J. Chem. Phys. 125, 154106 (2006); https://doi.org/10.1063/1.2360262 125, 154106

© 2006 American Institute of Physics.

Hole-particle characterization of coupled-cluster singles and doubles and relatedmodelsCite as: J. Chem. Phys. 125, 154106 (2006); https://doi.org/10.1063/1.2360262Submitted: 25 July 2006 . Accepted: 12 September 2006 . Published Online: 19 October 2006

A. V. Luzanov, and O. V. Prezhdo

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Hole-particle characterization of coupled-cluster singles and doublesand related models

A. V. Luzanova�

STC “Institute for Single Crystals,” National Academy of Sciences, Kharkov 61001, Ukraine

O. V. Prezhdob�

Department of Chemistry, University of Washington, Seattle, Washington 98195

�Received 25 July 2006; accepted 12 September 2006; published online 19 October 2006�

The hole-particle analysis introduced in the paper �J. Chem. Phys. 124, 224109 �2006�� is fullydescribed and extended for coupled-cluster models of practical importance. Based on operatorrenormalization of the conventional amplitudes tai and tab,ij, we present a simplified method forestimating the hole-particle density matrices for coupled-cluster singles and doubles �CCSD�. Withthis procedure we convert the first-order density matrix of the configuration interaction �CI� singlesand doubles �CISD� model, which lacks size consistency, into an approximately size-consistentexpression. This permits us to correctly estimate specific indices for CCSD, including the hole andparticle occupation numbers for each atom, the total occupation of holes/particles, and theentropylike measure for effective unpaired geminals. Our calculations for simple diatomic andtriatomic systems indicate reasonable agreement with the full CI values. For CCSD and CISD wederive special types of two-center indices, which are similar to the charge transfer analysis ofexcited states previously given within the CIS model. These new quantities, termed charge transfercorrelation indices, reveal the concealed effects of atomic influence on electronic redistribution dueto electron correlation. © 2006 American Institute of Physics. �DOI: 10.1063/1.2360262�

I. INTRODUCTION

The interpretation of the internal structure of multicon-figurational states is an important part of the quantum-chemical machinery, and various methods have been pro-posed for approaching this nontrivial conceptual problem.1–5

In particular, Ref. 6 introduced the measure for multiconfigu-rational nature of states for the simplest configuration inter-action �CI� model, taking into account only single excita-tions, CI singles �CIS�. This measure is simply an averagerank of the corresponding excitation operator and is termedthe collectivity number or collectivity index. Analogs of thecollectivity index for the full CI model �FCI� were presentedin Ref. 7 and independently in Ref. 8. A detailed study maybe found in Ref. 9, in which logarithmic collectivity indices�a measure of entropylike quantities� were regarded as thespecific complexity measures of the many-electronic systemunder study �for fuller details of the notion of “complexity”and complexity measurement see Ref. 10�. The first defini-tion of entropy measures within the coupled-cluster �CC�theory was obtained in Ref. 11. Unfortunately, in practicemost indices of this kind are restricted to essentially the samelimited range of applicability as the FCI method.

A simpler approach which makes use of the first-orderreduced density matrix �1-RDM� is outlined in Ref. 12. Asalient feature of these papers is the definition of the so-called unpaired electron density, which provides a specificmeasure of the departure of a 1-RDM from idempotency �see

also Ref. 13 for some critical points of the approach�. Theanalysis of open-shell states proposed in Ref. 14 also relieson the 1-RDM spectral properties; it was modified to someextent in Ref. 9. Furthemore, an original analysis of electronattachment and detachment densities for CIS excited stateswas proposed in Ref. 15. This work actually generated theCIS hole and particle distributions from 1-RDM.

Quite recently a general description of hole-particle�h-p� distributions for arbitrary multiconfigurational wavefunctions was developed.16 It is based on a purely operatorialtreatment of CI models,17 in conjunction with the principaldefinitions and techniques of the RDM theory.18–21 However,some practically important models such as CI singles anddoubles �CISD� and coupled-cluster singles and doubles�CCSD� models are only briefly considered in Ref. 16 interms of the corresponding h-p indices.

In this paper we provide a more extensive presentationof the hole-particle picture for the CISD and CCSD approxi-mations. The analysis in Ref. 16 is augmented below in twoways. First, we present new entropylike measures that aremore feasible than those in Refs. 8 and 9. Second, we intro-duce special two-center h-p indices that describe interatomiccorrelation interactions. Moreover, the procedure given inRef. 16 for renormalizing the conventional CCSD ampli-tudes is carefully studied and extended in some respects.Adopting a general viewpoint, we extend the terminology ofRef. 22 concerning quadratic CI and regard CCSD as a non-linear size-consistent counterpart of CISD. CISD and CCSDare treated below as related models, with each model deter-mined by appropriate one- and two-electron excitation opera-tors.

a�Electronic mail: luzanov@xray.isc.kharkov.comb�Electronic mail: prezhdo@u.washington.edu

THE JOURNAL OF CHEMICAL PHYSICS 125, 154106 �2006�

0021-9606/2006/125�15�/154106/14/$23.00 © 2006 American Institute of Physics125, 154106-1

The paper is organized as follows. Section II introducesnotations and a representation of the CI wave function interms of appropriate h-p generators. In Sec. III the maindefinitions of h-p indices16 are given for the CISD model.Section IV presents the interatomic h-p indices reminiscentof charge transfer indices from electronic excitationtheory.6,23 A simple approximation involving renormalizedh-p CCSD amplitudes is given in Sec. V, providing a basisfor the computation of h-p indices for CCSD and relatedschemes, such as coupled cluster with double excitations�CCD� and CCD with the Brueckner condition �the so-calledBD scheme�.24 Entropylike measures in the framework ofCCD and CCSD schemes are defined in Sec. VI. Section VIIdescribes a spin-free formulation for singlet states. Some nu-merical demonstrations are given in Sec. VIII at the FCI,CCSD, CISD, and Moller-Plesset second-order �MP2� levelsfor several simple molecular systems, and at the CCSD andMP2 levels for more complex problems. We conclude in Sec.IX. The appendix provides working expressions for calcula-tions on singlet states.

II. CISD WAVE FUNCTION IN TERMS OF h-pGENERATORS

We begin with the conventional representation of theCISD wave function for the given N-electron problem,

��CISD� = C0��� + �i

�a

Cai��i→a�

+ �i�j

�a�b

Cab,ij��i→aj→b

� , �1�

where ��� is a reference one-determinant wave function builtup from the occupied spin orbitals

���i�1�i�N. �2�

The ��i→a�, etc. are obtained from ��� by substituting ��i�with the corresponding vacant spin orbitals taken from thecomplementary spin-orbital basis

���a�N+1�a�dim. �3�

The quantities C0, Cai, Cab,ij are the so-called configurationalcoefficients which represent, in terms of h-p formalism, theappropriate h-p amplitudes.25,26 Within the variational CISDmodel these may be obtained through a familiar technique�for more details see, e.g., Ref. 27�. By neglecting Cai in �1�one obtains the CI doubles �CID� method.

Now let us introduce the matrices composed of the h-pamplitudes above,

C1 = Cai, C2 = Cab,ij . �4�

The corresponding one- and two-electron operators can benaturally associated with Eq. �4�

C1 � C1�1� = �i

�a

Cai��a�1����i�1�� , �5�

C2 � C2�12� = �i�j

�a�b

Cab,ij��ab�12����ij�12�� , �6�

where ��uv�12�� is the Slater determinant comprised of ��u�and ��v�. Thus we make no principal distinction betweenoperators and the corresponding matrices.

This notation permits us to rewrite Eq. �1� via �5� and �6�as follows:

��CISD� = CNCISD��� , �7�

CNCISD = C0I + �

1�k�N

C1�k� + �1�k�l�N

C2�k,l� . �8�

As in Ref. 28, the individual sums in Eq. �8�, e.g.,�1�k�NC1�k�, can be viewed as generators of the excited-configuration superposition of the corresponding multiplicity.For simplicity we extend the same terminology to the initialh-p operators �1� and �2� as well, thus designating C1 and C2

as 1 and 2 generators. It is essential that the correspondingRDMs of arbitrary order k denoted by Dk

CISD are explicitlyrepresented via C1 and C2 or their Hermitian combinationsC1+C1

+ and C2+C2+.16,29,30 By including the high-order h-p

generators C3, C4, etc., in Eqs. �7� and �8�, one arrives at theextended CI models up to FCI.

III. HOLE-PARTICLE DISTRIBUTIONS FOR CISD

At this point we will introduce the key objects of ourstudy in a manner prescribed by Ref. 16. The k-order h-pdistributions for CISD are readily determined by computingthe appropriate projections of Dk

CISD. For example, the corre-sponding hole density matrix D1

h�1��D1h is defined as

D1h = � − �D1

CISD� �9�

�see the general definition �5.7� in Ref. 16�, where � is theDirac-Fock 1-RDM for the reference, that is,

� = �i=1

N

��i���i� . �10�

The particle density matrix �1-RDM for electrons excitedabove the Fermi sea� is computed as the full projection ofD1

CISD onto a vacant spin-orbital space �3�,

D1p = �I − ��D1

CISD�I − �� . �11�

In accordance with Ref. 16 the above h-p RDMs may berecast in a more suitable form,

D1h = C1

+C1 + 2 Tr�2�

C2+�12�C2�12� , �12�

D1p = C1C1

+ + 2 Tr�2�

C2�12�C2+�12� . �13�

The second-order h-p RDMs are computed similarly. Forexample, the CISD mixed hole-particle RDM is written as

D2h-p�1T2� = P12�C1�1�C1

+�2� + 4 Tr�3�

C2�13�C2+�23� , �14�

where P12 is the transposition operator. Note that in Eq. �14�the partially transposed operator is defined by the generalrule

154106-2 A. V. Luzanov and O. V. Prezhdo J. Chem. Phys. 125, 154106 �2006�

Z�1T2���, = Z�12��,� �15�

for arbitrary Z. Using the h-p distributions �12�–�15�, we canderive some useful numerical characteristics �indices� ofelectron correlation. The occupation number of holes/particles arising due to many-electron correlations may bedetermined as follows:

N1h = N1

p = Tr D1h. �16�

Equation �16� can be decomposed into contributions fromindividual atoms through application of the unity operator

I = �A

UA, �17�

where the local atomic projectors UA are given by Refs. 31and 32

UA = ���A

�������� . �18�

Equation �18� employs Lowdin’s orthonormal atomic spinorbitals �����1���dim. It follows that the indices

DAh = Tr D1

hUA, DAp = Tr D1

pUA �19�

are simply the average hole/particle occupation of the givenatom A. Moreover, the total number of holes and particles

Neff�h-p� � N1h + N1

p = 2 Tr D1h �20�

may be used to determine the effective number of unpairedelectrons within the given hole-particle analysis of the corre-lated many-electron functions.

The mixed h-p density matrix D2h-p allows one to esti-

mate the probability of the simultaneous occurrence of holesand particles

N2h-p = Tr

�12�D2

h-p. �21�

This index is also used when calculating a mean h-pdistance,16 approximated by the expression

rh-p = N2h-p/ Tr

�12�g�1,2�D2

h-p, �22�

where g�1,2� is the Coulomb repulsion operator. While ingeneral N2

h-p and rh-p are not size-consistent quantities, theyare quite helpful in practice for describing electron correla-tion in intuitive terms.

IV. CHARGE TRANSFER CORRELATION INDICES

With the estimated h-p RDMs above, only the totalatomic h-p densities �19� may be obtained. We now proposea new kind of correlation index that permits the extraction ofinteratomic hole-particle interactions, termed charge transfercorrelation �CTC� indices. They are somewhat analogous tothe CT indices previously proposed for analyzing the local-ization of electronic excitations at the CIS level.6 Note thatfor the random phase approximation, quantities rather similarto the CT indices were independently introduced, and arewidely used.23,33 Determining a rigorous counterpart to the

CT notion for electron correlation is a rewarding problem,which can be resolved in a natural way through a more at-tentive analysis of h-p densities.

The main difficulty with this task lies in obtaining thehigh-order h-p indices that are normalized to the total hole/particle occupation number N1

h �16�. The situation parallelsthat of multicenter bond index theory.3,32 In the latter, aproper two-electron operator 2

� is introduced � 2� is con-

nected to the so-called exchange RDM due to Ruedenberg34�,along with the usual 1-RDM D1

� �more exactly, its spin-freecomponent�, so that

− Tr�2�

2� = D1

�. �23�

Note that 2� cannot be reduced to a 2-RDM since these are

normalized in a different way than D1� �for details see Ref.

32�. Here we have a similar situation, namely, that we mustconstruct a new mixed h-p 2-RDM different from D2

h-p suchthat it satisfies the reduction equations of type �23�. We de-note this operator by �2

h-p and impose on it the reductionrequirements

Tr�2�

�2h-p�12� = D1

h, �24�

Tr�2�

�2h-p�21� = D1

p, �25�

leading to the size-consistent normalization

Tr�12�

�2h-p�12� = N1

h. �26�

Here we employ the standard definition Z�21�= P12Z�12�P12.The mixed h-p density �14� does not obey Eqs.

�24�–�26�. This is easy to see when performing the relevantcalculations based on the reduction rules for the transpositionoperator P12,

Tr�2�

P12Z�2� = Z�1�, Tr�3�

P13Z�23� = Z�21�, . . . ,etc. �27�

�see Refs. 17 and 32�. The results at the CISD level �12� and�14� are

N1h = Tr

�12�D1

h = C12 + 2C22, �28�

N2h-p = Tr

�12�D2

h-p = C12 + 4C22. �29�

Note that just a slight alteration of coefficients in Eq. �14�will make it satisfy Eqs. �24�–�26�; the �2

h-p we require is ofthe form

�2h-p�1T2� = P12�C1�1�C1

+�2� + 2 Tr�3�

C2�13�C2+�23� . �30�

For the extended CI models additional terms must be added,

P12�3 Tr�34�

C3�134�C2+�234�

+ 4 Tr�345�

C4�1345�C4+�2345� + ¯

�compare with Eqs. �5.18�, �5.4�, and �5.5� for the exact D2h-p

in Ref. 16�. The conversion D2h-p→�2

h-p preserves semidefi-niteness, so �2

h-p�0 in the operator sense.

154106-3 Hole-particle characterization J. Chem. Phys. 125, 154106 �2006�

Now we are ready to introduce the CTC indices for thegiven atoms A and B via diagonal elements of �2

h-p in theatomic orbital �AO� basis. We define them by the quantities

lA→Bh = Tr

�12��2

h-p�12�UA�1�UB�2� , �31�

lA→Bp = Tr

�12��2

h-p�21�UA�1�UB�2� � lB→Ah , �32�

or explicitly

lA→Bh = �

��A���B

�������2h-p������ , �33�

etc. From the symmetry relation �32� it follows that the “par-tial” atomic h-p densities coincide,

lAh � lA→A

h = lAp . �34�

The “gross” atomic h-p indices �19� are constructed from�33� as follows:

DAh = lA

h + �B

�A

lA→Bh , �35�

DAp = lA

p + �B

�A

lA→Bp � lA

h + �B

�A

lB→Ah . �36�

We observe that the nonzero DAh �or DA

p� arise due to thelocalization of holes �particles� on the atom as well as h-ptransfer processes from other atoms. The additional physicalinterpretation of these CTC indices is based on their connec-tion with the main contribution, �DA, to the change inatomic charge caused by electron correlation,

�DA = Tr�1�

�D1h − D1

p�UA,

or explicitly

�DA = �B

�lA→Bh − lB→A

h � = �B

�lA→Bh − lA→B

p � , �37�

where the atomic charge is measured in absolute electroncharge units �e�. The CTC indices �33� give a more intimateh-p analysis than those based solely on DA

h and DAp . For ex-

ample, for diatomics A2 we have only �DA=0 and DAh =DA

p

=N1h /2 as a result of normalization �16�. But with the aid of

lA1→A2

h a hidden hole-particle interaction is observed.

V. SIMPLIFIED COMPUTATIONS FOR CCSD

Now we must address the problem of how to computethe quantities defined above within CCSD and relatedschemes. The CC theory25,35,36 typically furnishes high-quality estimates of the electronic energy even by rathersimple approximations such as CCD and BD.24 But withother physical quantities, and RDMs in particular, the situa-tion is more difficult. While there are several practical tech-niques available—such as perturbation theory,25 the Z-matrixmethod,37,38 or the finite-perturbation scheme,16 etc.—theyare all time-consuming and computationally intensive, espe-cially for computing 2-RDM components. Further, the resultsmay violate the semipositive definiteness of the RDMs ob-

tained; this error cannot be removed in general since anyrealistic CC scheme is not variational. This is why we pre-viously suggested a fairly reasonable approximation for cal-culating the hole-particle RDM components in the frame-work of typical CC models such as CCD, BD, and CCSD.16

Here we present the more detailed consideration of thismethod. �For the sake of simplicity the abbreviated notationCCSD//appr of the corresponding computational scheme16

will not be used here.�

A. A suggestive example of the Slater determinants

First we will motivate the procedure outlined below byanalyzing two determinants within the CI theory: the refer-ence determinant ��� and a different determinant ��� �. Wecan define the latter as the Thouless transformation �see Ref.39�,

��� � = exp �1�k�N

z�k����� , �38�

where z is a certain 1 generator of type �4�. The intermediatenormalization �� ��� �=1 is assumed. Since � is the Dirac-Fock 1-RDM for ���, the corresponding projector �� for thestate �38� with the usual normalization ��� ��� �=1 may becalculated by the elegant McWeeny formula40 �for furtherprogress of such presentations see Refs. 41 and 42�

�� = �� + z�I

I + z+z�� + z+� . �39�

For what follows it is useful to rearrange �39� in the follow-ing manner:

�� = � + zI

I + z+z+

I

I + z+zz+ + z

I

I + z+zz+ − z+z

I

I + z+z.

�40�

Now we turn to the related CI problem by retaining thefirst two members in Ea. �1�, that is, CIS with an admixtureof the reference

��� = C0��� + �i

�a

Cai��i→a�

= �C0I + �1�k�N

C1�k����� . �41�

It is easy to write down the 1-RDM for �41� via C1 �Refs. 16and 30� �see also Ref. 43�,

D1� = � + �C1 + C1

+�C0 + C1C1+ − C1

+C1. �42�

A comparison of �40� and �42� suggests the introduction of aC1-like generator for �38�, which we will denote as C� 1. Wedefine C� 1 as the operator-renormalized generator z,

C� 1 = z�I + z+z�−1/2 �43�

or equivalently

C� 1 = zc0, �44�

where

154106-4 A. V. Luzanov and O. V. Prezhdo J. Chem. Phys. 125, 154106 �2006�

c0 = ��I + z+z�−1/2 �45�

is the operator counterpart of the scalar quantity C0. Hereand elsewhere the standard definition of the arithmetic rootof the positive-definite operator is implied.

With this notation one obtains the CIS-like presentationof Ea. �40� as 1-RDM for the exponential function �38�,

D1�� � �� = � + C� 1c0 + c0C� 1

+ + C� 1C� 1+ − C� 1

+C� 1. �46�

B. Operator-renormalization procedure fortwo-electron problem

We now give a physically plausible argument for apply-ing a similar renormalization procedure to the CC theorytreating it as a nonlinear size-consistent extension of CI ap-proach. In particular, CCSD can be viewed as the irreduciblecounterpart of the CISD approximation.16 Consider a simplebut quite revealing example of the two-electron system in theCCD scheme. �Note that with the Brueckner orbitals, boththe CCD solution and the equivalent CID solution providethe exact result for the two-electron problem�. Since we arenow using the CC methodology, we will replace the varia-tional h-p amplitudes Cai ,Cab,ij, etc. with the correspondingirreducible h-p amplitudes tai , tab,ij, etc. In the case of CCSDthese can be written as follows:

t1 = C1/C0, �47�

t2 = C2/C0 − A2t1�1�t1�2� , �48�

where A2 is the antisymmetrizer

A2 = �I − P12�/2. �49�

The corresponding CID two-electron wave function is anexplicit example of Eqs. �7� and �8�, i.e.,

���12�� = �C0 + C2����12�� , �50�

where C2 is a rank-one operator of the form �see Eq. �6��

C2 = C*��**�12�����12�� . �51�

In �51�, ��**�12�� is the appropriate doubly excited configu-ration; C0 and C* are configurational coefficients obeying thenormalization condition

C02 + C*

2 = 1. �52�

The irreducible operator t2 defined by Eq. �48� now has astructure similar to Eq. �51�

t2 = t*��**�12�����12�� , �53�

where t*=C* /C0. The terms in the normalization �52� be-come

C0 = �1 + t*2�−1/2, �54�

C* = t*�1 + t*2�−1/2, �55�

so �51� is readily rewritten in a purely operatorial form

C2 = t2�I + t2+t2�−1/2. �56�

In �56� we have actually obtained the full two-electron ana-log of the operator-renormalization procedure �43�. Theequivalent representation

C2 = �I + t2t2+�−1/2t2 �57�

is generally less suitable for computations because it requiresmatrix operations in the vacant molecular orbital �MO�space, which is usually more extended than the occupied MOspace.

C. General CCD and CCSD cases

Formula �56� is exact for two-electron problems only.But this closed-operator expression motivates the idea of ex-tending the same expresion to the general N-electron case,which will give a C2 analog associated with the CCDscheme. It is important that the approximation generates onlya special C2 counterpart which has irreducible componentsonly, unlike C2 in the usual variational schemes. Hence, weshall distinguish the usual two-electron variational operatorC2 from an irreducible operator such as �56�, as in the aboveanalysis of two determinants. It is convenient to apply thenotation of Sec. V A and define this irreducible operator as

C� 2 = t2�I + t2+t2�−1/2 �58�

for any t2 produced by the general CCD scheme. Anotherform of this operator is

C� 2 = t2�12�c0�12� , �59�

where the two-electron counterpart of the reference coeffi-cient C0 is given by the operator

c0�12� = �2�I + t2+t2�−1/2, �60�

with the 2-RDM for the reference

�2 = �2�12� = A2��1���2� . �61�

It is essential that �58� automatically leads to the correctasymptotic solution: a decay of the even-numbered electronsystem into two-electron subsystems. Let us consider the so-called singular-value decomposition44 �SVD� for t2. In thiscase, the SVD procedure is performed by solving the eigen-value problem for t2

+t2, giving the eigenvalue set

�� j1�j�N�N−1�/2 �62�

and the associated eigenvectors

��� j�12��1�j�N�N−1�/2. �63�

Along with the geminal set �63� spanned by occupied states,the set of dual �“vacant”� geminals

��� j**�12��1�j�N�N−1�/2 �64�

is computed by the relation

�� j**�12�� = t2�� j�12��/�� j �65�

for all nonzero � j. Then the SVD becomes

154106-5 Hole-particle characterization J. Chem. Phys. 125, 154106 �2006�

t2 = �j

�� j�� j**�12���� j�12�� , �66�

where �� j are identified as the singular values of t2. Evi-dently, for the two-electron problem �50� the only term of set�62� is t*

2��*. Returning to �58�, we can determine the ap-propriate SVD in terms of the quantities defined above,

C� 2�12� = �j

� � j

1 + � j�� j

**�12���� j�12�� . �67�

In the case of molecular decay onto N /2 isolated two-electron subsystems, the geminals �63� and �64� are naturallylocalized in the individual two-electron subsystems, each al-lowing only one nonzero singular value. Therefore, only N /2singular values �� j survive, corresponding to the quantitiesanalogous to t* in our consideration of the single two-electron problem in Sec. V B.

For CCSD we propose a related approximation by recon-sidering the single two-electron problem at the CISD leveland assuming that normally the C1 contribution is suffi-ciently small in comparison to C2. Then, rewriting the exactrelation �48� as

C2 = �t2 + A2t1�1�t1�2�C0, �68�

we automatically produce the C� 2 generator appropriate toCCSD when replacing the scalar C0 by its counterpart c0�12�from Eq. �60�. This leads to

C� 2 = �2c0�12� , �69�

�2 = t2 + A2t1�1�t1�2� , �70�

c0�12� = �2�I + �2+�2�−1/2. �71�

Under the same assumptions, the “almost-irreducible” coun-terpart of C1 is defined as

C� 1�1� = t1�I + 2 Tr�2�

�2+�2�−1/2. �72�

Finally, we formulate a similar renormalization proce-dure for higher-order h-p amplitudes, say, tabc,ijk. In case oft3�123�� t3, we may in fact apply the same renormalizationprocedure �69� as above, setting

C� 3 = �3�c0�12� + c0�13� + c0�23�/3, �73�

�3 = t3 + A3�t1�1�t1�2�t1�3� + 3t2�12�t1�3��A3, �74�

where the three-electron antisymmetrizer A3 ensures theusual antisymmetry. Extending this procedure to higher-order amplitudes is clear. In our approach, when calculatingthe approximate hole-particle RDMs at the CCSD level,these renormalized almost-irreducible generators �69� and�72� should be used directly in Eqs. �12�–�14�. The corre-sponding contribution to RDMs from �73� is calculated viaEqs. �4.10�, �4.14�, and �5.1� in Ref. 16; e.g., to the holedensity �12� one must add the term

3 Tr�23�

C� 3+�123�C� 3�123� . �75�

This technique comes only to the approximately size-consistent estimations for D1

h and D1p �due to small factorized

terms such as t1�1�t1�2� appearing in Eqs. �69�–�72��. Thisdrawback is fully removed when performing the nearlyequivalent �with respect to energy� BD scheme rather thanCCSD. But as our practice shows �Sec. VIII�, even at theCCSD level the drawback is not so significant.

VI. ENTROPYLIKE MEASURES

In this section we consider some entropylike measureswhich generalize the quantities recently given in Ref. 45. Inaddition, mention should be made of papers8,9,11,46 givingother entropy measures. While a detailed discussion of thecomparative merits of different entropylike indices is beyondthe scope of this paper, we want to stress that the quantitiesdescribed previously in Refs. 8, 9, and 11 deal only withsmall molecular systems. They cannot be applied efficientlyto large-scale systems without reformulation and consider-able alteration. In constrast, we present below a straightfor-ward determination of the new entropylike complexity mea-sure in terms of hole-particle amplitudes of realistic CCschemes.

We begin with a simplified analysis of wave functiondissimilarity. For two state vectors ���, ����, dissimilarity iscommonly quantified by deviation of the squared overlapintegral

��������2 �76�

from 1. However, �76� provides multiplicative separabilityrather than additive separability. Taking the logarithm of �76�provides us the correct additive measure.

When treating electron correlation, it makes sense tocompare the reference determinant ��� with the multicon-figurational wave function ��� �see Eq. �1��. We note that�76� is simply �C0�2, and the properly scaled quantity

Neff�C0� = −2

ln 2ln�C0�2 = −

1

ln 2ln�C0�4 �77�

can serve as the sought-after entropylike measure. Fromother consideration this quantity was introduced in Ref. 45. Itis reminiscent of the index9

Neff� = −

2

ln 2ln�Tr X4� , �78�

where X is the operator counterpart of the N-electron singlet-state wave function. The additivity of �77� is ensured bymultiplicative separability of the correct wave functions and,consequently, that of C0. Employing the notation Neff in Eqs.�77� and �78� reflects the ability of the indices to estimate theeffective number of unpaired electrons. For example, in thedissociation of H2, C0→1/�2 and

Neff�C0� → 2. �79�

While the reference coefficient C0 is not given explicitlywithin the CC theory, one can obtain the approximate CCSDvalue using the technique given below.

First consider the CCD approximation. Invoking Eq.�54� for the two-electron problem and accounting for themultiplicative separability of C0, generally one can producethe CCD estimate

154106-6 A. V. Luzanov and O. V. Prezhdo J. Chem. Phys. 125, 154106 �2006�

C0CCD = Det�C� 0�12�� = �

j

�1 + � j�−1/2

=�Det��2�I + t2+t2���−1/2

which is equivalent to the expression

C0CCD = �Det��2�I − C� 2

+C� 2���1/2.

In the case of C0CCSD, inclusion of the similar contribution

from the t1 �47� yields the approximate quantity

C0CCSD = �Det���I − C� 1

+C� 1�� · Det��2�I − C� 2+C� 2���1/2, �80�

which turns out to be quite reasonable �see Sec. VIII�.Equation �80� affords us a simple method for calculating

index �77� for the CCD or CCSD models. But our experienceshows that for real molecular problems the Neff�C0� generallytoo small when compared to the h-p index �20�, althoughthey can serve as rough estimates. Therefore we suggest analternative entropylike invariant, K, and identify it as a com-plexity measure. We require that K vanish for �C0�=1 as wellas for �C0�=0, satisfying the so-called one-hump criterionwhich is typically attributed to complexity measures.10 Sta-tistical entropy and its modifications satisfy the criterion, soone possible way to produce K is to make a use of thesesimilar functions, for instance,

K = −4

ln 2�C0�2 · ln �C0�2, �81�

where the prefactor ensures suitable asymptotic behavior, asin �79�. But index �81� is not appropriate since it lacks addi-tivity. An improvement is easy to achieve by again consider-ing the two-electron solution and restoring a possible generalexpression. For the two-electron problem �see Eq. �54�� wehave

K = −4

ln 2� 1

1 + t*2 ln

1

1 + t*2� = −

4

ln 2� 1

1 + �*ln

1

1 + �*� .

Generally, with each member of the SVD �66� one can asso-ciate the effective geminal

�� j�12� + �� j� j**�12��/�1 + � j ,

so the general expression is

K = −4

ln 2�j

1

1 + � jln

1

1 + � j. �82�

Equation �82� also allows the equivalent representation interms of t2. To emphasize that �82� reveals the complexity ofelectron-pair correlations, we use the notation K2 for ourfinal complexity. This quantity is straightforwardly reducedto the matrix counterpart of Eq. �81�,

K2 � K�t2� =− 4

ln 2Tr�12�

�c0�12�2 ln c0�12�2�

�SVD �66� is used when transforming Eq. �82��. An equiva-lent representation as an explicit functional of t2 is given by

K2 =4

ln 2Tr�12�

�2ln�I + t2

+t2�I + t2

+t2. �83�

The complexity measure defined in �83� satisfies the one-hump criterion in the sense that

�K2�t2→0 = 0, �K2�t2→� = 0. �84�

The full entropy measure for t2 may also be introduced

K2full = −

2

ln 2�j� 1

1 + � jln

1

1 + � j+

� j

1 + � jln

� j

1 + � j� .

�85�

However, being somewhat less convenient in numericalpractice, Eq. �85� gives no clear benefits and will not be usedhere.

It is a simple matter to generalize Eq. �83� and obtain therelevant complexity measures Kk=K�tk� for every irreducibleh-p generator tk, so that a sum of K�tk� over all possible kmight serve as a total complexity measure. In this work,however, we calculate only K2 as physically more importantelectron-pair characteristics allowing clear asymptotic valuesunder molecular dissociation such as in �79� �see Sec. VIII�.Moreover, Eq. �83� can be applied not only to CCD orCCSD, but to FCI or simpler CI approaches such as CISD aswell, since the generator t2 corresponding to the FCI statevector is easy to reproduce by relation �48�. Thus, K2 shouldbe considered as the general electron correlation invariant atthe two-electron level. Note that K2 is size consistent only atthe CC and FCI levels.

VII. SPIN-FREE FORMULATION FOR SINGLETSTATES

Spin-free expressions for singlet states are more suitablefor practical computations. The key formulas are taken fromRefs. 30 and 47 where the spin structure of the generators C2

and t2 is presented. Following, Ref. 30 it is possible to writedown C1 and C2 using the corresponding spin-free amplitudematrices defined as

0C1 = 0Cai,0C2 = 0Cab,ij . �86�

Then

C1 = 0C10�0, C2 = A2

0C2, �87�

where �0 is the spin unity matrix from the standard set ofPauli matrices. Analogously,

t1 = 0t1�0, t2 = A20t2. �88�

Due to the conventional spin-purity requirement which ex-cludes any admixture of high-spin states, the bisymmetrycondition

0Cab,ij = 0Cba,ji,0tab,ij = 0tba,ji �89�

should be satisfied by any correct CISD/CCSD solution forsinglet states.

In separating spin variables the customary technique isused �e.g., see Refs. 20, 21, and 48�. As a result, the h-pdistributions �12� and �13� become

154106-7 Hole-particle characterization J. Chem. Phys. 125, 154106 �2006�

D1h =

1

20D1

h�0, D1p =

1

20D1

p�0, �90�

where the basic spin-free h-p components are defined as

0D1h = 2�0C1

+0C1 + Tr�2�

0C2+�12�0C̃2�12� , �91�

0D1p = 2�0C1

0C1+ + Tr

�2�

0C̃2�12�0C2+�12� , �92�

with 0C̃2�12���2− 0P12�0C2�12�. Similar manipulations lead

to the next spin-free component for the mixed h-p RDMs�14� and �30�,

0D2h-p�1T2� = 2�0P12

0C1�1�0C1+�2�

+ Tr�3�

0C̃2�23��0P12 + 0P13�0C2

+�23� , �93�

0�2h-p�1T2� = 20P12

0C1�1�0C1+�2�

+ Tr�3�

0C̃2�23��0P12 + 0P13�0C2

+�23� . �94�

We then obtain

N2h-p = Tr

�12�

0D2h-p�12�, rh-p = N2

h-p/ Tr�12�

g�1,2�0D2h-p. �95�

The spin-free representation of entropylike measure �83� isachieved similarly,

K2 =4

ln 2Tr�12�

0�̃2

ln�I + 0t2+0t2�

I + 0t2+0t2

, �96�

where 0� is the spinless projector on the occupied MOs ��= 0��0� and 0�̃2��2− 0P12�

0��1�0��2�.

TABLE I. Hole-particle indices �20�–�22�, reference coefficient C0, and correlation energy �corr at variouscorrelation levels. Correlation energy �corr is given for FCI in a.u., and for other schemes in percent relative tothe FCI value. Distances r0 �Eq. �100�� and rh-p �Eq. �22�� are given in angstroms.

System Scheme Neff�h-p� N2h-p rh-p C0 �corr

HF FCI 0.136 0.141 0.64 0.983 −0.132 28�r0=0.52 Å� CCSD 0.132 0.132 0.64 0.987 99%

MP2 0.121 0.121 0.64 0.988 97.3%CISD 0.120 0.120 0.64 0.985 96.0%

HF, 3 Re FCI 1.768 1.72 1.14 0.716 −0.302 67�r0=0.55 Å� CCSD 1.834 1.725 1.14 0.737 96%

MP2 1.106 1.106 1.10 0.882 85%CISD 1.329 1.2590 1.15 0.795 86%

BeH2 FCI 0.124 0.126 1.11 0.984 −0.041 23�r0=1.01 Å� CCSD 0.119 0.118 1.11 0.985 99%

MP2 0.057 0.057 0.11 0.993 71%CISD 0.113 0.113 1.11 0.986 98%

�BeH2�# FCI 1.276 1.321 1.57 0.828 −0.076 64�r0=1.05 Å� CCSD 0.927 0.918 1.48 0.881 95%

MP2 0.126 0.126 1.26 0.985 55%CISD 0.113 0.113 1.11 0.913 89%

CH2 FCI 0.321 0.334 0.96 0.960 −0.090 61�r0=0.80 Å� CCSD 0.282 0.282 0.94 0.965 98%

MP2 0.111 0.111 0.934 0.986 73%CISD 0.232 0.232 0.93 0.970 94%

H2O FCI 0.182 0.191 0.73 0.977 −0.136 67�r0=0.63 Å� CCSD 0.175 0.175 0.72 0.978 99%

MP2 0.151 0.151 0.74 0.981 94%CISD 0.156 0.155 0.72 0.980 95%

�H6�# FCI 0.252 0.262 1.07 0.969 −0.132 28�r0=1.44 Å� CCSD 0.245 0.245 1.05 0.975 99%

MP2 0.134 0.134 1.04 0.984 97%CISD 0.218 0.218 1.05 0.972 96%

TABLE II. Entropylike measure K2 �Eq. �83�� at various correlation levels.

System FCI CCSD BD MP2 CISD

HF 0.19 0.19 0.18 0.14 0.18HF, 3 Re 2.05 1.97 1.85 1.21 1.70BeH2 0.18 0.17 0.17 0.08 0.16�BeH2�# 1.72 1.22 1.24 0.18 0.96CH2 0.45 0.40 0.40 0.16 0.34H2O 0.26 0.25 0.25 0.21 0.23��H6�#� 0.34 0.34 0.35 0.19 0.33

154106-8 A. V. Luzanov and O. V. Prezhdo J. Chem. Phys. 125, 154106 �2006�

Finally, recall that the computation of h-p quantities�91�–�94� at the CCSD level should be performed with renor-malized spin-free generators 0C� 1 and 0C� 2 of the form

0C� 1 = 0t1�0� + Tr�2�

0�2+�12�0�̃2�12��−1/2, �97�

0C� 2 = 0�2�0��1�0��2� + 0�2+�12�0�2�12��−1/2, �98�

with 0�2�12�� 0t2�12�+ 0t1�1�0t1�2� and 0�̃2�12���2− 0p12�

0�2�12�. The working expressions that follow from theformulas above are given in the Appendix.

VIII. APPLICATIONS

We now apply our hole-particle analysis to systems ofdiatomics and polyatomic molecules, using various CImodels and the standard 6-31G basis set. The CISD,CCSD, and FCI algorithms30,47,49 were implemented inMATHEMATICA-5.50 The resulting code, while limited to treat-

ing systems of small complexity, was sufficient to demon-strate the capabilities of the method. All molecular geom-etries were taken from Ref. 51, except for the hydrogenclusters52 and the boron cluster �B3

−�.53

In Table I we compare the h-p analysis of various ap-proximations with that of FCI for generic diatomics andsome transition states �TSs�. The CCSD approximation per-forms particularly well almost in all cases; the main indexNeff�h-p� deviates from the “exact” FCI values by only 5%on average. Even the description of the unstable, semibondedTS for the insertion reaction

Be + H2 → BeH2 �99�

is permissible with CCSD, albeit at a semiquantitative level�see systems 3 and 4 in Table I; molecular geometry is foundin Ref. 9�.

Next, we observe that the index Neff�h-p��N2h-p even at

the FCI level, while in our scheme the identity Neff�h-p�

TABLE III. Hole-particle indices �20� and �22�, entropylike measure K2 �Eq. �83��, and reference coefficient C0

at various correlation levels for some diatomics and atomic clusters. Correlation energy �corr is given for CCSDin a.u., and for MP2 and CISD in percent relative to the CCSD value. Distances r0 �Eq. �100�� and rh-p �Eq.�22�� are given in angstrom.

System Scheme Neff�h-p� K2 rh-p C0 �corr

C2 CCSD 1.272 1.68 1.05 0.851 −0.271 65�r0=0.87� MP2 0.785 1.10 1.04 0.908 99%

CISD 0.733 1.16 1.04 0.901 87%

N2 CCSD 0.320 0.55 0.84 0.964 −0.227 75�r0=0.77� MP2 0.397 0.57 0.85 0.952 105%

CISD 0.309 0.47 0.84 0.960 93%

F2 CCSD 0.375 0.53 0.69 0.960 −0.269 97�r0=0.74� MP2 0.315 0.45 0.69 0.962 99%

CISD 0.305 0.45 0.69 0.961 95%

F2, 2 Re CCSD 1.858 2.05 0.93 0.747 −0.475 57�r0=0.86� MP2 2.484 2.02 0.93 0.776 108%

CISD 1.413 1.77 0.94 0.802 88%

BeO CCSD 0.428 0.51 0.98 0.948 −0.173 81�r0=0.75� MP2 0.335 0.48 0.93 0.959 105%

CISD 0.294 0.43 0.85 0.962 94%

CO CCSD 0.356 0.49 0.85 0.964 −0.208 86�r0=0.77� MP2 0.326 0.47 0.84 0.960 102%

CISD 0.286 0.43 0.85 0.962 94%

B3−�D3h� CCSD 1.039 1.44 1.35 0.887 −0.251 08

�r0=1.20� MP2 0.763 1.08 1.35 0.910 96%CISD 0.613 1.00 1.33 0.919 88%

C3 CCSD 0.613 0.84 108 0.937 −0.243 12�r0=1.09� MP2 0.503 0.72 1.10 0.939 96%

CISD 0.433 0.67 1.07 0.943 91%

O3 CCSD 0.852 1.11 0.86 0.890 −0.441 54�r0=0.97� MP2 0.835 1.17 0.85 0.902 107%

CISD 0.512 0.80 0.83 0.931 88%

�H10�# CCSD 0.445 0.64 1.16 0.657 −0.156 01�r0=1.92� MP2 0.282 0.40 1.24 0.966 80%

CISD 0.352 0.55 1.15 0.955 94%

154106-9 Hole-particle characterization J. Chem. Phys. 125, 154106 �2006�

=N2h-p is warranted only within the CCD and MP2 models.

Moreover, the hole-particle average distance rh-p shows acertain stability with respect to varying CI level. At the sametime though, the rh-p values themselves are sensitive tochanges in the electronic structure due to molecular rear-rangements �e.g., compare diatomics in equilibrium geom-etry and the stretched diatomics in Table I�. This is a markedcontrast to the rather stable behavior of the Hartree-Fockaverage radius, defined in Ref. 16 as

r0 =N�N − 1�

2 � Tr g�1,2��2. �100�

The rough scheme we propose for estimating the referencecoefficient C0 within CCSD generally agrees with FCI within3% �Table I�. Together with the satisfactory results forNeff�h-p� and entropylike index K2 �Table II�, this indicatesthat the renormalization scheme ��69�–�72�� is fairly reliableat the semiquantitative level for molecular problems whereCCSD works �systems that allow only elementary quaside-generate states such as breaking a simple bond�.

Table III shows the results for more complicated systemsthan those in Tables I and II. �Note that the FCI scheme isnot included in Table III; this approach is hardly realized byour codes for the diatomics and atom clusters considered,even at the 6-31G basis set level.� We expect that CCSDshould be quite reliable, and that MP2 will provide a usefulmimicry for CCSD in practice. In Table III we see that as arule MP2 estimates the correlation indices at a semiquantita-tive level, except for dissociative states as in the case ofmolecule F2 at 2 Re. The inability of MP2 to describe bondcleavage accurately is discussed in many works �see, e.g., theRef. 54�. Table III reflects this deficiency in terms of thehole-particle index Neff�h-p�.

A brief comment is desirable concerning the new indexK2 �83�. The latter can also be viewed as the entropylikemeasure for the effectively open-shell electron pairs that oc-cur due to electron correlation. Thus, since the given electronpair of this type is fully open if � j =1 �a la the Heitler-London geminal�, it contributes 2 to the index K2. FromTables II and III we observe that our expectation K2�2 isvalid for all systems with breaking a simple bond �systems 2in Table II and F2 at 2 Re in Table III�. We also note thatthere is essentially no principal difference in behavior be-tween K2 and the main h-p index Neff�h-p�. In particular,Neff�h-p��2 for bond cleavage, even though in practice

K2 � Neff�h-p� . �101�

The latter is quite reasonable for a stable molecule near equi-librium when t2 is small and t1�0. Then from Eqs. �20�,�28�, and �83� it follows that

Neff�h-p� � 4t22,

K2 � 4t22/ln 2 � Neff�h-p�/ln 2.

It is worth mentioning that �101� reflects the trend ob-served in Ref. 9 for the entropylike complexity measure Neff

�

�78� and the Neff�h-p� analog Neffav �Eq. 40 in Ref. 9�. We

suggest again that entropylike measures such as K2 and Neff�

reflect the electronic shell complexity at a greater level of

detail than do one-electron indices Neff�h-p� or Neffav. While

we maintain that the latter indices are more chemically intui-tive than the entropylike measures, K2 does give us an addi-tional viable �unlike Neff

� � method, in terms of effectivelyunpaired geminals, to interpret electron-correlation phenom-ena.

We now consider the h-p analysis in terms of inter-atomic CTC indices �31�; we begin with a diatomic A2, morespecifically, with the hydrogen molecule H2 in a minimal AObasis set. Using the ready expressions �51� and �55� one eas-ily finds in this case that the CTC matrix lA→B

h is of theform

lA→Bh =

t*2

2�1 + t*2��1 1

1 1� =

N1h

4�1 1

1 1� . �102�

As a result, all the CTC indices are equal, and they tend to1/4 when bonds are broken because N1

h→1 for H2 dissocia-tion. This description realizes in its own fashion the conven-tional interpretation of H2 dissociation.

Indeed, as pointed in Sec. IV, the CTC index lA→Bh gives

the redistribution of charge density caused by electron-correlation. Keeping this in mind we can display the CTCindices for diatomics in a pictorial manner with the diagram

In particular, the dissociation of H2 is described by the CTCdiagram given in Fig. 1�a�. The same is true for particle CTCindices lA→B

p �recall identity �32��. In other words, the aboveCTC processes fully annihilate ion structure fluctuations inthe initial MO state, thus leading to the pure covalent wavefunction.

Calculations on more complex systems allow us to un-derstand the typical behavior of CTC indices and comparethese with the simplest pattern �102�. The pertinent resultsare given in Table IV where for comparing with �102� thequantity N1

h /4=Neff�h-p� /8 is listed as well. From Table IVwe see that in the case of homonuclear molecules Eq. �102�is approximately true, thus reflecting a pairwise character ofelectron correlation. In the case of heteronuclear diatomicsCTC indices show also a quite important role of the imbal-ance lA→B

h � lB→Ah for going to the correct dissociation limit.

As a consequence, holes and particles are inhomogenously

FIG. 1. Pictorial diagrams for the charge transfer correlation indices �33�and �34�.

154106-10 A. V. Luzanov and O. V. Prezhdo J. Chem. Phys. 125, 154106 �2006�

distributed over atoms. By the pictorial CTC diagram �Fig.1�b��, we can demonstrate it for the stretched �3 Re� FHmolecule. Similar data for triatomics are presented in TableV. In particular, the CTC diagram for transition state �BeH2�#

is shown in Fig. 1�c�. We observe the CTC asymmetry againas well as a preference for localization of holes and particleson the beryllium atom. Finally, the topographical dissimilar-ity of spatial images for the distributions can be visualizedby three-dimensional �3D� graphics, as shown in Fig. 2 in thecase of dicarbene C3 and ozone O3.

IX. CONCLUDING REMARKS

In this paper we presented a simplified hole-particle de-scription of CCSD and related models. The core of themethod is based on the substitution of the effective RDMs�unknown, in fact, for CCSD� by a special size-consistentcounterpart derived from the CISD RDMs. In doing so, acertain operator-renormalization procedure for h-p genera-tors is used to build up the irreducible analog C� k of thevariational excitation operators Ck from the conventional

TABLE IV. CCSD hole �DAh� and particle �DA

p� atomic occupancies �19� and matrix lA→Bh of charge transfer

correlation indices �33� for some diatomics AB.

AB DAh DA

p DBh DB

p N1h /4 lA→B

h

HF 0.06 0.05 0.01 0.01 0.017 �0.049 0.010

0.005 0.003 �HF, 3 Re 0.63 0.33 0.29 0.59 0.229 �0.234 0.399

0.096 0.190 �C2 0.32 0.32 0.32 0.32 0.159 �0.159 0.159

0.159 0.159 �CO 0.07 0.10 0.11 0.07 0.045 �0.050 0.024

0.056 0.050 �N2 0.10 0.10 0.10 0.10 0.040 �0.053 0.045

0.045 0.053 �F2 0.09 0.09 0.09 0.09 0.047 �0.062 0031

0.031 0.062 �F2, 2 Re 0.46 0.46 0.46 0.46 0.258 �0.249 0.215

0.215 0.249 �

TABLE V. CCSD hole �DAh� and particle �DA

p� atomic occupancies �19� and matrix lA→Bh of charge transfer

correlation indices �33� for some symmetric triatomics AB2.

AB2 DAh DA

p DBh DB

p lA→Bh

BeH2 0.03 0.03 0.02 0.01 �0.016 0.006 0.006

0.009 0.006 0.000

0.009 0.000 0.006 ��BeH2�# 0.26 0.35 0.10 0.06 �0.199 0.031 0.031

0.075 0.024 0.024

0.075 0.024 0.024 �CH2 0.12 0.12 0.01 0.01 �0.099 0.008 0.008

0.008 0.005 0.000

0.008 0.000 0.005 �C3 0.12 0.07 0.09 0.12 �0.029 0.045 0.045

0.019 0.058 0.018

0.019 0.018 0.058 �O3 0.12 0.15 0.16 0.14 �0.055 0.030 0.030

0.048 0.073 0.034

0.048 0.034 0.073 �B3

− 0.17 0.17 0.17 0.17 �0.069 0.052 0.052

0.052 0.069 0.052

0.052 0.052 0.069 �

154106-11 Hole-particle characterization J. Chem. Phys. 125, 154106 �2006�

CCSD generators tk. The procedure is only approximatelysize consistent if t1�0. This deficiency can be remedied eas-ily if Brueckner orbitals are used, which means in fact ana-lyzing the BD scheme24 instead of CCSD and simultaneouslyapplying natural spin orbitals when comparing the resultswith FCI. Specific computations on typical molecular sys-tems and their transition states or bond breaking �both in-volving large correlation effects� show that when analyzingCCSD as described above, the results obtained are in a semi-quantitative agreement with the exact FCI results.

The atomic indices DAh and DA

p �Eq. �19�� are especiallysuitable for examining electronic states, since for each indi-vidual atom they give the average probabilities of holes andparticles arising from electron correlation. With this the totalh-p occupation number Neff�h-p� can be exploited as thequantity associated with the average number of unpairedelectrons. Additional useful information is obtained by cal-culating the charge transfer correlation indices introduced inSec. IV. Pictorial diagrams such as given in Fig. 1 reveal arather complicated picture of mutual atomic influence interms of the virtual correlation charges lA→B

h .In the search for an adequate measure of complexity of

the many-electron state, we arrived at the entropylike mea-sure K2, which specifically reflects the multiconfigurationalnature of the state vector under study. It is important that thisindex can be treated as twice the number of effectively un-paired geminals as well; this means that K2 reflects the open-ness of electron shells, thereby permitting the use of a morereadily available quantity rather than the entropylike measureNcorr

� �78� given previously in Ref. 9We will conclude with a few words regarding possible

applications of the renormalization procedure �69�–�72� tothe computation of two-electron properties other than themixed density D2

h-p�12� �Eq. �14��. A particular combinationof 1-RDM and 2-RDM �operator 2

� in Sec. IV� is defined inRef. 34 and is directly connected to the generalized bondindices and the valency of the atom �see Ref. 3 and 32 and

main references therein�. Another type of index is producedfrom spin components of 2-RDM when studying atomic spincorrelations �Refs. 7, 55, and 56 are but a few contributionsto this rich field of research�. A CCSD-level description of allsuch indices is particularly desired, and we hope a simplecomputational scheme based on Eqs. �69�–�72� may be pro-posed in these cases as well. Our preliminary calculationssupport these expectations.

ACKNOWLEDGMENTS

The authors thank Colleen F. Craig and Oleg. A. Zhikolfor comments on the manuscript. The financial support ofDOE Grant No. DE-FG02-05ER15755 and PRF Award No.41436-AC6 is gratefully appreciated.

APPENDIX: WORKING EXPRESSIONS

We assume that real spin-free amplitudes �86� and �87�are obtained by any existing CISD algorithm such as thosegiven in Refs. 30 and 47 or by CCSD algorithm as in Ref.47. Having at our disposal the initial CCSD amplitudes 0taiand 0tab,ij, we must first calculate the renormalized ampli-tudes �97�,

0C� a,i = �j

0taj0c ji, �A1�

0cij � �ij + �k;a,b

0�̃ab,ik0�ab,jk�−1/2

, �A2�

where 0�̃ab,ij =20�ab,ij −0�ba,ij and 0�ab,ij =

0tab,ij +0tai

0tbj.Analogously, Eq. �98� gives

0C� ab,ij = �k,l

0�ab,kl0ckl,ij , �A3�

FIG. 2. The relief map for the holedistribution �in left� and the particledistribution �in right� in plane of �a� C3

molecule and �b� O3 molecule. The or-dinate presents the �D1

h�r ,r��r=�x,y,0�and �D1

P�r ,r��r=�x,y,0� values.

154106-12 A. V. Luzanov and O. V. Prezhdo J. Chem. Phys. 125, 154106 �2006�

0cij,kl � �ik jl + �a,b

0�ab,ij0�ab,kl�−1/2

. �A4�

For calculating the inverse matrix roots �A2� and �A3�, onemay use the same algorithms as applied for S−1/2. Note thatas in Ref. 16 we do not apply the above renormalization forMP2 scheme, putting 0C� ab,ij =

0tab,ij instead.To obtain spin-free h-p RDMs �91� and �92� at the CISD

level, we compute the matrix elements of the forms

0Dijh = 2��

a

0Cai0Caj + �

k;a,b

0C̃ab,ik0Cab,jk� , �A5�

0Dabp = 2��

i

0Cai0Cbi + �

i,j;c

0C̃ac,ij0Cbc,ij� , �A6�

where 0C̃ab,ij =20Cab,ij −0Cba,ij. The same quantities at the

CCSD level are computed by replacing the elements 0Cai and0Cab,ij in Eqs. �A5� and �A6� with their counterparts �A1� and�A3�.

For the given atom A the gross atomic hole/particle den-sities �19� are

DAh = �

i,j

0Dijh Uij

+, DAp = �

a,b

0Dabp Uab

− , �A7�

where the hole/particle components of the local atomic pro-jector �17� are defined as

Uij+ = �

��A

ci�cj�, Uab− = �

��A

ca�cb�, �A8�

with ci�= �0�i �0��� the linear combination of atomic orbital

�LCAO� coefficient of spinless MO�0�i� with respect to theLowdin AO �0���.

The mixed h-p RDM �93� is determined by the elements

0Dia,jbh-p = 2�0Cai

0Cbj + �k;a,b

�0C̃ac,ik0Cbc,jk

+ 0C̃ac,ki0Cbc,kj�� . �A9�

The modified h-p density �94� and CTC indices �33� arecalculated in the same manner.

The entropylike measure �96� involves the computationof matrix logarithm. To avoid solving the eigenvalue prob-lem of order N2 /4 �difficult for large-scale problems�, onecan proceed as follows. Compute the four-index matrix

�0 = ��a,b

0tab,ij0tab,kl� · �2ik jl + �

a,b

0tab,ij0tab,kl�−1

�A10�

and utilize the iteration process

�k =2k − 1

2k + 1�0

2 · �k−1, �A11�

uk = uk−1 + 2�k−1, u0 = 0, �A12�

so that

limk→�

uk → 0�2 ln�I + 0t2+0t2� . �A13�

This process is simply the direct-matrix realization of thewell-known series,57 Eq. 4.1.29. For typical problems �when

the 0t2 Frobenius matrix norm,44 or its maximal singularvalue, is of order 1�, the iterative process converges after fiveto six iterations with accuracy of 10−7.

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