment of long-range motion in a dipole polar-ization potential. The calculation of Kim and Greene(1989) concurred with the earlier prediction by FroeseFischer et al. (1987) that Ca2 has a stable negative-ionstate 4s24p ; both of these early calculations overesti-mated the Ca electron affinity by about a factor of 2,judging from recent experiments (see, for example,Walter and Peterson, 1992). It is not surprising thattheory struggles to predict the Ca electron affinity accu-rately, since it is so small ('20 meV). The most success-ful theoretical calculation to date of the Ca2 bindingenergy is the semiempirical model-potential calculationof van der Hart et al. (1993), who include the dielec-tronic polarization term. After adjusting the ‘‘cutoff’’ ra-dius to obtain an accurate ground-state energy of neu-tral calcium, van der Hart et al. (1993) then used theresulting dielectronic term in their calculation of Ca2 toobtain a binding energy that agreed well with experi-ment. Kim and Greene (1989) also predicted that theheavier alkaline-earth atoms Sr, Ba, and Ra would havestable negative ions, of similar mos

2m0p2P designation,

whereas the corresponding Be2 and Mg2 states are un-stable shape resonances. New experimental results forBa2 show that most calculations overestimated thestrength of binding by about a factor of 2 (Petruninet al., 1995).This class of theoretical techniques was also extended

to treat the photodetachment spectra of the alkali nega-tive ions. Nonrelativistic close-coupling calculations byMoores and Norcross (1972, 1974) derived realistic re-sults for photodetachment of Li2 and Na2 up to theenergy of the first excited alkali energy level m0p . Forthe heavier alkali negative ions, the experiments ofLineberger and co-workers (Patterson et al., 1974; Slateret al., 1978) showed prominent effects associated withthe alkali m0p fine-structure splitting, indicating theneed for inclusion of spin-orbit effects into the theory.Lee (1975) developed a multichannel effective-rangetheory treatment that included fine-structure effects



through a frame transformation, along the lines of theearlier study of S2 photodetachment by Rau and Fano(1971). Lee’s semiempirical fit was able to reproduce themeasured (Patterson et al., 1974) Cs2 photodetachmentcross sections, and was influential because it was able tofit cross sections for other alkali negative ions. More-over, the resulting fitted parameters permitted predic-tion of other quantities, such as partial detachment crosssections, which were confirmed by subsequent measure-ments. The formulation of Lee (1975) neglected thelong-range potentials entirely. Taylor and Norcross(1986) calculated the smooth, short-range reaction ma-trix K0 that Lee (1975) had assumed to be energy inde-pendent across the 550 cm21 fine-structure splitting ofCs(6p), and discovered that K0 actually varied quiterapidly with energy. They showed, however, that the en-ergy variation could be expressed as an energy-dependent long-range phase shift in each channel, whichdid not affect the total or partial cross sections. Wa-tanabe and Greene (1980) used generalized quantum-defect theory to demonstrate that the strong energy de-pendence of K0 is caused by the huge polarizability(a;10 3 a.u.) of the m0p excited state. Moreover, a dif-ferent reaction matrix Kpol

0 could be defined that wasnearly energy independent and better suited to the ap-plication of a fine-structure frame transformation.Greene (1990a) combined a jj-coupled eigenchannel

R-matrix calculation with the generalized MQDT treat-ment of electron escape in a polarization potential in thefirst nearly ab initio calculation able to reproduce theobserved photodetachment spectra of the heavy alkali-metal negative ions. Resonant photodetachment spectraof Cs2, Rb2, and Fr2 were calculated near the lowestm0s and m0p thresholds, which are quite similar in all ofthese heavy alkali-metal negative ions. The calculationalso predicted the existence of three Cs2 6s6p 3PJ

0

states that are barely bound, in addition to the 6s2

ground state, in agreement with an earlier tentative con-clusion reached by Froese Fischer and Chen (1989). Asubsequent theoretical treatment of Cs2 bound statescarried out by Thumm and Norcross (1991, 1992) dis-puted the stability of these Cs2 6s6p 3PJ

0 levels.Thumm and Norcross calculated e-Cs scattering crosssections using a relativistic Wigner-Eisenbud-typeR-matrix calculation based on the Dirac Hamiltonian,also using a model potential. Electron affinities of theCs2 6s2 1S0

e , 6s6p 3PJo , and 6p2 3PJ

e levels were cal-culated in two different ways: with and without the di-electronic term that represents an additional electron-core-electron interaction [Eqs. (3.34) and (3.40)].Calculations carried out without the dielectronic termagreed with those of Greene (1990a), who omitted itfrom the model Hamiltonian. When the dielectronicterm was included, however, the negative-ion spectrumwas found to change dramatically: the 6s6p 3PJ

o levelswere shifted into the continuum, where they appeared asnarrow shape resonances instead of as bound levels.These states are exceedingly close to threshold, so thequestion of whether they are bound states or resonanceswill probably not be resolved until they are investigated

experimentally. For the present, though, the best calcu-lation performed thus far is probably that of Thummand Norcross (1991, 1992).Sadeghpour and Cavagnero (1993) investigated the

photoabsorption spectrum of He below the n=3 thresh-old with the LS-coupled eigenchannel R-matrix ap-proach; the motion beyond the reaction volume utilizedgeneralized quantum-defect theory (Greene et al . ,1982) for long-range potentials of Coulomb-plus-dipoleform. Sadeghpour et al . (1992) combined anLS-coupled eigenchannel R-matrix calculation with aquantum-defect description of electron motion in a di-pole field to predict the photodetachment spectrum ofH2, including resonant structures up to the n54 hydro-genic threshold. Higher-lying doubly excited states ofH2 up to the n56 threshold were calculated by Pan etal . (1994), who adopted an LS-coupled eigenchannelR-matrix calculation. The reaction volume in this studyextended out to r05110 a.u., but long-range couplingcaused by the large core multipole moments requireddirect solution of the close-coupling equations in theouter region, instead of the economical (but approxi-mate) quantum-defect method. This calculation ac-counted well for the experimental results of Harris etal . (1990). High-lying doubly excited states of Li2 weretreated using the same procedures and found to be re-markably similar to the H2 resonance pattern, althoughsome conspicuous differences were apparent.This review has not explicitly discussed electron-ion

and electron-atom scattering processes, although theyhave been studied extensively in Wigner-Eisenbud-typeR-matrix calculations. Burke and Berrington (1993)published a survey of calculations carried out with theBelfast R-matrix packages. Relatively few calculationshave been carried out using the eigenchannel R-matrixapproach. One eigenchannel study was that of Pan(1991a, 1991b), who calculated electron collisions withBe1, Mg1, and Ca1 ions at low energies. Pan devel-oped a systematic and comprehensive analysis of theprominent and complicated series of double excitationof the collision complex. Pan’s calculations also resolveda long-standing discrepancy between theory and experi-ment in the near-threshold excitation of the Be1 reso-nance transition by electron impact.More recently, Robicheaux et al . (1994) studied the

controversial problem of e-H2 inelastic scattering abovethe triple-electron escape threshold. The main purposeof this investigation was to see whether H22 resonances(if they exist) could influence the scattering process. Ex-periment (Walton et al., 1970, 1971; Peart and Dolder,1973) and theory (Taylor and Thomas, 1972) had longago presented evidence in support of this claim. To dealwith this problem, it was necessary to extend the eigen-channel R-matrix method to describe approximately theeffects of multiple-electron escape. The R-matrix calcu-lation, conducted in parallel with a configuration-interaction study of the dependence of the 2s22p 2Presonance on the nuclear charge Z , led to a predictionthat the previously observed (and calculated) H22 reso-nances could not exist. This prediction has since been



confirmed by a new experiment of Andersen et al.(1995), who observed no evidence for resonances inelectron-scattering from the deuterium negative ion.(D2 is expected to behave the same as H2 in such elec-tron scattering experiments.)Following the lead of Robicheaux et al . (1994) in ex-

tending R-matrix methods to handle the double con-tinuum H+e1e , Meyer and Greene (1994) calculateddouble photoionization of He by one-photon absorptionin the energy range from the He11 double-escapethreshold up to photon energies near 200 eV. The stan-dard LS-coupled eigenchannel R-matrix method, ap-plied previously to single photoionization only, wasadapted to describe a class of two-electron escape pro-cesses. Although the boundary conditions imposed inthe calculation appeared not to allow the possibility ofdirect electron escape into the double continuum, a re-alistic description of the He double-photoionizationcross section, and of the ratio of double- and single-photoionization cross sections of He was obtained. Thecentral idea was to distinguish among the closed-typeorbitals used to represent the inner electron He1 in aclose-coupling expansion: those that had negative ener-gies (relative to the double-escape threshold) were inter-preted differently from those having positive energies.The He1 eigenfunctions that had positive energies wereviewed as representing a discretized continuum state ofHe, whereby all flux escaping in channels associatedwith a positive-energy inner electron was interpreted ascontributing to double photoionization. All flux escapingin channels with a negative-energy inner electron wassimilarly associated with single photoionization. Pseu-doresonances appeared in the double-photoionizationcross section, owing to the artificial discretization of theHe1 continua. These artifacts of the finite-volume cal-culation were eliminated from the final spectrum by us-ing two averaging techniques: (i) the weakly closedchannels were treated as though they were open whenthe MQDT equations were solved to determine thephotoionization cross section; (ii) the spectrum was cal-culated for a number of different values of theR-matrix box size r0, and these results were averaged todetermine the final spectrum that was compared withexperiment. The basic ideas have been improved in asubsequent study by Meyer et al. (1995), through the in-troduction of a frame-transformation-type projection ofthe ‘‘box eigenstates’’ of the residual ion onto the truecontinuum states of the ion. Other methods have beenproposed through the years to describe two-electron es-cape. One of these, the R-matrix theory for two activeelectrons proposed by Burke et al . (1987), might be ableto describe two-electron continua and high-lying doublyexcited states. An important step in its development wasthe Le Dourneuf et al . (1990) implementation of a two-dimensional R-matrix propagation procedure for ans-wave model (Temkin, 1962; Poet, 1978) of e-H scatter-ing. Two formulations that have shown great promiseare the convergent close-coupling approach of Bray andStelbovics (1992, 1995) and the hyperspherical close-coupling approach developed by Kato and Watanabe

(1995). Further theoretical effort is still needed, how-ever, to overcome limitations of all these methods andto develop an efficient R-matrix method or other ap-proach that can handle two escaping electrons.Another source of extremely rich spectra that have

been successfully described using theoretical multichan-nel spectroscopy is the system of a one- or two-electronatom in an external magnetic or electric field. The hy-drogen atom in a magnetic field has attracted much at-tention. This and related systems have served as proto-types for the study of ‘‘quantum chaos,’’ i.e., the study ofnonseparable quantum systems whose classical dynamicsare chaotic. Nonseparability of the Schrodinger equationfor hydrogen in a magnetic field becomes paramountnear the zero-field ionization threshold. For states inthat energy range, the cylindrically symmetric diamag-netic term (associated with the Lorentz force) achieves astrength comparable with the spherically symmetricCoulomb potential. In this situation, neither perturba-tive nor adiabatic treatments (Starace and Webster,1979; Wang and Greene, 1989) can describe the ex-tremely complicated diamagnetic spectra. However, theadiabatic treatment in cylindric coordinates by Wangand Greene (1989) correctly described some qualitativefeatures of the quasi-Landau resonance spectra, such asthe 3

2vc spacing of resonances observed by Garton andTomkins (1969a) and the existence of ‘‘one-dimensional’’ Rydberg states (Iu et al., 1989) convergingto the vc-spaced Landau ionization thresholds. Anyconvincing theoretical treatment must account for thestrong interactions between these Rydberg series. Asshown by Wintgen and Friedrich (1986, 1987) and byWang and Greene (1991a), the quasi-Landau ‘‘reso-nances’’ can be viewed as broad perturbers embeddedamong densely packed high-Rydberg states that belongto series converging to lower-lying Landau thresholds;these quasi-Landau resonances interact with high-Rydberg states to form complex resonances. Moreover,destructive interferences occur periodically, resulting inthe occurrence of resonances whose width nearly van-ishes. Hydrogen photoionization in an astrophysicalstrong magnetic field was described by Greene (1983)and by Wang and Greene (1991b) using an eigenchannelR-matrix calculation augmented by a multichannelquantum-defect description of the asymptotic wavefunctions in cylindrical coordinates. The latter of thesepapers used a mixed-symmetry basis set that includedspherical basis functions that could describe the impor-tant departures from cylindrical symmetry close to thenucleus. The R-matrix calculations carried out within acylindrical R-matrix box determined the smooth short-range reaction matrix and dipole matrix elements thatwere then used to calculate the photoionization crosssection on a fine energy mesh.Wang and Greene (1991b) calculated photoionization

spectra for superstrong magnetic fields B>103 T. Thecalculation of MQDT parameters, needed to account forspectroscopic observations in magnetic fields relevant toterrestrial experiments (B;6 T), is far more difficultbecause of the large number of interacting channels and



the large volume over which the Hamiltonian is non-separable, a cylinder of radius '10 3 a.u. and length'10 4 a.u., for B=6 T. A variety of different methodshave been developed to calculate the diamagnetic Ry-berg spectra of atoms in a magnetic field of laboratorystrength. Among them, the methods developed byDelande et al . (1991), O’Mahony and Mota-Furtado(1991), Watanabe and Komine (1991) or Halley et al.(1992, 1993), were used to interpret recent observations.Two of these calculations (O’Mahony and Mota-Furtado, 1991; Watanabe and Komine, 1991) deter-mined a smooth short-range reaction matrix that charac-terizes the solutions in the cylindrical asymptotic region.This information was then used to calculate the photo-ionization cross section. Both of these approaches intro-duced a small spherical inner region where the Rydbergelectron motion could be described by field-free MQDTparameters; they both solved the Schrodinger equationover a large region in spherical coordinates, and thenmade a spherical-to-cylindrical two-dimensional frametransformation at large distances. O’Mahony and Mota-Furtado (1991) connected the outer asymptotic region tothe inner region using an R-matrix propagation scheme.Watanabe and Komine (1991) implemented thediabatic-by-sector method of Launay and Le Dourneuf(1982) to construct channel functions that were treateddiabatically within each radial sector. Both methodsachieved impressive agreement with the high-resolutionspectra of Li in a magnetic field of 6 T (Iu et al . , 1989,1991). Delande et al . (1991) calculated the photoioniza-tion cross section of hydrogen in a magnetic field byreplacing r→ re iu, p→pe2iu in the Hamiltonian and di-agonalizing once and for all the complex rotated Hamil-tonian H(u) built on a large basis of roughly 105 Stur-mian functions. Finally, Halley et al . (1992, 1993)combined the complex-coordinate method with a vari-ant of the R-matrix method (Schneider, 1981) to inter-pret the high-resolution diamagnetic spectra of Li (Iuet al . , 1989, 1991) and of the heavier alkaline earths Srand Ba (Lu et al . , 1978). This combination of theR-matrix and complex-coordinate methods was also suc-cessfully applied to interpret high-resolution diamag-netic spectra of He (Delande et al . , 1994) and Stark anddiamagnetic spectra of Na (Seipp and Taylor, 1994).Furthermore, it remains the only method presentlyavailable to treat atoms in much more complex situa-tions, such as in crossed electric and magnetic fields,where the asymptotic solution is not accurately known.The complex-coordinate method (Ho, 1983) is emerg-

ing as one of the simpler alternatives that currently existfor calculations of the total photoabsorption spectrum,or of resonance positions and total decay widths. If aconvenient basis set can be found for a problem withrelatively few degrees of freedom, such as the Sturmianbasis set for hydrogen in a magnetic field, it is relativelystraightforward to form an enormous Hamiltonian ma-trix and solve for many of its eigenvalues and eigenvec-tors. Moreover, if the matrix turns out to be sparse, as isagain true of the Sturmian basis set for hydrogen dia-magnetism, the Lanczos algorithm (see Ericsson and

Ruhe, 1980, and references therein) or its competitorscan efficiently solve the eigenvalue problem even whenthe basis-set size grows to enormous dimensions. Themain limitations of this method are: (i) It has been usedalmost exclusively for the calculation of total photoab-sorption cross sections (see, for example, Berzinsh et al.,1995). A limited number of attempts have been made toextract other observables, such as partial cross sections(Han and Reinhardt, 1995), but the ability of thismethod to compete with the techniques of multichannelspectroscopy for calculations of partial cross sections,angular distributions, and other observables has not yetbeen demonstrated. (ii) It is difficult to interpret the dy-namics of the resulting wave functions, as the complexrotation of the coordinates modifies the continuum solu-tions in a nontrivial manner. It is possible that, as expe-rience with this theoretical tool grows, the interpretivecapabilities of this approach will improve. (iii) A finaldisadvantage of the complex coordinate method and itsrelatives is that they do not produce intermediate quan-tities, such as the smooth short-range scattering matrixof multichannel spectroscopy. As this review has shown,these tools can be decisive in the interpretation of thespectrum; they can readily predict any observables thatit is possible to measure, and they can be utilized topredict the outcome of entirely new experiments, suchas one with a new external field applied. The examplesin the following paragraph describe this last capability.Nevertheless, when one is interested in a calculation ofthe total photoabsorption spectrum, say for comparisonwith an experimental measurement, the complex-coordinate approach is certainly one to consider, owingto its ease of implementation.The problem of a nonrelativistic hydrogen atom in a

static electric field can be solved exactly (Luc-Koenigand Bachelier, 1980a, 1980b) because the time-independent Schrodinger equation separates in para-bolic coordinates. However, the Hamiltonian becomesnonseparable for a more complex atom in an electricfield, since the potential of an electron departs from apure Coulomb field within the core of a nonhydrogenicatom. The Stark spectra of autoionizing levels of nonhy-drogenic atoms exhibit tremendous complexity and canbe viewed as multichannel spectra (Harmin, 1984;Sakimoto, 1986). Extensive measurements and calcula-tions of Ba autoionizing Stark spectra around the 5djionization thresholds were reported recently (Arm-strong et al . , 1993; Armstrong and Greene, 1994). Thespectra were recorded by photoionization of the laser-excited Ba 5d6p 3D1 level in the presence of an electricfield. The theoretical description combined informationfrom jj-coupled eigenchannel R-matrix calculations,performed for zero external field, with a Harmin-Fano-type field-dependent frame transformation (Harmin,1984, and references therein). The calculations success-fully reproduced the complex resonance patterns ob-served over broad spectral regions that included asmany as six interacting Stark manifolds attached to thetwo different 5d3/2 and 5d5/2 ionization thresholds. Theaverage widths of autoionizing Stark states were found



to scale approximately as n24, n being the relevant ef-fective quantum number. However, large departuresfrom this scaling law were observed for individual reso-nances, which could be important for the theoretical de-scription of other processes in an electric field, such asdielectronic recombination (Jacobs and Davis, 1979; Na-har and Pradhan, 1992).The statistics of resonance distributions in multichan-

nel atomic and molecular spectra has been a subject ofincreasing interest in connection with ‘‘quantum chaos’’studies. Statistical analyses of quantum-mechanical spec-tra have attempted to ascertain the extent to which theresonance positions and widths follow the predictions ofrandom-matrix theory. Specifically, the statistical distri-butions of energy-level spacings (nearest-neighbor spac-ings) have been compared with the Wigner distribution,which applies when strong level repulsion results in fewclosely spaced levels. The resonance width distributionswere compared with the Porter-Thomas distribution,whose form implies that small widths are by far the mostprobable (for references see Brody et al . , 1981, and De-lande and Buchleitner, 1994). While portions of field-free autoionizing spectra were found to exhibit theWigner and Porter-Thomas distributions of positionsand widths (see, for example, Connerade et al . , 1990,and Miecznik et al . , 1995) there remain at the sametime striking regularities and Rydberg periodicities inthe pattern of resonance energies and decay widths. Thiswork suggests that caution should be exercised beforeautomatically interpreting ‘‘classical chaos’’ as reflecting‘‘quantal irregularity.’’ Similar results were found fordiamagnetic spectra: in the regions in which the classicaldynamics is chaotic, the statistical properties of quantumspectra are accurately described by random-matrixtheory (Gremaud et al . , 1993; Delande and Buchleitner,1994); again, however, regular Rydberg progressionshave been locally observed (Iu et al . , 1989, 1991) orpredicted (Wang and Greene, 1991a, 1991b). In multi-channel spectra, the distribution of nearest-neighborspacings evolves from the Poisson to the Wigner distri-bution as the channel coupling increases (Draeger andFriedrich, 1991). Moreover, extremely narrow reso-nances typical of the Porter-Thomas distribution appearperiodically in the vicinity of each broad interloper ofRydberg series, owing to destructive interferences be-tween autoionization pathways associated with differentclosed channels (Wintgen and Friedrich, 1986, 1987;Wang and Greene, 1991a, 1991b). This implies that com-plex spectra (which may or may not reflect classicalchaos) need not be totally irregular, random, and unpre-dictable, but can possess similar statistical properties. Infact, as discussed recently (Ivanov et al . , 1995; Zakrze-wski et al . , 1995) for two-electron atoms and hydrogenin a magnetic field with classical chaotic dynamics, evenclose to an ionization threshold, there remain regions inthe classical phase space where an approximate adia-batic separation of the motion along two coordinatesexists that can induce deviations from random-matrixtheory.

This review has primarily treated photoionizationspectra associated with single-photon absorption fromthe ground state or from a laser-prepared excited state.One striking difference between the one-photon andmultiphoton cross sections is the greater prominence ofresonances in the latter process. In multiphoton ioniza-tion cross sections, Rydberg series of resonances canarise at each intermediate or final step of the process.Interest in the description of multiphoton processes inmultielectron atoms continues to grow. Most theorieshave dealt with nonresonant processes and have calcu-lated multiphoton ionization cross sections using pertur-bation theory at the lowest order. Fink and Zoller(1989) proposed a multichannel quantum-defect param-etrization of perturbative two-photon ionization crosssections, in the presence of intermediate- and final-stateresonances, suitable for use with ab initio calculations.Retaining the ideas of Fink and Zoller (1989), Ro-bicheaux and Gao (1991, 1993) developed an efficientmethod for calculating perturbative two-photon ioniza-tion cross sections based on variations of the eigenchan-nel R-matrix method and multichannel quantum-defecttheory. The method was applied to two-photon ioniza-tion of the Mg and Ca ground states. A relativistic time-dependent Dirac-Fock approach combined with multi-channel quantum-defect theory was used by Fink andJohnson (1990) to calculate two-photon ionization crosssections of the rare gases. One of the first results ob-tained by the Belfast group with the Wigner-Eisenbudform of the R-matrix method concerned its applicabilityto the calculation of the infinite summations over inter-mediate states involved in dynamical dipole polarizabi-ties (Allison et al . , 1972). Later, Shorer (1980) and thenSmith et al . (1992) extended the codes of the Belfastgroup to the calculation of two-photon excitation andionization. Applications to Ne, Be, and C atoms werepresented. A configuration-interaction formulation, suit-ably extended to incorporate field-induced effects, wasused by Lambropoulos et al . (1988) to interpet experi-mental observations on multiphoton ionization in Sr.Configuration-interaction approaches using L2 basis setswere used to calculate multiphoton ionization (detach-ment) cross sections in H2, He, Be, and Mg (Mocciaand Spizzo, 1989; Chang and Tang, 1992; Sanchez etal . , 1995, and references therein). The study of atomicsystems interacting with intense laser fields has attractedconsiderable interest in recent years. Among the non-perturbative methods developed to treat multiphotonprocesses accounting for strong-field effects, theR-matrix-Floquet theory developed by Burke and co-workers (Burke et al . , 1991; Dorr et al . , 1992, 1993)looks very promising.Photoionization spectra investigated with the eigen-

channel R-matrix approach have been restricted totreatments of valence-shell photoionization. However,the subject of atomic inner-shell photoionization contin-ues to be of great interest to both experimenters andtheorists. For instance, inner-shell photoabsorptionspectra of alkaline-earth elements exhibit variousanomalies that reveal strong correlation and spin-orbiteffects (Connerade et al . , 1990; Connerade and Sarpal,



1992). The great majority of eigenchannel R-matrixphotoionization calculations used a frozen-core modelpotential to account for the effects of the closed shells.Theoretical investigation of inner-shell photoionizationin alkaline-earth atoms would require an ab initio de-scription of all electrons, as was used in eigenchannelR-matrix calculations of the chalcogen (Chen and Ro-bicheaux, 1994) and Al (Miecznik et al . , 1995) outer-shell photoionization cross sections. Many successful cal-culations of photoionization spectra involving inner-shell excitation have been carried out using powerfulmethods such as the relativistic random-phase approxi-mation, many-body perturbation theory, or the Wigner-Eisenbud-type R-matrix approach (for references seeStarace, 1982; Kelly, 1987, and Burke and Berrington,1993).We address now the description of relativistic effects.

As shown in Secs. IV and V, the jj-LS frame-transformation treatment was found to be very success-ful in describing numerous nonperturbative spin-orbiteffects in some atoms as heavy as barium or as compli-cated as titanium. Despite the clear success of the ap-proximate frame-transformation treatment, exampleswere found for two-electron systems in which thejj-coupled eigenchannel R-matrix calculations (whichinclude explicit spin-orbit terms in the Hamiltonian)gave improved agreement with measurements. Al-though among the relativistic effects only the spin-orbitterms are included explicitly, we believe that numerousother relativistic effects, such as the mass-velocity andDarwin terms, are implicitly taken into account (at leastapproximately) through the use of a semiempirical one-electron model potential (Greene, 1990a; Greene andAymar, 1991). The jj-coupled eigenchannel R-matrixapproach has not yet been extended to treat open-shellatoms, though we anticipate that it should give improvedresults, for instance in heavy open p-shell atoms. Thepossibly surprising success of this non-Dirac descriptionof relativistic valence-electron dynamics probably stemsfrom the fact that photoabsorption (or photodetach-ment) mainly probes the details of the wave functionsbeyond ; 1 a.u. from the nucleus. Poorer theoreticalresults are anticipated for short-range observables suchas hyperfine splittings. A more correct, but also muchmore complex, way to treat relativistic effects is to usethe Dirac Hamiltonian. A relativistic version of theeigenchannel R-matrix method based on the DiracHamiltonian was formulated by Hamacher and Hinze(1991), but no application was reported. Wigner-Eisenbud-type R-matrix approaches based on the DiracHamiltonian have been developed by Chang (1975,1977), Norrington and Grant (1981), and by Thumm andNorcross (1991, 1992). To our knowledge, these meth-ods have been applied only to electron-scattering calcu-lations. A relativistic R-matrix treatment would be use-ful in its own right, and to check the validity of thepredictions obtained for Fr2 (Greene, 1990a) and Ra(Greene and Aymar, 1991). This would also be a naturaltool for investigations of photoionization and electron-scattering processes involving multicharged positive

ions. In this review, we have considered electric dipoletransitions only; the study of magnetic and higher-orderelectric dipole transitions in heavy multicharged positiveions could also be profitably studied using a relativisticR-matrix description.Despite the great amount of progress in multichannel-

spectroscopy theory, improvements are still desirable ina number of directions: (i) To reach higher energies veryclose to thresholds for escape of two or more electrons,for instance, one must adopt a very large reaction vol-ume. The convergence of calculations deteriorates rap-idly as the size of the reaction volume grows, which ren-ders intractable the type of basis-set-based variationalcalculations emphasized in this review. New conceptswill be required to handle that regime, such as the prom-ising hyperspherical close-coupling method (Tang et al.,1992), which has obtained accurate results for helium.(ii) Titanium is the most complicated atomic system thathas been studied using the eigenchannel R-matrix ap-proach. More complicated spectra, such as those involv-ing an open 4f (lanthanides), 5f (actinides), or 4d sub-shell, present a serious challenge for theory. Thedescription of these elements remains a difficult under-taking for the future. (iii) The use of these techniques totreat molecular species, i.e., combining ab initioR-matrix calculations with multichannel quantum-defecttheory, have been rare (Stephens and McKoy, 1992;Greene and Yoo, 1995) and still require substantial im-provement in order to develop a scheme for calculationsthat is spectroscopically accurate. (iv) The developmentof improved methods for interpretation and visualiza-tion of the quantum dynamics of multichannel spectraremains a key task as well. Tremendous insight has beengained in some classes of few-body systems through theuse of adiabatic hyperspherical methods (Fano and Rau,1986; Lin, 1995) yet their extension to each new type ofquantum-mechanical system still requires great effort.

ACKNOWLEDGMENTS

We would like to thank Jean-Marie Lecomte andGregory Miecznik for their careful reading of parts ofthe manuscript and for their helpful comments. Some ofthe numerical calculations were carried out on the Cray98 belonging to the ‘‘Institut du Developpement et desRessources en Informatique Scientifique’’ of the CentreNational de la Recherche Scientifique and on the CrayYMPEL of the computer center ‘‘Paris Sud Informa-tique.’’ The laboratoire Aime Cotton is associated withthe Universite Paris-Sud. This work was supported inpart by the National Science Foundation.

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Multichannel Rydberg spectroscopy of complex atoms