Study of electron polarization and correlation effects in resonant and background electron scattering off CF3Cl

Study of electron polarization and correlation eectsin resonant and background electron scattering o CF3Cl

Thomas Beyer *, Bernd M. Nestmann, Sigrid D. Peyerimho

Institut fur Physikalische und Theoretische Chemie der Universitat Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany

Received 28 December 1999

Abstract

We report on ab initio calculations for electron±CF3Cl scattering based on the R-matrix approach. Eigenphases and

cross-sections are presented for collision energies up to 15 eV in static exchange and static exchange plus polarization

(SEP) approximation. The SEP results for integral cross-sections agree satisfactorily with experimental total cross-

sections. Moreover, the partial cross-sections con®rm the symmetry assignment of the lowest three resonances as has

been found in experiments carried out by Mann and Linder.

Electron polarization and correlation eects are studied separately for resonant and background scattering. For this

separation, the Feshbach projection operator technique formulated within the R-matrix framework is applied. We

found that the description of resonant scattering is considerably aected by con®guration interaction. The background

scattering appears to be appropriately described on the SEP level. Ó 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

The models used to explain electron±moleculecollision processes on the basis of the properties ofthe target molecule are dierent for resonant andnonresonant scattering. In the latter case, thescattered electron is considered to be aected bythe static ®eld of the molecule and by short-rangerepulsion due to an exchange interaction. This ef-fect is incorporated in the static exchange (SE)approximation. The additional polarization of theelectronic target system by the charge of the scat-tered electron is taken into account in the staticexchange plus polarization (SEP) approach. Incontrast, resonances in electron±molecule scatter-

ing are considered to be associated with unoc-cupied molecular orbitals. In particular, thelow-lying shape resonances usually correspond toanti-bonding orbitals, temporarily occupied by thescattered electron. These molecular orbitals areobtained by linear combination of atomic orbitals(LCAO). Due to this picture, resonances are as-sumed to be caused by metastable anionic states.The distinction between resonant and nonresonantscattering has an analogy in the separation ofmolecular bound states into more localized valenceand united atom-like Rydberg states.

A separation of the complete scattering infor-mation into a background part, describing non-resonant scattering, and into resonances, isprovided by the Feshbach projection operatorformalism. In other words, the resonances arerepresented by metastable states interacting withthe background. Due to the dierent physics of

Chemical Physics 255 (2000) 1±14

www.elsevier.nl/locate/chemphys

* Corresponding author. Fax: +49-228-739066.

E-mail address: [email protected] (T. Beyer).

0301-0104/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 1 - 0 1 0 4 ( 0 0 ) 0 0 0 6 2 - 8

resonant and background scattering, such a sepa-ration seems to be the appropriate way to comparethe electron scattering cross-sections for similarmolecules. Moreover, the parametrization of thecross-section and phases may be easier for back-ground and resonances separately than for theentire problem. Such a parametrization, however,may also be useful in applications such as model-ing plasma properties.

Due to the long lifetime of the metastable statecompared to the collision time in backgroundscattering, polarization and correlation eects areof dierent importance in the two cases. In thecorresponding calculations, the separation intoresonant and background scattering opens theopportunity to account for electron correlation forthese two cases on dierent levels.

In the present work, we report results forlow-energy elastic electron scattering o CF3Clobtained from R-matrix calculations. Ample in-formation about this system is available in theliterature. Illenberger et al. [1] investigated nega-tive ion formation due to fragmentation of CF3Clby electron attachment. The ion yield for Clÿ

shows maxima at 1.3 and 4.8 eV collision energy.For the fragments Fÿ, FClÿ and CF2Clÿ theyfound maxima at 4.1, 3.9 and 4.2 eV, respectively.Absolute total cross-sections for electron scatter-ing, obtained from electron-transmission experi-ments between 0.6 and 50.0 eV were published byJones [2]. He found structures at 2.0 and 5.94 eV.By comparison with the results of Illenbergeret al., he assigned the lower peak to a resonance ofA1 symmetry and the higher to an E resonance.Mann and Linder [3] con®rmed this assignment bycrossed-beam experiments on vibrational excita-tions. In addition to the A1 resonance at 2.0 eV,they found the E resonance at 5.5 eV. They ob-served a further A1 resonance at 8.5 eV. Inthe same reference, measured integral elasticcross-sections were compared with the total cross-sections of Jones in order to estimate the totalinelastic contributions. Underwood-Lemons et al.[4] published absolute total cross-sections for anenergy range of 0.3±12 eV. They reproduced thestructures in the cross-sections of Jones, but theirabsolute values for the cross-sections are system-atically about 7 10ÿ16 cm2 smaller. Below 0.6 eV

they found a steep increase in the cross-sectionsfor decreasing energies. Calculated dierentialcross-sections on SE level using the Schwingermultichannel method (SMC) were publishedby Natalense et al. [5,6] in the energy range of5±30 eV.

We consider the eÿ CF3Cl system as an ex-cellent candidate to apply the resonance/back-ground separation discussed at the beginning. Thenegative ion formation observed by Illenbergeret al. can be explained to a great extent as a resonantprocess. The theoretical investigations of dissoci-ation dynamics by Lehr and Miller [7] and byWilde et al. [8] were based on this assumption. Intheir studies, resonant scattering was representedby the potential curve and the decay width for theionic system, estimated from experimental ®nd-ings. On the other hand, the estimated total in-elastic cross-sections by Mann and Linder, whichare even larger for nonresonant scattering, indicatethat the eect of background scattering to thenuclear motion is not negligible. We consider®xed-nuclei scattering calculations, separatedproperly into the resonant and background com-ponent, as a ®rst step for a strict ab initio study ofthe dissociative electron attachment process.

Considering the 2p electrons of the ¯ourine at-oms, the 3p electrons of chlorine and the 2s and 2pelectrons of the carbon atom as of valence type,the total number of valence electrons in the mol-ecule is 32. Due to this large number, a propernumerical treatment of electron correlation eectsin electron scattering o CF3Cl is a particularchallenge.

The cross-sections presented in this paper arebased on R-matrix theory, and the partitioningprocedure applied is based on the Feshbach for-malism. Both methods are well documented in theliterature, R-matrix [9,10] and references therein,Feshbach formalism [11,12]. The implementationof the Feshbach formalism into the R-matrix ap-proach is described in Ref. [13]. A brief introduc-tion to the methods is given in Section 2. Theresults presented and discussed in Section 3 includeSE and SEP calculations and the application of theseparation procedure and con®guration interac-tion (CI) calculations for the resonance position.A conclusion is given in Section 4.

2 T. Beyer et al. / Chemical Physics 255 (2000) 1±14

2. Theory

The R-matrix procedure provides a discreterepresentation of the scattering problem byconsidering the scattered particle within a ®niteregion surrounding the target system. In electron±molecule scattering, this region is usually chosen asa sphere X, de®ned by its radius r0 and its centerlocated inside the molecule. This sphere should belarge enough to contain completely the targetsystem, resulting in a negligible electron density ofthe target outside this sphere.

In order to preserve the hermitecity of the Xcon®ned Hamiltonian HX, its component repre-senting the kinetic energy has to be modi®ed. Interms of a one-particle basis representation fuig,this can be achieved by replacing the huijDjujiXmatrix elements by ÿhruijrujiX. The subscript Xreminds that these matrix elements (as well as allthe others) are obtained by an integration con®nedto X. The eigenfunctions and eigenvalues of theresulting equation

HX ÿ EkWk 0 1contain the complete scattering information, apartfrom a long-range electron molecule interaction.

In general, the wave function for electron scat-tering o an n-electron target system, character-ized as a regular solution of the dierentialequation

H ÿ EWqE; x1 . . . xn1 0 2can be expressed in the form,

WqE; x1 . . . xn1 Xilm

A Uix1 . . . xnY ml /; h1r filmE; r

ÿ HE; x1 . . . xn1;

3

where Ui are the energetically allowed target statesconsidered, r;/; h, the spherical coordinates ofxn1, film are the radial functions of the scatteredelectron, corresponding to the spherical quantumnumbers l and m and the target state Ui and E isthe collision energy plus the energy of the initialtarget state. A denotes the anti-symmetrizationoperator. The index q reminds that Wq is a par-ticular solution of Eq. (2). Polarization and cor-relation eects between the scattered electron and

target system are represented by the n 1-electron function H which is assumed to be zerooutside X.

Inside X, the solutions Wk of Eq. (1) are theregular solutions of Eq. (2) with E Ek and cantherefore be expressed in the form:

Wkx1 . . . xn1 Xilm

A Uix1 . . . xnY ml /; h1r fk;ilmr

ÿ HEk ; x1 . . . xn1; r 6 r0:

4

They are speci®ed by the boundary conditions,

dfk;ilmrdr

rr0

0: 5

The relationship between the scattering wavefunction and the solutions of Eq. (1) is the subjectof standard R-matrix theory. We shall thereforerestrict our discussion only to facts which are rel-evant for our present approach.

Scattering cross-sections, eigenphases, as well asthe normalization of the wave functions (3) areobtained from their asymptotical behavior forr!1. Due to the vanishing of H, the asymptoticscattering wave function is given by filmE; r,which can be obtained from a system of close-coupling equations de®ned by the long-range in-teraction between the scattered electron and thetarget system. The dimension N of the solutionvector is equal to the number of open channelsconsidered, while the number of linear indepen-dent solutions is 2N . The selection of N, the lin-early independent solutions which correspond to(regular) scattering functions is provided by thelogarithmic boundary conditions,

filmE; r0 Xi0l0m0

Rilm;i0l0m0 E dfi0l0m0 E; rdr

rr0

6

de®ned by the solutions of Eq. (1) via the R-matrix

Rilm;i0l0m0 E 1

2

Xk

fk;ilmr0fk;i0l0m0 r0Ek ÿ E

: 7

Due to their appearance in Eq. (7), the eigenvaluesof Eq. (1) are referred to as poles of the R-matrix.The values fk;ilmr0 are called the amplitudes of Wk

T. Beyer et al. / Chemical Physics 255 (2000) 1±14 3

with respect to the channels characterized byfilmg.

In order to calculate the interaction of somemetastable ionic state Ures with the decay contin-uum, the corresponding scattering wave functionhas to be determined. Since Ures is assumed tovanish outside X as long as the decay interaction isof short range, we only need a description of thescattering wave function inside X. In general, theR-matrix formalism provides an expression tocontinue the wave function from the outer regionof X to the inner:

WqEjX X

k

Wk1

2Ek ÿ E

Xilm

f qk;ilmr0 df q

ilmE; rdr

rr0

: 8

The index q distinguishes the dierent scatteringwave functions at the same energy E. Within theframework of the Feshbach projection operatorformalism, the energy-dependent resonance widthis given by

CE 2pXN

q1

hWbg;qEjHXjUresi: 9

The Wbg;qE are scattering states with respect tothe X-con®ned background Hamiltonian

H bgX 1ÿ jUresihUresjHX1ÿ jUresihUresj: 10

They are assumed to be mutually orthogonal andenergy normalized. Provided an expansion

Ures X

k

ckWk 11

is available, the calculation of CE and the cor-responding level shift DE is now a matter ofstandard R-matrix theory. For details in the nu-merical procedure the reader is referred to Ref. [13]and references therein.

Before discussing the question of how the co-ecients ck in Eq. (11) may be obtained from thesolutions of Eq. (1) we want to mention some re-lationships between these solutions and the scat-tering eigenphases. First of all, since the energies inEqs. (7) and (8) appear only in form of dierencesthey can be replaced by the corresponding relative

energies with respect to some reference energy Eref .In the following, Eref is set to the energy of theinitial target state. Then, E represents the collisionenergy. The R-matrix pole positions Ek now be-came comparable for dierent target systems andin particular for the case that no target exists at all.In that case, the radial dependence of the solutionsof Eq. (1) is given by the Bessel functions obeyingthe boundary condition (5). The pole positions aredetermined by the corresponding kinetic energy. Aweak interaction between target and scatteredelectron will cause a shift of these poles and thescattering eigenphases. If the poles are shifted tolower energies, the phases change to higher valuesand vice versa. The scaling of this interaction bysome factor k will provide a smooth transition ofthe poles corresponding to the free particle prob-lem into that of an electron±molecule scatteringproblem. The corresponding eigenfunctions and inparticular their amplitudes will also changesmoothly as long as no avoidable crossing of theR-matrix poles is caused by some singularity in thecomplex k plane. Our basic assumption is that thisis the case in nonresonant scattering. Then a one-to-one correspondence between the free particlepoles and the poles of the real problem in the en-ergy region considered exists, and the amplitudesof the corresponding eigenstates are similar. Theappearance of a resonance is indicated by an extraR-matrix pole which obscures an assignment ofstates between the resonant and the free particleproblem.

The construction of the ck in Eq. (11), as de-scribed in Ref. [13], is based on the R-matrix polepositions Ek and amplitudes fk;ilmr0 correspond-ing to the eigenstates Wk of the electron±moleculescattering problem. The considered Wk are re-stricted to those having the same symmetry as theresonance and a corresponding energy below someupper limit Emax;res. The construction of Ures isbased on the requirement that the complementarybackground subspace represents to a good ap-proximation the corresponding solutions of Eq. (1)for the free-particle problem. For this reason, thenumber nilm of free-particle solutions related toeach channel filmg has to be determined. Thesenumbers have to be found out by inspecting theamplitudes and pole positions of the considered Wk


and comparing them with the pole positions of thefree particle problem (one should notice that Ures

does not contribute to the amplitudes of the Wk).With increasing energy range for the Ek consid-ered, the determination of the nilm becomes moreand more ambiguous. This diculty can beavoided by an appropriate choice of Emax;res.

3. R-matrix calculations

3.1. Numerical realization of the R-matrix method

The crucial quantities in the numerical realiza-tion of the R-matrix approach for scatteringproblems are the eigenvalues Ek and the corre-sponding eigenstates Wk of Eq. (1). As can beobserved from their appearance in the Eqs. (7) and(8), the resulting phases and cross-sections aresensitive to these quantities if the corresponding Ek

is close to the total energy E and only weakly af-fected by energetically high-lying states. Thereforeone can restrict the in®nite series in Eq. (7) to a®nite sum. This restriction enables us to treatEq. (1) in a manner similar to bound-state calcu-lations for molecules. In the present case, themulti-reference singles- and double-excitationcon®guration interaction (MRDCI) concept isapplied in a modi®ed manner. This techniquerepresents the electronic wave function by ®nitebasis set expansions. The molecular orbitals (MOs)of the target system are obtained from a self-consistent ®eld (SCF) calculation. For the repre-sentation of the continuum we use nine s, seven p,seven d (published in Ref. [14]) and six f (fromRef. [15]) Gaussian type functions. These func-tions are con®ned within a sphere X with its centerlocated at the origin of the coordinate system. Inthe present calculation, the radius of X is set to 10bohr. The functions describing the scattered elec-tron are orthogonalized with respect to the targetMOs. We refer to these orthogonalized and nor-malized functions as continuum orbitals (CO).Since only elastic scattering is considered, thenumber of partial waves represented by the COsdetermine the number of scattering channels ap-pearing in the calculation.

Electron correlation is taken into account bycon®guration expansion. In the calculations dis-cussed below, the electron correlation is treated atdierent levels of sophistication. The SE approachincludes no correlation at all. The correspondingcon®gurations contain the occupied orbitals of thetarget ground state and an additional orbital,which may be an MO unoccupied in the target orany CO. The SCF energy of the target system istaken as reference energy Eref . The SEP calculationallows the electronic target system to relax in re-sponse to the approaching electron. This can beconsidered as a correlation between the scatteredelectron and the electronic target system as awhole. At CI level all electrons involved in thescattering process are correlated. Details of theSEP and CI approaches are described in the rele-vant subsections.

3.2. Representation of the target

At equilibrium geometry the CF3Cl molecularsymmetry point group is C3v. The center of mass islocated at the origin of the coordinate system andthe threefold molecular axis coincides with thez-axis. The correspondence between the partialwaves and the irreducible representations of thetarget point group is shown in Table 1. In ourcalculation, we use RC±Cl 3:332 bohr andRC±F 2:465 bohr as internuclear distances. TheFCF angle is taken to be 109:3°. These equilibriumvalues are obtained in MCSCF calculations for theneutral CF3Cl using the Gaussian code. Bartelland Brockway [16] estimated the equilibrium ge-ometry by electron diraction experiments:RC±Cl 3:309 0:009 bohr, RC±F 2:510 0:004 bohr and \FCF 108:6° 0:4°. We ®nd

Table 1

Classi®cation of real spherical harmonics ~Y lm for l < 4 abbre-

viated by l;m; according to the irreducible representations

of the C3v point group

IR l;m;A1 (0,0), (1,0), (2,0), (3,0), (3,3,+)

A2 (3,3,ÿ)

E (1,1,+), (2,1,+), (2,2,+), (3,1,+), (3,2,+)

(1,1,ÿ), (2,1,ÿ), (2,2,ÿ), (3,1,ÿ), (3,2,ÿ)


the agreement between these data and our calcu-lations to be acceptable for our purposes.

The molecular orbitals of the target are repre-sented in a correlation-consistent polarized valencedouble zeta (cc-pvdz) basis [17,18]. The groundstate occupation of the target is

1a211e42a2

13a214a2

15a212e46a2

13e47a218a2

19a214e4

10a215e46e47e41a2

2;1A1:

To reduce the computational eort of SEP and CIcalculations, we consider the K-shell of the carbonand ¯ourine atoms and the K- and L-shells of thechlorine, i.e.

1a211e42a2

13a214a2

15a212e4

as frozen (18 electrons).

3.3. Static exchange results

The SE results presented in the following pro-vide a reference for estimating polarization andcorrelation eects in the SEP and CI calculations.The number of symmetry adapted functions(SAFs) occurring in a particular symmetry speciesis de®ned by the sum of MOs unoccupied in thetarget and COs corresponding to this symmetry(Table 2). These numbers re¯ect the quality of thebasis for describing the scattered electron. The SEresults mainly depend on this description. There-fore SE results may be used to compare continuumrepresentations of dierent calculations.

In Fig. 1, the full curve shows the calculatedintegral cross-sections in the SE approximation,rSE

integralE. They are of the same order of magni-tude as the experimental cross-sections (pluses

and crosses) and exhibit resonant structuresaround 4.6, 9.4 and 13.2 eV. The symmetriesof these resonances are 2A1, 2E and 2A1, respec-tively, as can be observed in the related partialcross-sections (Fig. 2). In the energy range that isconsidered, the 2A2 cross-sections are smaller than0:04 10ÿ16 cm2 on the SE level, as well as on SEPand CI level. Therefore, this symmetry will not bediscussed any further. The two resonances in 2A1

and the one in 2E can also be observed as a steepincrease by p in the corresponding eigenphases(Fig. 3). The structures in the two low-lying 2A1

eigenphases can be explained as avoided crossingscaused by the resonance. This becomes obvious byconsidering the equivalent phases which dierfrom the original by an amount of p.

In comparison with the data of Mann andLinder, the resonance position in the SE approxi-mation lies roughly as 2.5 eV too high for the

Table 2

The number of symmetry-adapted functions (SAFs) for the symmetry species considered in the R-matrix scattering calculations at

dierent levels of approximationa

Symmetry SE SEP CI

Total T1 T2

2A1 38 19504 16134064 234404 1486532A2 7 18989 4380385 94197 666262E 40 19504 17417231 295534 186048

40 19009 17599988 294972 185656

a The two numbers for the 2E symmetry refer to the twofold degeneracy of that symmetry. T1 and T2 refer to the thresholds used (see

text).

σSEP;dipolintegral

σSEPintegral

σSMCSEintegral

σSEintegral

σtotal (exp. [4])

σtotal (exp. [2])

E [eV]

σ[10

16cm

2 ]

14121086420

50

45

40

35

30

25

20

15

10

5

0

Fig. 1. A comparison of calculated integral cross-sections (SE

and SEP) with experimentally determined total cross-sections.


lower, and 4.7 eV for the higher 2A1 resonance.The 2E resonance energy is found to be 4 eV abovethe experiment in the SE approach. A systematicoverestimation of the resonance energy on the SElevel has already been found in the previous cal-culations (e.g. Refs. [14,15,19,20]).

Fig. 1 shows the integral cross-sections obtainedfrom dierential cross-section data published byNatalense et al. [5,6]. These authors apply theSMC on the SE level employing pseudopotentials.The continuum is represented by partial waves upto an angular momentum quantum number ofl 7. Our results are in reasonable agreement forenergies above 8 eV, but show signi®cant devia-tions for lower energies. Dierential cross-sectionsfor a collision energy of 10 eV obtained from ourcalculations agree very well with the results byNatalense et al. (Fig. 4). For lower collision ener-gies, noticeable dierences can be observed.

3.4. Static exchange plus polarization results

To include polarization eects the con®gurationspace of the SE calculation is extended by singleexcitations of target electrons into virtual SCFMOs. These single excitations are restricted tothose con®gurations in which only MOs and not

Fig. 3. A comparison of scattering phases: 2A1 symmetry (top),2E symmetry (bottom), SE results on the left-hand side, SEP

results on the right-hand side. The dashed lines in the upper cut-

out are obtained from the lowest two phases by a shift of p.

E = 10:0 eV

θ150o120o90o60o30o

10

1

E = 8:0 eV

150o120o90o60o30o

10

1

E = 7:0 eV

θ

dσ dΩ(E;

θ)[1

0

16cm

2 ]

150o120o90o60o30o

10

1

E = 5:0 eV

dσ dΩ

(E

;

θ)[1

0

16cm

2 ]

150o120o90o60o30o

10

1

Fig. 4. Angular-dependent dierential cross-sections calculated

in SE approximation in comparison with SMC at the SE level

[5]: full curve R-matrix (this work), dashed curve R-matrix

background (this work) and dotted curve SMC.

σSEP2E

σSEP2A1

σSE2E

σSE2A1

E [eV]

σ[10

16cm

2 ]

14121086420

50

45

40

35

30

25

20

15

10

5

0

Fig. 2. 2A1 and 2E partial cross-sections calculated in the SE

and SEP approximation.


COs are occupied. The reference energy, i.e.the energy of the target system, is the same as inthe SE calculation due to Brillouin's theorem. Thenumber of resulting SAFs is also shown in Table 2.The calculated eigenstates are restricted to thelowest 15 of 2A1 symmetry and to the lowest nineof 2E. The energy of the highest 2A1 state is 16.8eV and that of the highest 2E state is 8.4 eV.Higher-lying R-matrix poles are approximated atthe SE level according to the selected statesR-matrix method (SSRM) [19]. Consequently, theSEP results approach the SE results for energieslarger than 9 eV in 2E symmetry.

The integral cross-sections in the SEP approxi-mation, rSEP

integralE, are depicted by the long-dashedcurve in Fig. 1. Partial cross-sections are shown inFig. 2. In 2A1 the resonances appear at 1.8 and8.6 eV, the 2E at 6.2 eV. The energetic position ofthe 1.8 eV resonance is roughly 0.2 eV below theexperimentally determined value. The 6.2 eVstructure is 0.7 eV above the experimental reso-nance position and the 8.6 eV resonance appears0.1 eV too high. The widths of all these resonancesare smaller than those in the experiment. We ®ndthe agreement of our results and experimental datasucient to con®rm the symmetry assignments ofMann and Linder. The nonresonant scattering inthe SEP approximation closely resembles experi-mental data for energies higher than 1 eV. TheSEP cross-sections for energies below 1 eV arereduced considerably in comparison with the SEresults.

In the results discussed above, the electron±target interaction outside X has been neglected.However, for a correct description of the scatteringprocess for energies below ca. 1 eV the long-rangepotential of the target has to be included in theclose-coupling equations mentioned in Section 2.The integral cross-section in the SEP approxi-mation including a dipole moment of 0.5 D [21] isdepicted by the short-dashed curve in Fig. 1. Now,the threshold behavior is in qualitative agreementwith the experimental data from Underwood-Lemons et al. For energies higher than 1 eV we®nd that the in¯uence of the long-range potentialsis negligible. Since we are particularly interested inresonant scattering we do not consider thesepotentials any further.

The resonant increases of the scattering eigen-phases for 2A1 and 2E symmetry as shown in Fig. 3are steeper than in the SE approach according tothe smaller resonance width. The structures in thenegative 2A1 phases can be explained in the sameway as in the SE case. In general, for a given en-ergy the SEP eigenphases are higher than thecorresponding SE phases. This is due to the factthat polarization acts as an attractive force ingeneral.

3.5. Resonance/background separation

In order to determine the metastable states andthe corresponding background scattering problemalong the concept mentioned in Section 2, one hasto specify the energy region in which the reso-nances are expected. In the present case, this en-ergy region was chosen to have an upper limit ofEmax;res 13:5 eV. The R-matrix poles Ei for thesystem considered, as well as for the free-particleproblem Efree

k are shown in Figs. 5 and 6. By in-specting the amplitudes corresponding to the Ei wefound that the appropriate free-particle problem,approximately represented by the 2A1 background,is spanned by three states of 0; 0 symmetry(Table 1), three of 1; 0, two of 2; 0, two of 3; 0and two of 3; 3;. In 2E symmetry, the corre-sponding free-particle problem is spanned by three

2f

1f

2d

1d

3p

2p

1p

3s

2s

1sEiEresEbgjEfreek

12

10

8

6

4

2

0

2f

1f

2d

1d

3p

2p

1p

3s

2s

1s

E[eV]

EiEresEbgjEfreek

12

10

8

6

4

2

0

Fig. 5. 2A1 R-matrix pole positions. Left-hand side, SE ap-

proach; right-hand side, SEP approach.


poles of 1; 1;, two of 2; 1; and 2; 2; andtwo of 3; 1; and 3; 2;.

The resonant scattering phases and the resultingbackground phases for the 2A1 symmetry areshown in Fig. 7 (top) and for 2E symmetry in Fig. 8(top). The background phases for both symmetriesin SE as well as in SEP approximation appearsmooth and structureless. This indicates that ourspeci®cation of the nilm is meaningful.

Figs. 7 and 8 show the energy-dependent decaywidth together with the level shift for both sym-metries and SE and SEP approximations. Thecrossing point of the straight line E±Eres with DEde®nes the null of the real part in the denominatorof the resonant T-matrix element. This energy canbe taken as an approximation for the resonanceposition. Table 3 lists the resonance positions forthe resonances in 2A1 and 2E symmetry on SE andon SEP level. The lower 2A1 resonance in SEPapproximation is 0.32 eV below the experimentallydetermined position, the maximum of the higher2A1 resonance lies 0.18 eV below the experiment ofMann and Linder. The 2E resonance on SEP levellies 0.34 eV above the corresponding experimentalvalue.

For a better understanding of the characteris-tics in resonant as well as background scatteringwe analyze Eres and the poles of the backgroundproblem Ebg

j shown in the Figs. 5 and 6. Each Ebgj is

connected with that free-particle pole positionsEfree

kj which represents the channel with the largestamplitude of the wave function corresponding toEbg

j . For comparable energies and l P 1 we ®ndthat the perturbation of free particle states de-creases with an increasing angular momentum.This is re¯ected in the background scatteringphases in a manner discussed in Section 2. Withinthe same angular momentum quantum number theperturbation of the free-particle states becomeslarger for higher energies. In addition, the ampli-tudes corresponding to the background poles showa stronger coupling of partial waves. This makesthe speci®cation of the nilm more and more dicultfor higher energies. To avoid this problem an ap-propriate choice of the energy region considered isnecessary.

2f

1f

2d

1d

3p

2p

1p

EiEresEbgjEfreek

12

10

8

6

4

2

0

2f

1f

2d

1d

3p

2p

1p

E[eV]

EiEresEbgjEfreek

12

10

8

6

4

2

0

Fig. 6. 2E R-matrix pole positions. Left-hand side, SE ap-

proach; right-hand side, SEP approach.

Fig. 7. Separated scattering phases and energy-dependent de-

cay width and level shift for both resonances of 2A1 symmetry.

The two pictures in the middle row refer to the low-energy 2A1

resonance. The two pictures in the last row refer to the higher-

energy resonance with that symmetry.


Comparing the SE results with those obtainedin the SEP approximation, one ®nds dierences inthe magnitude of polarization eects for thebackground poles and the resonances. The back-ground R-matrix pole positions are lowered sys-tematically by no more than 0.9 eV whenpolarization is included. In contrast, the Eres areshifted by more than 2.6 eV.

Dierential cross-sections derived from thebackground R-matrix in SE approximation areincluded in Fig. 4. Surprisingly, the results ofNatalense et al. agree much better with our back-ground results than with the complete cross-sections containing the resonant process. This is astrong indication that the dierences between our

work and that of Natalense et al. lies in the de-scription of the resonances, which is sensitive tothe choice of the MO basis.

3.6. Con®guration interaction approach for thelower 2A1 and the 2E resonance

According to the previous discussion of thepolarization eects there is a strong interactionbetween the scattered electron and the target sys-tem in the resonant case, but only a weak inter-action for background scattering. Consequently,one expects negligible electron correlation eectsfor the background, but noticeable eects for theresonances. Therefore, the SEP approach seems tobe sucient for describing nonresonant scattering.However, in order to estimate Eres, an inclusion ofelectron correlation is desirable. It is possible toreconstruct the R-matrix poles Ei and the corre-sponding amplitudes from the resonance/back-ground separated problem, where Eres is shifted bya certain amount. Therefore, electron correlationin resonant scattering can be taken into account bya separate CI calculation of Eres.

For a CI calculation, the SAFs are generatedfrom the SE con®guration space by single anddouble excitations. The maximum number ofelectrons which have to be correlated is 33. Thisleads to a very large con®guration space (Table 2).In order to make the calculation feasible, thenumber of con®gurations has to be reduced. Weapply a selection procedure based on the estimatedenergy contribution ei;j of the jth con®guration tothe ith root of the reference con®guration space(the SE con®guration space in our case). Thecon®guration j is discarded if jei;jj is below a giventhreshold T for all roots considered. The contri-bution of electron correlation which is neglectedby that selection with respect to the energy of rooti can be estimated by

eiT X

j:jei;jj<T

ei;j: 12

The selection threshold used in the presentcalculation is T1 2:8 lhartree. Table 2 showsthe (still large) number of selected SAFs for eachof the species of the point group C3v. Depending

Table 3

Calculated resonance positions

Symmetry SE (eV) SEP (eV)

2A1 4.24 12.56 1.68 8.322E 8.72 5.84

Fig. 8. Separated scattering phases and energy-dependent de-

cay width and level shift for 2E symmetry.


on the particular R-matrix states considered, theenergy of the neglected con®gurations in the orderof 16 million SAFs is small in each case, but thesum adds up to ei T1 2:8 lhartree between 4.4and 11.3 eV. These energy contributions musttherefore be taken into account in an appropriateapproximation. In most of the cases, the eiT overestimates the neglected correlation contribu-tion and has to be multiplied by appropriatescaling factors ki. In the extrapolation procedureintroduced by Buenker and Peyerimho [22,23]these factors are obtained from two CI calcula-tions using dierent selection thresholds T1 and T2

along

ki kiT1; T2 ÿECIi T1 ÿ ECI

i T2eiT1 ÿ eiT2 ; 13

where ECIi T1 and ECI

i T2 are eigenvalues relatedto the con®guration spaces obtained from the

selection thresholds, respectively. The energy,which contains the estimated contributions of thediscarded con®gurations, is then given by

Eexti 0 ECI

i T1 kiT1; T2 eiT1: 14

This correction procedure requires a one-to-onecorrespondence between the CI wave functionsobtained from both the con®guration spaces andthe eigenfunctions of the reference space, i.e.WCI

i WSEj , based on the overlap matrix ele-

ments cij hWCIi jWSE

j i.In order to describe the lower 2A1 resonance we

calculated the lowest seven R-matrix states for thissymmetry on the CI level. In the case of 2E reso-nance we considered the lowest nine states. Thematrix elements cij listed below convincinglydemonstrate that there is no unambiguous corre-spondence between WCI

i and WSEj . Therefore, an

extrapolation as in a manner described abovecannot be applied.

Representation of CI scattering states by SEstates for 2A1 symmetry is as shown:

Representation of CI scattering states by SEstates for 2E symmetry is as shown:

The restriction of our calculation to an esti-mation of the Eres may help to avoid this problem.In the following we denote the transformationmatrix which represents Ures and the backgroundstates Wbg

k in the basis of the original states Wk by

I hWCIi jWSE

j i

0:90 ÿ0:14 ÿ0:15 ÿ0:06 ÿ0:01 0:02 0:12ÿ0:04 ÿ0:62 0:06 ÿ0:01 0:62 0:16 ÿ0:19

0:08 0:07 0:64 ÿ0:59 ÿ0:11 0:27 ÿ0:140:15 0:04 0:55 0:71 ÿ0:06 0:15 0:000:08 0:58 0:10 ÿ0:04 0:68 ÿ0:09 0:190:06 0:31 ÿ0:32 0:12 0:02 0:65 ÿ0:46ÿ0:11 ÿ0:07 ÿ0:06 ÿ0:03 ÿ0:01 0:50 0:74

0BBBBBBBB@

1CCCCCCCCA:

II hWCIi jWSE

j i

0:93 0:00 0:00 0:06 0:04 0:11 ÿ0:04 0:01 ÿ0:02ÿ0:01 0:72 0:10 0:30 ÿ0:50 ÿ0:03 ÿ0:11 ÿ0:01 ÿ0:02

0:03 0:18 ÿ0:85 ÿ0:34 ÿ0:11 ÿ0:01 0:00 0:09 0:01ÿ0:06 0:00 ÿ0:35 0:77 0:40 ÿ0:05 ÿ0:01 0:10 ÿ0:04

0:00 ÿ0:56 ÿ0:14 0:28 ÿ0:67 0:04 0:03 0:06 ÿ0:07ÿ0:07 0:02 ÿ0:10 0:08 0:02 0:59 0:04 ÿ0:57 0:27ÿ0:01 0:11 0:06 0:02 ÿ0:02 0:40 0:72 0:40 0:05ÿ0:09 ÿ0:03 0:04 ÿ0:04 0:03 0:55 ÿ0:59 0:46 0:03ÿ0:03 0:03 ÿ0:01 ÿ0:03 0:02 0:22 0:03 ÿ0:16 ÿ0:87

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA:


SEF in the SE case and CIF in the CI case. Then theoverlap of the resonance/background basis in theSE representation and the corresponding basis inthe CI representation can be expressed by

~C SEFT C CIF: 15In the scheme in Figs. 9 and 10 the squares of thecoecients of ~C correspond to the size of the dots.These ®gures show that electron correlation causesa strong interaction between the SE backgroundstates, while the wave function representing theresonance is only mildly aected by correlation.This opens the possibility to apply the extrapola-tion procedure described above to Eres.

On the analogy of Eq. (14), we obtain

Eextres0 ECI

resT1 kresT1; T2 eresT1 16with

ECIres

Xi

CIf 21;i ECI

i ; 17

eresT1 X

i

SEf 21;i eiT1; 18

kresT1; T2 ÿP

iCIf 2

1;i ECIi T1 ÿ ECI

i T2Pi

SEf 21;i eiT1 ÿ eiT2 ;

19CIf1;i and SEf1;i are the matrix elements of CIF andSEF, respectively, the ®rst rows of CIF and SEF areassumed to correspond to Ures.

The results of this extrapolation in our presentcalculation are summarized in Table 4. Because theEext

res0 for the two components of the 2E symmetryare obtained in separate calculations, the degen-eracy is not exact. However, the energy splittingis satisfactorily small (less than 0.04 eV). Relativeto the ground state of the neutral system(ECI

neuT1 2:8 lhartree ÿ796:377904 hartree,Eext

neu0 ÿ796:424289 hartree) the obtained po-sition of the resonant state is

DEextres 1:45 eV 20

for the 2A1 symmetry and

DEextres 5:24 eV 21

Fig. 9. Pictorial representation of ~C for 2A1 symmetry. First

row corresponds to the discrete component of the resonance in

SE, further rows to the background R-matrix states in SE ap-

proximation; the ®rst column corresponds to the discrete

component in CI, further columns to background states in the

CI approximation.

Fig. 10. Pictorial representation of ~C for 2E symmetry. First

row corresponds to the discrete component of the resonance in

SE, further rows to the background R-matrix states in SE ap-

proximation; the ®rst column corresponds to the discrete

component in CI, further columns to background states in CI

approximation.


for the 2E symmetry. The 2A1 resonance position is0.55 eV, the 2E resonance position roughly 0.26 eVbelow the experimentally determined positions [3].

In the case of the lowest scattering state in 2A1

as well as in 2E symmetry there is a reason-able agreement between its SE and its CI repre-sentation as in I and II and the extrapolationformula (14) can be applied. The energies obtainedare Eext

1 0 ÿ796:407559 hartree for 2A1 andEext

1 0 ÿ796:376991 hartree for 2E symmetry.Since these two states contribute only little to theresonances (less than 3%), they can be consideredas background states. The energy dierence withrespect to the corresponding SEP states providesan estimation of the correlation energy which isnot taken into account by target polarization. Forthe lowest 2A1 state this correlation contribution isÿ19.78 eV and for the 2E state ÿ19.58 eV. Theamount of electron correlation energy for thetarget state is ÿ19.67 eV. The dierence betweenbackground and target correlation energy esti-mates the contribution provided by the nonreso-nantly scattered electron. These contributions areÿ0.11 eV for the 2A1 and 0.09 eV for the 2E state.The positive value in the latter case indicates thathere the correlation of the target electrons is low-ered by polarization. In both cases, the correlationcontribution of the scattered electron is small,which con®rms the assumption that for back-ground scattering SEP is an appropriate model. Inthe resonant case, the correlation contribution ofthe scattering electron is ÿ0.63 eV for the low lying2A1 resonance and ÿ0.55 eV for the 2E resonance.These values are in agreement with the assumptionthat the resonance position noticeably depends onelectron correlation.

The underestimation of the resonance energiesin CI calculation may be due to an overestimationof the correlation obtained by the extrapolation.The corresponding energy contributions (Eext

res0ÿECI

resT1 ÿ6:86 eV for 2A1 and ÿ6:56 eV for 2E

resonances) are large compared with the values forthe two background states (Eext

1 0ÿ ECI1 T1

ÿ6:13 eV for 2A1 and ÿ5:28 eV for 2E). In addi-tion, one should take into account that in thepresent CI calculations contributions of electroni-cally excited (open or closed) channels are in-cluded, while the T-matrix is restricted to elasticscattering. We expect that the quality of the cal-culation can be improved by a systematical sup-pression of these contributions.

4. Conclusion

R-matrix results for elastic electron scatteringo CF3Cl are an example in which the SEP ap-proach reproduces satisfactorily the experimentalcross-sections. The resonance positions appearingin the calculation dier by no more than 0.4 eVfrom the positions taken from experiment. Ananalysis of the target polarization reveals large ef-fects for resonant collisions, but only moderatechanges in the background scattering. Conse-quently, one expects noticeable electron correlationeects in the resonant case. For the background,however, it is assumed that the correlation of thescattered electron with the electrons of the targetsystem is well described by the target polarization.This assumption is con®rmed by our CI calcula-tion. The resonance positions on the CI level aretoo low and will be subject to further investiga-tions. Since the background scattering phases arevery smooth, resonance/background separationappears to be helpful when a parametrization of thescattering information is desired.

Acknowledgements

The authors wish to thank Professor M. Pericfor helpful discussions. The ®nancial support of

Table 4

Results of the extrapolation scheme (Eq. (16)) for Eres

Symmetry ECIres2:8 lhartree (hartree) kres2:8; 4:4 lhartree (hartree) eres2:8 lhartree (hartree) Eext

res 0 (hartree)

2A1 ÿ796:118921 0:697900 ÿ0:361256 ÿ796:3710412E ÿ795:990657 0:690825 ÿ0:348772 ÿ796:231598


the Deutsche Forschungsgemeinschaft is gratefullyacknowledged.

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Documents

Study of electron polarization and correlation effects in resonant and background electron scattering off CF3Cl