Dyson orbitals for ionization from the ground and ...iopenshell.usc.edu/pubs/pdf/jcp-127-234106.pdf · Dyson orbitals for ionization from the ground and electronically excited states

Dyson orbitals for ionization from the ground and electronicallyexcited states within equation-of-motion coupled-cluster formalism:Theory, implementation, and examples

C. Melania Oana and Anna I. Krylova�

Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA

�Received 14 August 2007; accepted 11 October 2007; published online 21 December 2007�

Implementation of Dyson orbitals for coupled-cluster and equation-of-motion coupled-cluster wavefunctions with single and double substitutions is described and demonstrated by examples. Bothionizations from the ground and electronically excited states are considered. Dyson orbitals arenecessary for calculating electronic factors of angular distributions of photoelectrons, Comptonprofiles, electron momentum spectra, etc, and can be interpreted as states of the leaving electron.Formally, Dyson orbitals represent the overlap between an initial N-electron wave function and theN−1 electron wave function of the corresponding ionized system. For the ground state ionization,Dyson orbitals are often similar to the corresponding Hartree-Fock molecular orbitals �MOs�;however, for ionization from electronically excited states Dyson orbitals include contributions fromseveral MOs and their shapes are more complex. The theory is applied to calculating the Dysonorbitals for ionization of formaldehyde from the ground and electronically excited states.Partial-wave analysis is employed to compute the probabilities to find the ejected electron indifferent angular momentum states using the freestanding and Coulomb wave representations of theionized electron. Rydberg states are shown to yield higher angular momentum electrons, ascompared to valence states of the same symmetry. Likewise, faster photoelectrons are most likely tohave higher angular momentum. © 2007 American Institute of Physics. �DOI: 10.1063/1.2805393�

I. INTRODUCTION

Photoelectron spectroscopy is a powerful technique forprobing the electronic structure of molecules and ions. Bymeasuring kinetic energy of the ejected electrons, one candetermine the electronic and vibrational energy levels of theionized system. The ensuing Franck-Condon progressionscontain information on the ionization-induced structuralchanges from which changes in electronic wave functionscan be inferred, e.g., long progressions suggest the removalof an electron from bonding or antibonding orbitals, etc.More direct probe of the initial and target electronic wavefunctions is possible by measuring angular distributions ofthe photoelectrons �PADs�,1,2 which can be recorded in es-sentially molecular frame by employing coincidencespectroscopy.3 Furthermore, changes in the electronic struc-ture in the course of a chemical reaction can be monitored byintroducing time resolution in photoelectron spectroscopy.4–7

However, retrieving the electronic structure information fromthe experimentally measured PADs is not a straightforwardtask, despite impressive progress in experimental techniques.

Different aspects of theoretical framework for calculat-ing PADs have been developed by several groups.8–22 Moti-vated by resonance enhanced multiphoton ionization�REMPI� experiments in molecular beams, the dependenceof PADs on the initial and target rotational states has beenworked out in meticulous details. Molecular-frame PADs ap-propriate for the experiments when the orientation of the

system is fixed in the laboratory frame have also beendiscussed.10 Recently, simulations of time-resolved PADs,which take into account nuclear dynamics, have beenreported.23–25

A central quantity in these studies is a so-called photo-electron matrix element, which connects the initial and finalwave functions,22,26

DkfIF = ��IF

d �r��kfel �r��r, �1�

where � is the dipole moment operator, �kfel �r� is the final

�continuum� state of the ejected electron with kinetic energydetermined by its momentum k, and �IF

d �r� is a one-electronquantity �called Dyson orbital� containing all the necessaryinformation about N and N−1 electronic wave functions ofthe initial and the ionized states of the system,

�IFd �1� = �N �I

N�1, . . . ,n��FN−1�2, . . . ,n�d2 . . . dn . �2�

Dyson orbitals are also necessary for calculations of Comp-ton profiles,27 electron momentum spectra,28,29 and interpre-tation of other orbital imaging experiments.30–32

For atomic ionization, the electron continuum states�kf

el �r� are well described by spherical waves �often referredto as partial waves�, either freestanding or Coulomb, whichare the exact solutions for �unbound� motion of electrons incentrally symmetric fields.33 For molecular ionization, thepotential created by the ionic core is noncentral and thepartial-wave description holds only in the asymptotic limitwhen the electron is far from the ionized core.18,34a�Electronic mail: [email protected].

THE JOURNAL OF CHEMICAL PHYSICS 127, 234106 �2007�

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http://dx.doi.org/10.1063/1.2805393

http://dx.doi.org/10.1063/1.2805393

http://dx.doi.org/10.1063/1.2805393

http://dx.doi.org/10.1063/1.2805393

While considerable attention9,18–22 has been given to cal-culating final states of the ionized electron, �kf

el �r�, the mo-lecular electronic factor �IF

d �1� has been treated very ap-proximately, most often within Hartree-Fock/Koopmansframework. Effects of correlation on Dyson orbitals havebeen discussed,35 mostly within Green function framework.36

The focus of this work is calculating Dyson orbitalsfrom high-level ab initio calculations of the initial and ion-ized electronic wave functions by equation-of-motioncoupled-cluster �EOM-CC� methods. The departure from theuncorrelated Koopmans approximation becomes particularlyimportant when ionization of electronically excited andopen-shell species is considered. The former is relevant to awide range of experiments,1–3,5–7 e.g., REMPI, in which nphoton excitation of the initial molecule is followed by mphoton ionization, or time-resolved pump-probe experimentsemploying ionization probe to monitor the dynamics of anelectronically excited system.

It should be noted that the Dyson orbitals of Eq. �2� canbe computed for any initial and target many-electron wavefunction, and their definition does not invoke propagatorformalism.37,38 In the latter, the Dyson orbitals are solutionsof an effective one-electron equation yielding correlated ion-ization energies and pole strengths.36,39,40

The structure of the paper is as follows. The next sectiondiscusses the theoretical framework for electronic factors,Eqs. �1� and �2�, their relationship to PADs, and partial-waveanalysis of the outgoing electrons. Then EOM-CC methodsare briefly described, and the calculation of Dyson orbitalswithin EOM-CC formalism is presented. The new methodol-ogy is applied to calculate Dyson orbitals for photoionizationof the ground and electronically excited states of formalde-hyde producing ground and excited states of the ion. Thepartial-wave analysis, albeit approximate, reveals interestingqualitative relationships between the character of the ionizedstate and the final states of the photoelectrons.

II. THEORY

A. Molecular-frame PADs and electronic wavefunctions

In this section, we summarize the formalism behindPADs �see Ref. 1 for an excellent review� focusing on elec-tronic factors involved. Our presentation is done in the mo-lecular frame and averaging over molecular orientations isdiscussed later.

Consider ionization of an N-electron system describedby the wave function �I

N�1, . . . ,n�, where i denotes the spa-tial and spin coordinates of ith electron, i�xi ,yi ,zi ,�i�= �ri ,�i ,�i ,�i�, and the transformation between polar andCartesian coordinates is given in the Appendix. Assumingthat ionization is fast, i.e., within sudden ionizationapproximation,8,11 the final state can be described by the in-dependent cation and continuum wave functions,�F

N−1�1, . . . ,n−1� and �kfel �r ,��, respectively, with no corre-

lation between the outgoing and the remaining N−1 boundelectrons. The wave vector k is defined by kinetic energy ofthe ejected electron:

k =p

�=

�2mE

�. �3�

By applying time-dependent perturbation theory to de-scribe ionization by electromagnetic field and within the di-pole approximation, the probability of producing the ejectedelectron in the final state �kf

el is26,33

IIFkf = �IN�1, . . . ,n��F

N−1�1, . . . ,n − 1�

��kfel �n�d1 . . . dn 2

, �4�

where � is a dipole moment operator associated with ioniz-ing field, and all the functions and operators are expressed inthe molecular frame r. Invoking strong orthogonality condi-tion, which is usually justified by vanishing overlap betweenthe core and the ionized electron,27,31 and by using the anti-symmetric properties of the electronic wave functions andintegrating over the spin coordinates, we arrive at

IIFkf = �IFd �r��r��kf

el �r�dr 2

= �DkfIF�2, �5�

where �IFd �r� is a Dyson orbital defined by Eq. �2� and Dkf

IF isa photoelectron matrix element from Eq. �1�. Thus, thesquares of the dipole matrix elements, �Dkf

IF�2, give the prob-ability to find the ionized core in state F and the ejectedelectron in state f for initial state I of the system.

Equation �5� may be used as a starting point for derivingPAD. The probability of finding an electron at the point�� ,��, IIFkf�� ,��, can be obtained from Eq. �5� by changingthe three-dimensional integration over r by the integrationover the radial coordinate r only. Thus,

IIFkf��,�� = �DkfIF��,��2. �6�

Alternatively, IIFKf�� ,�� can be computed as1

I�k,�,�� = ��kfel �r,�,��*�kf

el �r,�,�� · r2dr . �7�

The spatial part of the wave function of a free electron�kf

el can be expanded over the basis of spherical waves,1

�kfel �x,y,z� = �kf

el �r,�,�� = �lm

CklmRkl�r�Ylm��,�� , �8�

where Ylm�� ,�� are familiar spherical harmonics,

Ylm��,�� = �lm�� · m�� 9�

and Rkl�r� is a radial part, which will be discussed below.The angular and radial basis functions are orthonormal,

0

2

m*��m� ��d� = �mm�

0

Ylm* ��,��Yl�m��,��sin �d�d� = �ll��mm� �10�

0

�

r2Rkl�r�Rk�l�r�dr = 2��k� − k� = ��E� − E� .

234106-2 C. Melania Oana and A. I. Krylov J. Chem. Phys. 127, 234106 �2007�


Eq. �7� then becomes

I�k,�,�� = �ll�mm�

Cklm* Ckl�m�Ylm

* ��,��Yl�m��,��

� Rkl�r�Rk�l��r�r2dr . �11�

If the basis functions RklYlm of expansion �8� are theeigenstates of a field-free Hamiltonian describing the ionizedelectron �in the potential due to the core�, then the squares ofthe corresponding expansion coefficients �Cklm�2 are just thesquares of the following photoelectron matrix elements:

�Cklm�2 = � �IF�d��r��r�Rkl�rn�Ylm��,��dr�2

�DklmIF �2.

�12�

Their phases, the so-called scattering-induced phase shifts,1

depend on the interaction of the outgoing electron with theionized core and cannot be determined without solving somekind of a Schrödinger equation for the outgoing electron.1

They can be defined as

Cklm = �DklmIF �ei�l. �13�

Thus, in this case, the overall probability of Eq. �11� assumesthe following simple form:

I�k,�,�� = �ll�mm�

�DklmIF � · �Dkl�m�

IF �ei��l�−�l�

� Rkl�r�Rkl��r�r2dr · Ylm* ��,��Yl�m��,�� .

�14�

This is the basis of partial-wave decomposition analysis ofthe final continuum states. Under the above conditions, onecan easily calculate the probabilities of finding the ionizedelectron in a certain angular momentum state from the Dysonorbital by using Eq. �12�, however, the resulting partial-wavedecomposition coefficients �Dklm

IF � are not sufficient for calcu-lating the PAD, unless interference between different outgo-ing waves can be neglected, i.e., within random-phase ap-proximation. This is generally not justified.41,42

The above assumption, i.e., that RklYlm are the eigen-states of the field-free Hamiltonian describing the ionizedelectron, is valid only for central potentials in which exactseparation of radial and angular motions is possible. This isthe case of atomic ionization. The choice of Rkl depends onthe system being ionized, e.g., for ionization of anions, thepotential due to the ionized core, which is neutral, is weakand standing free waves �see the next section� are expectedto be a good approximation.34 For ionization of neutrals, amore appropriate choice of Rkl is the eigenfunctions of theCoulomb potential.33 When molecular ionization is consid-ered, two approaches are possible. Expansion �8� can be re-written using basis functions corresponding to molecularpoint group symmetry,15 as has been done for diatomics byPark and Zare.18 Alternatively, one may still use partialwaves to represent the state of the ionized electron, with theunderstanding that Eq. �12� and, consequently, Eq. �14� are

no longer rigorously correct. Nevertheless, a small size of thecore relative to the continuous final state of the electron sug-gests using Eqs. �12�–�14� as an approximation. This is es-sentially equivalent to neglecting the interaction between thecore and the electron.34 Note that even under this approxi-mation, Eq. �12� will give rise to more partial waves in thecase of molecular ionization as compared to atoms, wherethe selection rules are strictly l= ±1. In this work, we applypartial-wave decomposition for qualitative analysis of Dysonorbitals and selection rules of molecular ionization. We em-ploy both the freestanding and Coulomb waves �see nextsection� and discuss the consequences of different choices ofthe radial functions.

The Dyson orbital �d�1� introduced by Eq. �2� is ananalog of the reduced one-electron transition density matrixand can be described as an overlap between the N and N−1 electron wave functions of, for example, the neutral andthe cation. In the second quantization form,

�d�1� = �p

�p�p�1� , �15�

�p = ��N�p+��N−1� , �16�

where ��p� are the reference molecular orbitals �MOs�.Dyson orbitals are not normalized, and their norms are pro-portional to the one-electron character of the ionization tran-sition. For the Hartree-Fock description of the neutral, andwithin the Koopmans approximation for the cation, the cor-responding Dyson orbitals are just the cannonical Hartree-Fock orbitals, and their norm is one. As Eqs. �5� and �12�suggest, one may interpret Dyson orbitals as wave functionsof the ejected electron. For example, in a hydrogenlike atom,Dyson orbitals are just electronic wave functions of the statesthat are being ionized.

B. Radial functions Rkl

In this section we discuss two obvious choices of radialfunctions Rkl in expansion �8�, i.e., the freestatnding and theCoulomb waves. In the former case, Rkl�r� are expressedthrough either the half-integer or spherical Bessel functions,Jl+1/2 and jl,

33 respectively,

Rkl�r� =�2k

rJl+1/2�kr� = 2kjl�kr� . �17�

The first few members of the series are

Rk0 = 2ksin�kr�

kr, �18�

Rk1 = 2ksin�kr��kr�2 − 2k

cos�kr�kr

, �19�

Rk2 = 6ksin�kr��kr�3 − 2k

sin�kr�kr

− 6kcos�kr��kr�2 . �20�

The Rkl functions contain sin�kr� / �kr�n and cos�kr� / �kr�n

terms and have the overall form of damped oscillations. TheRkl functions corresponding to kinetic energies of 0.25 and1 eV and l=0–5 are plotted in Fig. 1. As one can see, the

234106-3 Dyson orbitals for ionization J. Chem. Phys. 127, 234106 �2007�


higher l is, the lower is the value of the function at smalldistances. Also, lower kinetic energy corresponds to longeroscillations.

It is instructive to compare Rkl with typical atomic orbit-als, as done in Fig. 2, which displays the radial parts of the s,p, and d Gaussian-type orbitals with the exponents represen-tative of the carbon valence shell. These orbitals extend toonly about 5 Å, where only the lowest l radial functions havesignificant density. By virtue of Eq. �12�, one can expect thationization of Rydberg states yields higher angular momen-tum photoelectrons, as compared to ionization of valencestates of similar symmetry. High-l states also become moreprobable for higher-energy electrons due to tighter oscilla-tions in the Rkl functions with larger k values �i.e., Wignerlaw�.

As kr becomes very small �kr� l�, the radial functionshave the following asymptotic behavior:43

Rkl = 2k�kr�l

�2l + 1�!!. �21�

The Coulomb waves have more complicated form thatinvolve confluent hypergeometric functions of complexarguments.33 In this work, we consider only the limit of verysmall k values �k→0�, in which the radial functions can beexpressed33 through the Bessel functions,

�Rkl/��k��k→0 = ��4/r�J2l+1��8r�� . �22�

The Coulomb waves for k=0.01 eV are plotted in Fig. 3. Themost obvious difference between the Coulomb and the free-standing waves �shown in the inset� is much tighter oscilla-tions of the former, which can be rationalized in terms ofstrong attractive character of the Coulomb potential. Thus,for the same Dyson orbital, the expansion in the Coulombwaves will contain higher angular momentum terms, and thedependence of the corresponding �Cklm�2 on kinetic energy ofthe electrons will be less pronounced.

C. Averaging over molecular orientations

Depending on the experimental setup, different distribu-tions of molecules are sampled by ionization, which givesrise to different types of PADs. The analytic expressions forseveral experimental setups have been developed using an-gular momentum algebra, as summarized in Ref. 1. We adopta more flexible numerical approach that enables us to aver-age over any spatial distribution as described below.

By introducing Euler transformations R�� and R��−1 be-

tween the molecular �x� ,y� ,z�� and the laboratory �x ,y ,z�frames �see Appendix�, and by considering probability�� ,� ,�� of different molecular orientations, Eq. �12� be-comes

�Cklm�2 = �Cklm��,�,��2��,�,��d�d�d� , �23�

�Cklm��,�,��2

= �d�R��r��r�Rkl�rn�Ylm��n,�n�dr 2

, �24�

where �Cklm�2 are the �Cklm�2 coefficients averaged over allmolecular orientations in the laboratory frame. If all molecu-lar orientations are equally probable, i.e., �� ,� ,��=1 /4,most of the information about the Dyson orbital is lost. Toavoid the loss of information, the molecules could be pre-aligned prior to ionization event, e.g., by electronic excita-tion or by selecting a particular rotational state. Finally, incoincidence experiments,3,44 the photoelectrons can be re-corded in the recoil frame, thus allowing one to retrieve theinformation about the Dyson orbital essentially in the mo-lecular frame. Below we discuss averaging for two commonexperimental setups.

Let us consider photoionization from electronically ex-cited states, when the molecules are selected by the first laser�assume the first laser is polarized along z axis�. The prob-

FIG. 1. Radial parts of the freestanding spherical waves, Rkl�r� for k=0.271 a.u. �upper panel� and k=0.068 a.u. �lower panel�, which correspondto photoelectron kinetic energies of 1 and 0.25 eV, respectively.

FIG. 2. Radial parts for the s, p, and d Gaussian-type orbitals with theexponent �=0.1, a typical value for the carbon valence shell.



ability of a molecule to be excited is proportional to thescalar product between the laser dipole �l=z and the dipolemoment of an electronic transition �el,

��,�,�� = �R��−1 �el · z�2. �25�

For diatomic molecules, the above yields two simple expres-sions, for the parallel and perpendicular electronic transi-tions,

�par��,�,�� = cos2�� , �26�

�per��,�,�� = sin2�� . �27�

In coincidence experiments, if molecules are aligned along zaxis, only averaging over the azimuthal angle � should beconducted, as well as averaging between parallel and anti-parallel �with respect to z axis� orientations.

D. Ionized, electron-attached, and electronicallyexcited states within EOM-CCSD formalism

Conceptually, EOM approach45–52 is similar to configu-ration interaction �CI�: Target EOM states are found by di-agonalizing the so-called similarity transformed Hamiltonian

He−THeT,

HR = ER , �28�

LH = LE , �29�

LR = 1, �30�

where T, R, and L+ are general excitation operators withrespect to the reference determinant �0�. Due to the non-

Hermitian nature of H, its left eigenstates L are not Hermit-ian conjugates of R. Regardless of the choice of T, the spec-

trum of H is exactly the same as that of the originalHamiltonian H—thus, in the limit of the complete many-electron basis set, EOM is identical to full configuration in-teraction. In a more practical case of a truncated basis, e.g.,when T and R are truncated at single and double excitations,the EOM models are numerically superior to the correspond-ing CI models,53 because correlation effects are “folded in”in the transformed Hamiltonian. Moreover, the truncatedEOM models are size consistent �or, more rigorously, size

intensive�, provided that the amplitudes T satisfy the CCequations for the reference state �0�,

��H − E�0� = 0, �31�

where � denotes �-tuply excited determinants, e.g.,�i

a ,ijab� in the case of coupled-cluster singles and doubles

�CCSD�.By combining different types of excitation operators and

references �0�, different groups of target states can be ac-cessed, as explained, for example, in Refs. 52 and 54. In theionized/electron-attached EOM models,55–58 operators R arenot electron conserving �i.e., include different numbers ofcreation and annihilation operators�,

�N−1 = RIP�CCSDN = RIPeT1+T20,

�32�

RIP = �i

rii +1

2�ija

rija a+ji ,

where �CCSDN and 0 are the reference CCSD wave functions

and the reference Slater determinant of the neutral, respec-tively. Thus, for the ionization from the electronic groundstate, Dyson orbitals are defined as

�pR = �0e− �T1+T2��1 + ��p+�RIPeT1+T20� ,

�33��p

L = �0e− �T1+T2�LIP�p�eT1+T20� .

Electronically excited states can be described when thereference �0� corresponds to the ground state wave func-tion, and operators R conserve the number of electrons andthe total spin,47,49,59

�exN = REE�CCSD

N = REEeT1+T20, �34�

REE = r0 + �ia

riaa+i +

1

4 �ijab

rijaba+b+ji , �35�

and the corresponding Dyson orbitals are thus

�pR = �0e− �T1+T2�LEE�p+�RIPeT1+T20� ,

�36��p

R = �0e− �T1+T2�LIP�p+�REEeT1+T20� ,

FIG. 3. Radial parts of the Coulomb waves, Rkl�r� fork=0.002 71 a.u. �Ek=0.01 eV�. Inset: The correspond-ing freestanding spherical waves in the same r range.



Similarly to one-particle transition density matrices inEOM-CC for excitation energies �EOM-EE� theory,49 the“right” and “left” Dyson orbitals, �p

R and �pL, are not identi-

cal, and Eq. �12� should be rewritten as

�Cklm�2 = �d,R�r��r�Rkl�rn�Ylm��n,�n�dr

· �d,L�r��r�Rkl�rn�Ylm* ��n,�n�dr . �37�

Since the same reference CCSD wave function is em-ployed in both EOM-IP and EOM-EE calculations, theevaluation of Dyson orbitals is very straightforward and doesnot require to deal with nonorthogonal MO sets.

We implemented Dyson orbitals for the CCSD/EOM-EE-CCSD/EOM-SF-CCSD and EOM-IP/EA-CCSD wavefunctions of the initial and ionized/electron-attached states,respectively. EOM-EE-CCSD includes single �1h1p, 1-hole-1-particle� and double �2h2p� excitations in the EOM part,see Eq. �35�, and is therefore appropriate for electronicallyexcited states that are dominated by single-electron excita-tions. The SF variant allows us to consider initial states of adiradical or triradical character, as well as some doubly ex-cited states.54 EOM-IP-CCSD, which includes 1h and 2h1poperators in the EOM part, see Eq. �32�, is capable of de-scribing states dominated by Koopmans-type �1h� excita-tions including the states with several singly ionized configu-rations strongly mixed in the target wave function. EOM-EA-CCSD includes 1p and 1h2p operators and allows accessto other types of open-shell states.

Our implementation of Dyson orbitals within EOM usesa simple modification of our general EOM-CCSD code51 fol-lowing the elegant idea first employed by Stanton whopointed out that simply by adding a very diffuse orbital �e.g.,the size of Earth� to the basis and generating guess vectors

that include excitations to/from this orbital is equivalent toionization/attachment. Thus, Dyson orbitals are simply thecorresponding row/column in the EOM-EE transition densitymatrices.51,60 Their norms are proportional to the one-electron character of the ionizing excitation. The two-electron character of the ionization can be estimated onlyfrom the two-particle transition density matrices,

�pqsR = �0e− �T1+T2�LEE�p+q+s�RIPeT1+T20� , �38�

�pqsL = �0e− �T1+T2�LIP�s+qp�REEeT1+T20� . �39�

For ionization transitions that have mainly two-electron char-acter, the Dyson orbital norm is close to zero.

E. Implementation details

Our implementation consists of the two major parts: �i�Calculation of Dyson orbitals, and �ii� the partial-wave de-composition analysis and averaging. The first part, which isbased on Eqs. �16�, �33�, and �36�, requires calculation ofcorrelated wave functions of the initial and target states andis implemented within the EOM-CC suite of codes in theQ-CHEM electronic structure package.61 The Dyson orbitalsare expressed in terms of the reference MOs, see Eq. �16�,and the normalized coefficients �p, as well as the norms ofthe orbitals, are printed in the output. The values of theDyson orbital on a cubic grid can also be computed andplotted using visualization software, e.g., VMD.62

The second part, which is a separate computer program,calculates �Cklm�2, see Eq. �24�, for specified k values and upto a maximum l and specified polarization of ionizing radia-tion. The calculations are performed either in the molecularframe, or, if averaging over the molecular orientations is re-quested, over the ��i ,� j ,�k� grid with subsequent averaging,Eq. �23�. The evaluation of integral �Eq. �24�� is done nu-merically on the cubic �rlmn= �xl ,ym ,zn�� grid in the labora-tory frame as follows. At each grid point rlmn, the values of

TABLE I. Vertical ionization energies �eV� and leading contributions toDyson orbitals for the ionization of CH2O in its ground state producingvarious states of the cation, EOM-IP-CCSD /6-311G�2+ ,2+ �G**.

CH2O+ state IE �eV� Dyson orbital

X 2B1 �n�22n

1 � 10.63 98.7% �2b1

1 2B2 �n�21n

2 � 14.35 99.8% �1b2

1 2A1 �n�12n

2 � 15.90 99.1% �5a1

TABLE II. Vertical excitation energies for selected valence �V� and Rydberg�R� excited states of formaldehyde, calculated at the EOM-CCSD /6-311�2+ ,2+ �G** level of theory �Ref. 68�.

State Eex �eV�

5 1A1 �V� 10.001 1A2 �V� 3.942 1A2 �R� 8.223 1A2 �R� 9.631 1B1 �R� 7.051 1B2 �V� 9.252 1B2 �R� 10.75

FIG. 4. Relevant MOs of formaldehyde.



spherical waves are calculated using analytical expressions�see below�, and the value of the Dyson orbital in the labo-ratory frame is computed by making the transformation tothe molecular frame, Eq. �A4�, and calculating �d�R��r�from the MO coefficients and analytic expressions of theatomic basis functions.

The position of RklYlm and, consequently, the decompo-sition coefficients �Cklm�2 depend on the origin of the labora-tory coordinate system, which is unabigously defined only inthe spherically symmetric systems. For molecular systems,we chose the center of density of the �left� Dyson orbital asthe origin of the laboratory and molecular frames,

FIG. 5. Dyson orbitals for the 1A2 and 1 1B2 excitedstate ionizations of formaldehyde. The leading elec-tronic configurations are shown for each state. Only theexcited electron MOs are shown for each state; the restof the electrons occupy the same MOs as in the corre-sponding state of the cation.

FIG. 6. Dyson orbitals for the 1 1B1 and 2 1B2 excitedstate ionizations of formaldehyde. The leading elec-tronic configurations are shown for each state. Only theexcited electron MOs are shown for each state; the restof the electrons occupy the same MOs as in the corre-sponding state of the cation.



x = �d�r�x�d�r�dxdydz ,

y = �d�r�y�d�r�dxdydz , �40�

z = �d�r�z�d�r�dxdydz .

In the case of atomic systems, the above �x , y , z� correspondto the atom position. This choice is equivalent to orthogonal-ity condition of the state of ionized electron to the Dysonorbital from the pont of view of Eq. �1�.

The Bessel functions that enter the spherical waves ex-pression, Eq. �17�, are implemented in the GNU SCIENTIFIC

LIBRARY.63 Alternatively, they can be easily calculated usingrecurrence relations.43 At the small intramolecular r distancesused to calculate Dyson orbitals, often kr� l, and theasymptotic formula in Eq. �21� should be used. Our numerictests show that there is no general threshold value �kr / l�min

for using Eq. �21�. For l=3–11, the kr / l threshold decreasesalmost linearly as l increases, and one could fit a functionf�l�=al+b for the calculation of �kr / l�min. However, for pre-cision the exact threshold for each l was used in our calcu-lations. The Coulomb waves are also implemented in theGNU SCIENTIFIC LIBRARY,63 with the l�kr restriction, whichdoes not always hold for our applications, especially whenl=0. Thus, we restricted our implementation of the Coulombwaves to very small k values, as in Eq. �22�. The final �Cklm�2values are stored in a binary file, for subsequent PAD mod-eling, which gives a flexibility of calculating PADs in differ-ent representations.

III. FORMALDEHYDE EXAMPLE

In this section we present the results for ionization offormaldehyde from its ground and electronically excitedstates of different characters. Our goal is to develop qualita-tive understanding of the relation between the electron con-figurations of the neutral molecule and its cation and the

shape of the corresponding Dyson orbital. We consider ion-izing transitions of the following three types: �i� Same initialstate producing different final states; �ii� same final statesobtained from different initial states; �iii� different initial andfinal states. We calculate the probabilities of different l, mangular momentum states of the photoionized electron andrationalize the results by extending the atomic selection rulesto delocalized MOs using Eq. �12�. Detailed data on thecomputed Dyson orbitals are given in EPAPS.64 CorrelatedDyson orbitals for inner and outer valence ionizations of theground-state formaldehyde have been reported and employedto interpret electron momentum and Compton profiles.65–67

A. Dyson orbitals

The frontier MOs of formaldehyde are depicted in Fig.4. The highest occupied orbitals have n�, , and n bondingcharacters, while lowest unoccupied molecular orbital is the* MO. Removing an electron from highest occupied mo-lecular orbital �HOMO� �n ,2b1� leads to the ground state

FIG. 7. Dyson orbitals for the 5 1A1 excited state ion-izations of formaldehyde.

FIG. 8. Probabilities of finding photoelectrons in different l, m states forionization of the 1 1A2 excited state of formaldehyde producing the 1 2B1ground state. The corresponding Dyson orbital is also shown. Ionizationpolarization is along the y axis. Bars clustered around an l value correspondto different m=−1¯ l values.



�X 2B1� of the CH2O+ cation, while ionizations of the lowerMOs give rise to the 1 2B2 and 1 2A1 excited states �see TableI�.

For Hartree-Fock �HF� wave functions and within theKoopmans approximation, the Dyson orbitals for each of theabove ground state ionizations are just the corresponding ca-nonical MOs of CH2O. When correlation is taken into ac-count at the CCSD and EOM-IP-CCSD levels of theory �us-ing the 6-311�2+ ,2+ �G*+ basis�, the Dyson orbitals remainessentially unchanged and retain almost 100% Koopmanscharacter, as summarized in the last column of Table I, whichcontains leading contributions to the corresponding Dysonorbitals in terms of the canonical Hartree-Fock MOs. This isnot surprising for a well-behaved, closed-shell moleculewhose ground-state wave function is dominated by a singleelectronic configuration and the cation states are nondegen-erate. However, for ionization from excited states, in whichmultiple configurations are present, we expect significant dif-ferences with respect to the Koopmans picture.

The electronically excited states of formaldehyde aresummarized in Table II. The lowest valence singlet excitedstate 1 1A2 corresponds to the n→* excitation. Fromabout 7 eV onward, several valence and Rydberg excitedstates appear.68 We consider only one-electron transitions, forwhich the norm of Dyson orbitals is relatively large.

The wave function of the lowest valence excited state

1 1A2 contains two major configurations, corresponding tothe 2b1�n�→3b2�*� and 2b1→5b2 �Rydberg� excitations.The lowest state of the cation, which can be derived fromone-electron ionization of this excited state, is the 1 2B1ground state. The calculated Dyson orbital is shown in Fig. 5and consists of the 3b2 MO �71.4%�, 5b2 �92.4%�, as well assmaller contributions from other b2 MOs. This compositionreflects the relative weights of Slater determinants that formthe initial 1 1A2 excited state �64.0% and 21.1%, respec-tively�, as calculated from the EOM-EE amplitudes.

The next valence excited state is 1 1B2, which corre-sponds to excitations to the same pair of MOs �3b2 and 5b2�,but from the lower 5a1�n�� orbital. Ionization leads to the1 2B2 excited state of CH2O+. The Dyson orbital MO com-position for this 1 1B2→1 2A1 transition is almost the sameas for the above 1 1A2→1 2B1 ionization, despite differentcharacters of the initial and final states. What the two transi-tions have in common is the “wave function” of the leavingelectron: In both cases it is a combination of the 3b2 and 5b2

MOs. These Dyson orbitals are also similar to the 3b2 MObecause 3b2 and 5b2 MOs have approximately the same an-gular distribution and differ only by their radial dependence.For one-electron ionizations, Dyson orbitals can be estimatedsemiquantitatively by considering the ionization of each ofthe electronic configurations of the initial state wave func-tion, without relaxing the remaining occupied orbitals. Thus,

FIG. 9. Probabilities of obtaining different l, m states ofphotoelectrons for ionization of the 1 1A2 excited stateof formaldehyde producing the 1 2B1 ground state, inthe limit of small kinetic energy �0.01 eV�, calculatedusing the freestanding �upper panel� and the Coulomb�lower panel� spherical waves. The correspondingDyson orbital is also shown. Ionization polarization isalong the y axis. Bars clustered around an l value cor-respond to different m=−l¯ l values.



a Koopmans-type picture applies to the ionization of multi-configurational wave functions, which provides useful guide-lines for rationalizing the shapes of complicated Dyson or-bitals for excited state ionization.

The Koopmans-type picture also emerges from consid-ering ionization of Rydberg excited states. The X 1B1 and2 1B2 states consist mainly of excitations to the 9a1 and 7a1

Rydberg MOs from the 2b1�n� and 1b2�� orbitals, respec-tively, as shown in Fig. 6. The Dyson orbitals for these1 1B1→X 2B1 and 2 1B2→1 2B2 ionization transition containapproximately the same combination of a1 MOs due to thesimilarity of the excited electron wave functions.

Let us now analyze the Dyson orbitals for the transitionsbetween the different excited states of the neutral and thesame target state of the cation. The first three 1A2 states offormaldehyde represent excitations from the HOMO to dif-ferent b2 MOs; the resulting cation is in the ground 1 2B1state. The 1 1A2 state corresponds to excitations to the al-ready discussed 3b2 and 5b2 MOs, while the next two statesare the excitations to the plus and, respectively, minus com-binations of 2b2 and 4b2 Rydberg MOs. As seen from Fig. 5,the shapes of the Dyson orbitals for these n 1A2→1 2B1�n=1–3� ionizations are very different indeed. This is easilyrationalized because the orbital levels of the leaving electronare not the same: 3b2−5b2, 2b2+4b2, 2b2−4b2. For the lasttwo Dyson orbitals, the 2b2 :4b2 ratio is directly related tothe relative weights of the corresponding amplitudes in the2 1A2 and 3 1A2 wave functions: 63.7% 2b2+16.2% 4b2, and,respectively, 69.1% 2b2–19.5% 4b2.

The norms of Dyson orbitals, as discussed in Sec. II, arerelated to one-electron character of the transition. In the caseof formaldehyde, the norm of all Dyson orbitals for one elec-tron ionizations is found to be about 0.12. Our calculationsof other small molecules show that the norm of Dyson orbit-als for one-electron ionization is approximately constant fora given molecular species. An interesting case is the 5 1A1excited state of formaldehyde �Fig. 7�, which involves twodifferent types of excitations: 1b2��→3b2�*� and2b1�n�→5b1 �Rydberg�. Thus, ionization could lead to ei-ther the 1 2B2 or X 2B1 cation states. The norms of the

TABLE III. The largest �Cklm�2 coefficients ��10−4� for three formaldehydeionization transitions: �A� 1 1A2�V�→X 2B1, molecular frame, �B� 1 1A2�V�→1 2B1, averaging around y, �C� 2 1A2�R�→1 2B1, molecular frame. Ioniza-tion polarization is along the y axis, perpendicular to the molecular plane,and k=0.271 a.u. �Ek=1 eV�.

l �m�

�Cklm�2

A B C

0 0 7.8 7.1 5.3�101

10 3.7 2.2 2.0�10−3

1 ¯

a 1.1 ¯

0 1.6�10−1 8.3�10−2 1.2�102

2 1 ¯ 2.5�10−3¯

2 1.2�10−1 1.2�10−1 1.8�102

3

0 3.1�10−2 1.9�10−2 3.8�10−2

1 ¯ 1.6�10−3¯

2 2.9�10−2 1.7�10−2 2.3�10−2

3 ¯ 2.5�10−2¯

4

0 6.8�10−2

1 ¯

2 10−5–10−6b 10−5–10−6 4.1�10−2

3 ¯

4 7.7�10−2

5 012 10−6–10−8 10−7–10−8 10−5–10−7

345

6

0 2.2�10−3

1 ¯

2 1.6�10−3

3 10−7–10−8 10−7–10−8¯

4 8.4�10−4

5 ¯

6 3.3�10−3

aZero coefficients, due to symmetry.bFor �Cklm�2�10−4, only the order of magnitude of the highest coefficients isgiven.

FIG. 10. Probabilities of finding photoelectrons in dif-ferent l, m states for ionization of the 1 1A2 excited stateof formaldehyde producing the 1 2B1 ground state. Ion-ization laser is polarized along the y axis, perpendicularto the molecular plane, and cylindrical averagingaround the y axis is conducted. Bars clustered aroundan l value correspond to different m=−l¯ l values.



5 1A1→X 2B2 and 5 1A1→X 2B1 Dyson orbitals are 0.078and, respectively, 0.042, the sum of the values being 0.12.

B. Partial-wave decomposition: The �Cklm�2coefficients

The squares of the Cklm coefficients give the probabilityof obtaining a photoelectron with l, m angular momentum,and wave number k, under the assumption that RklYlm areeigenstates of the ejected electrons �see Sec. II A�. For ion-ization of atoms the allowed states are l= ±1 with respectto the Dyson orbital l quantum number. In the molecularcase, Dyson orbitals have lower symmetry, and the selectionrules are less strict.

Depending on the experimental setup, averaging overmolecular orientations is often necessary, see Eq. �23�. Todemonstrate the effect of the averaging on the Cklm, we con-sider azimuthal averaging around the y molecular axis,which would correspond to an experiment where the mol-ecules are oriented perpendicular to the laser ionization po-larization. Cylindrical averaging is conducted, for example,in Stolow’s experiments on the NO dimer,44,69 which moti-vated the present work. We also calculated the �Cklm�2 coef-ficients for non-averaged CH2O Dyson orbitals, in order tounderstand qualitatively the major l, m contributions.

Consider again ionization of formaldehyde from its low-est valence excited state to the ground state cation, 1 1A2→1 2B1 �Fig. 5�. Figure 8 shows the probabilities of produc-ing different l, m angular momentum states. The highest co-efficients are obtained for l=0, 1, 2, and 3. This is consistentwith the shape of the orbital: This b2 Dyson orbital could bedecomposed approximately into d and py orbitals. By exten-sion of the atomic selection rules, the allowed Ylm stateswould be indeed s, d, and p, f . The coefficients for other lstates are several orders of magnitude lower, although theyare not zero as would be in the atomic case. Thus, by decom-posing the delocalized Dyson orbitals into different same-center orbital contributions, one can rationalize the l, m an-gular momentum states of the photoelectron. However, thisis possible only when Dyson orbitals have relatively simpleshapes.

It is interesting to compare the �Cklm�2 coefficients ob-tained using the freestanding and Coulomb wave representa-

tions of the photoelectron wave function. The l, m decompo-sition for the two cases, in the limit of very small k �0.01 eV�are shown in Fig. 9. For the free-standing spherical waves,there is a large difference among the s, p, and d contributionsdue to the low density of the l�0 waves in the small rDyson orbital region. For the Coulomb waves, the l, m ex-pansion extends to higher angular momentum and the l=0–2 coefficients are within one order of magnitude. Thus,one can expect that ionization of neutral species will producehigher angular momentum photoelectrons than that of an-ions, assuming similar shapes of the corresponding Dysonorbitals.

After cylindrical averaging around the y axis, one ob-tains the average coefficients �Cklm�2 in Fig. 10, using spheri-cal wave representation. The highest coefficients correspondto the l=0–3 angular momentum, but more m states becomepopulated. By rotating the molecular frame, the x ,y ,z pro-jections of the electron angular momentum in the laboratoryframe change.

The highest �Cklm�2 contributions �above 10−4� are givenin Table III. Note that even the coefficients that give themajor contributions differ by a few orders of magnitude. Thisis due to the smaller overlap of Rkl radial functions with highl at smaller intramolecular distances �see Sec. II B�. There-fore, high angular momentum states of the electron becomemore accessible for ionization of the Rydberg states. Forcomparison, Fig. 11 displays the �Cklm�2 decomposition of theb2 Dyson orbital corresponding to the 2 1A2→X 2B1 ioniza-tion. The angular distribution of this Dyson orbital is similarto the previous one, but the increase in number of radialnodes and the higher density at large radius result in largercontributions from high l states, especially 4 and 6. Finally,Fig. 12 compares the �Cklm�2 values computed using the free-standing and the Coulomb waves for this Dyson orbital. Theresulting trend, higher angular momentum states for the Cou-lomb waves, is similar to that observed for the valence1 1A2→X 2B1 ionization.

The data shown in Figs. 8–12 are given in the EPAPSsupplement.64

FIG. 11. Probabilities of finding photoelectrons in dif-ferent l, m states for ionization of the 2 1A2 excited stateof formaldehyde producing the 1 2B1 ground state. Thecorresponding Dyson orbital is also shown. Ionizationpolarization is along the y axis. Bars clustered aroundan l value correspond to different m=−l¯ l values.



IV. CONCLUSIONS

We report the implementation of Dyson orbitals withinthe EOM-CC family of methods. Formally, Dyson orbitalsrepresent the overlap between an N-electron wave functionof the molecule and the N−1 /N+1 electron wave function ofthe corresponding cation/anion, and are required for calculat-ing PADs, Compton profiles, electron momentum spectra,and are relevant for other orbital imaging experiments. In thecase of ionizing transitions, they can be interpreted as statesof the leaving electron and are required for calculating elec-tronic factors of angular distributions of photoelectrons. Cor-related Dyson orbitals are especially important for ionizationof electronically excited and open-shell species. As demon-strated by the formaldehyde example, Dyson orbitals of theground-state closed-shell molecules are well approximatedby the corresponding HF molecular orbitals. For ionizationof species with multiconfigurational wave functions, theshapes of the Dyson orbitals are more complex, however,they still can be rationalized within Koopmans-type picture,i.e., by considering leading EOM amplitudes in the Hartree-Fock MO representation.

A complete PAD modeling requires the wave function ofthe ionized electron to account for interference effects andcore-electron interactions, and will be pursued in a futurework. Under certain approximations, however, the Dyson or-

bitals can be employed in partial-wave decomposition, thatis, for evaluating the probabilities of finding the ionized elec-tron in different angular momentum states. We discusspartial-wave analysis and apply it to formaldehyde. Thewave function of the ionized electron is approximated by anexpansion in partial waves, either freestanding or Coulomb.For relatively simple molecular Dyson orbitals, the �Cklm�2contributions can be rationalized by decomposition of theDyson MO into different l, m same-center orbitals and theextension of the atomic selection rules. Due to low density ofspherical waves at small r, ionization of Rydberg states ismore likely to produce higher angular momentum electrons.Likewise, faster electrons are more likely to have higher an-gular momentum. Anion ionization leading to neutral speciesshould favor lower angular momentum electrons due to thelonger oscillations of the freestanding versus the Coulombspherical waves. Thus, although quantitative simulation ofPADs involves more sophisticated treatment, simple partial-wave decomposition of Dyson orbitals reveals interestingtrends that can be applied to the qualitative interpretation ofexperimental PADs.

ACKNOWLEDGMENTS

Support from the National Science Foundation �GrantNo. CHE-0616271� is gratefully acknowledged. C.M.O. is

FIG. 12. Probabilities of finding photoelectrons in dif-ferent l, m states for ionization of the 2 1A2 excited stateof formaldehyde producing the 1 2B1 ground state, cal-culated using the freestanding �upper panel� and theCoulomb �lower panel� spherical waves, in the limit ofsmall kinetic energy �0.01 eV�. The correspondingDyson orbital is also shown. Ionization polarization isalong the y axis. Bars clustered around an l value cor-respond to different m=−l¯ l values.



Wise Postdoctoral Fellow and acknowledges support fromthe Wise Research Fund �USC�. We thank Professor HannaReisler and Professor David Yarkony for stimulating discus-sions and insightful comments. This work is conducted underauspices of the iOpenShell Center for Computational Studiesof Electronic Structure and Spectroscopy of Open-Shell andElectronically Excited Species supported by the NationalScience Foundation through the CRIF:CRF CHE-0625419�0624602�0625237 grant.

APPENDIX: TRANSFORMATION BETWEEN POLARAND CARTESIAN COORDINATES

Transformation from polar to Cartesian coordinates isgiven by

x = r cos��sin�� , �A1�

y = r sin��sin�� , �A2�

z = r cos�� . �A3�

An arbitrary rotation of a rigid body can be described bythe three sequential transformations, i.e., Euler rotations,70

r� = R��r = Rz��Ry��Rz��r , �A4�

where r denotes the coordinates in a fixed laboratory frameand r�—in a molecular frame rotating with the body. Opera-tors Rxi

�� describe rotation by angle � around laboratoryaxis xi and assume the following simple form:

Rz�� = � cos�� − sin�� 0

− sin�� cos�� 0

0 0 1� , �A5�

Ry�� = � sin�� cos�� 0

0 0 1

cos�� − sin�� 0� . �A6�

The inverse transformation from the molecular to thelaboratory-fixed frame is readily obtained as

R��−1 = Rz

−1��Ry−1��Rz

−1�� = Rz�− ��Ry�− ��Rz�− �� .

�A7�

The ranges for � , �, and � are 0 . . .2, 0 . . . and0. . .2, respectively.

The orientation of a diatomic molecule is uniquely de-fined by one angle describing the orientation of molecularaxis with respect to, for example, z axis, and the correspond-ing Euler transformation is thus Ry��. For nonlinear mol-ecules in coincidence experiments, when one of the molecu-lar axes is fixed, e.g., as in �NO�2 with the NN axiscoinciding with the laboratory axis z, there remains only onedegree of freedom, rotation around z axis, Rz��.

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Dyson orbitals for ionization from the ground and ...iopenshell.usc.edu/pubs/pdf/jcp-127-234106.pdf · Dyson orbitals for ionization from the ground and electronically excited states