Analytical energy gradients for local coupled-cluster methodsy

Guntram Rauhut and Hans-Joachim Werner

UniversitaÈt Stuttgart, Institut fuÈr Theoretische Chemie, Pfaffenwaldring 55,

70569 Stuttgart, Germany

Receivved 12th June 2001, Accepted 20th September 2001First published as an Advvance Article on the web

Analytical expressions for evaluating energy gradients for local coupled-cluster wavefunctions are derived, andrelations between the conventional and local coupled-cluster theories are elaborated. In the more general localcase additional terms arise from the geometry dependence of the localization transformation and the

non-orthogonality of the projected atomic orbitals (PAOs) which are used to span the virtual space.Furthermore, if the excitations from a given orbital pair are restricted to subsets (domains) of PAOs, new termsarise from the geometry dependence of these subspaces. The gradient theory is also generalized to the case in

which weakly correlated electron pairs are treated by local second-order M�ller±Plesset theory (LMP2) whilethe contributions of strong pairs are computed at the coupled-cluster level. A number of test calculations arepresented in which optimized equilibrium structures are compared for local and conventional calculations, and

it is concluded that the local approximations hardly a�ect the accuracy.

1. Introduction

During the last decade local electron correlationmethods1±25 havebecome e�cient tools for the treatment of electron correlation inlargemolecules.Most notably, linear scaling of the computationalcost with molecular size has recently been achieved in our groupfor the whole range of closed-shell single reference ab initio

methods, ranging from second-order local M�ller±Plesset per-turbation theory (LMP2),17,18 over local coupled-cluster withsingle anddouble excitations (LCCSD),14,19 up to theperturbativetreatment of triple excitations in LMP4(SDTQ) orLCCSD(T),20,21 and even for iterative variants like LCCSDT-1a=b.22 The implementation of open-shell methods is also inprogress.23 These new methods lead to enormous savings ofcomputation time if applied to larger molecules, and very muchextend the applicability of ab initio quantum chemistry.So far, most of these methods are only applicable to energy

calculations, but not yet for more general molecular proper-ties, which require evaluation of the response of the wave-function with respect to external perturbations. Only for theLMP2 case have energy derivatives been developed,26 andmore recently a pilot implementation of LMP2=GIAOshielding tensors has been described.27 First results obtainedwith these methods are encouraging. The present paper is the®rst step in generalizing these methods to higher levels. We willpresent the general spin-free closed-shell theory for computinganalytical energy gradients for local CCSD and variants likequadratic con®guration interaction (QCISD).28 Since the fulllocal implementation of such methods is very demanding, wehave developed a preliminary gradient program for localQCISD by introducing appropriate projections into a (newlywritten) conventional gradient program. Even though thismethod does not yet exploit all advantages of the local trun-cations to achieve low-order scaling, it already leads to sig-ni®cant savings and allows one to access the feasibility andaccuracy of geometry optimizations using higher-level localcorrelation methods.

The e�ciency and low-order scaling properties of local cor-relation methods rely on two approximations: ®rstly, the exci-tations are restricted to (non-orthogonal) correlation functionsin the spatial vicinity of the occupied localized orbitals (LMOs)fromwhich the electrons are excited. Thus, for each electron paira speci®c correlation space, which is independent of the mole-cular size, is used. Secondly, a hierarchical treatment of di�erentelectron pairs is introduced,14,16,19 in which the level ofapproximation depends on the distance of the two correlatedLMOs. For this purpose, to each LMO a subset of atoms isassigned according to a criterion proposed by Boughton andPulay.7 The classi®cation of electron pairs depends on theminimum distance of any pair of atoms associated to the twoLMOs. The usual classi®cation distinguishes strong pairs, weakpairs, and very distant pairs. The strong pairs share at least oneatom and typically account for more than 90% of the valenceshell correlation energy for a given basis set. These pairs aretreated at the highest level, e.g., LCCSD or LCCSD(T). On theother hand, very distant pairs, in which the closest distanceexceeds about 15 a0 , canbe entirely neglectedwithout signi®canterror. The correlation energy of the remaining weak pairs iscomputed at the LMP2 level. The treatment of di�erent electronpairs at di�erent correlation levels requires some modi®cationsof the gradient theory, which will be discussed in the presentwork. Furthermore the e�ect of theweak pair approximation onthe accuracy of computed equilibrium structures will benumerically investigated.In section 2 we brie¯y outline conventional coupled-cluster

energy gradient theory in general terms. In order to be con-sistent with our previous contributions to local correlationtheory14,26 we used the same form of presentation althoughother formalisms may be more concise in some aspects.29

Section 3 describes the di�erences between the conventionaland local theories and introduces additional terms that ariseonly in the local case. In the local case a set of coupled per-turbed localization (CPL) equations has to be solved, in orderto account for the geometry dependence of the localizedorbitals. In section 4 the modi®cations due to the weak pairtreatment at the LMP2 level are discussed. Finally, testcalculations for all methods are presented in section 5.

y Electronic Supplementary Information available. See http:==www.rsc.org=suppdata=cp=b1=b105126c=

DOI: 10.1039/b105126c Phys. Chem. Chem. Phys., 2001, 3, 4853±4862 4853

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2. Canonical coupled-cluster gradient theory

2.1. General gradient expression

This section outlines the basic concepts of coupled-clustergradient theory30,31 in order to introduce those quantitieswhich are necessary for understanding the following sections.Throughout this paper, indices i, j, k, and l refer to occupiedmolecular orbitals (MOs), a, b, c, and d denote virtual (cano-nical) orbitals, and r, s, t, and u describe any orbitals. Greekindices m, n, . . . denote atomic orbitals (AOs). For the sake ofsimplicity, we will consider only double excitations, but singleexcitations will be included in the ®nal working equations,which are given in the appendices. The coupled-cluster doubles(CCD) correlation energy, can be written as

Ecorr �Xij

�K ijT ji �

; �1�

where h i denotes the trace of a matrix, and T ijab�T ji

ba are thedouble excitation amplitudes. The two-electron exchange inte-grals over two occupied and two virtual orbitals are collectedin the exchange matrices

K ijab � �aijbj� and �K ij

ab � 2 �aijbj� ÿ �ajjbi�: �2�The amplitudes can be determined by solving the (non-linear)CC equations.

�V ij�T��ab � h �Fabij jHÿ EjCCC�T�i � 0 for all i5 j; a; b:

�3�The CC amplitudes T� {T ij} are optimized iteratively untilthe residual matrices V ij vanish. The explicit form of the resi-duals in matrix notation has been given elsewhere32 and isnot repeated here.The energy gradient depends on derivatives of the integrals

in the AO basis, derivatives of the amplitudes, T ij; q, andderivatives of the MO coe�cients, C q. The superscript q

denotes a derivative with respect to the nuclear coordinate q.The amplitude derivatives T ij ;q can be formally obtained fromthe condition that the residuals vanish at any geometry,i.e., their derivatives must be zero. This yields the coupledperturbed coupled-cluster (CP-CC) equations31

V ij ;q�T q� � G ij ;q �Xkl

h ij ;klT kl ;q � 0; �4�

where hij; klab; cd� (qV ij

ab=qTklcd)0 are the ®rst derivatives of

the residuals V ijab with respect to the amplitudes. Here, we

considerh ij ;kl to be a supermatrix, and the product with T kl ;q

is de®ned as (h ij ;klT kl ;q)ab�P

cdhij ;klab ;cdT

kl ;qcd . Of course, in

practice h is very sparse and never computed explicitly. Thematrices Gij,q comprise all terms that are independent ofthe amplitude derivatives. They depend on the derivatives ofthe orbitals and AO integrals.The energy gradient can formally be written as

Eq �Xij

�K ijT ji ;q �� T ij �K ji ;q

�� �: �5�

The explicit dependence of the energy gradient on the ampli-tude derivatives T ij; q can be eliminated by substituting theformal solutions T q� ÿhÿ1G q of the CP-CC eqn. (4) intoeqn. (5). This yields after some rearrangements

Eq �Xij

ZijG ji ;q �� T ij �K ji ;q

�� � �6�

where Zij are the solutions of the coupled-cluster Z-vector(CC-Z) equations33

R ij�Z� � �K ij �Xkl

hkl;ijyZ kl � 0: �7�

Thus, only one set of linear equations has to be solved, inde-pendent of the number of degrees of freedom q. At this pointit should be noted that for variational linear configurationexpansions (CISD) as well as for MP2 the residuals eqn. (3)can be expressed as

V ij�T� � K ij �Xkl

hij ;klT kl � 0: �8�

Since in these cases the matrixh ij ;kl is symmetric, eqn. (7) and(8) are equivalent, and eqn. (7) is fulfilled with Zij� 2T ijÿT ji,so that they need not to be solved iteratively.As mentioned above, the matrices Gij,q and K� ij,q in eqn. (6)

depend on the AO integral derivatives and the derivatives ofthe MO coe�cients. Separating both contributions yields

Eqcorr �

Xij

Z ijG ji ;�q� �� T ij �K ji ;�q� �� �� CqyY � �9�

where the superscript (q) indicates that the quantity dependsonly on derivatives with respect to the AO integrals. The pre-factors of the orbital derivatives C q

mr are collected in the matrixYmr . The treatment of the terms involving the orbital deriva-tives C q will be discussed in the next section.

2.2. Contributions of orbital derivatives

It is convenient to express the derivatives of the canonicalorbitals in the basis of MO vectors C�C(0) at the referencegeometry (q� 0), i.e.,

C�q� � C�0�U�q�; C q � CU q: �10�By di�erentiating the orthonormality condition CySC� 1 itfollows that U q can be written as

U q � R q ÿ 12CyS qC �11�

with R q being an antisymmetric matrix.26 Since the SCFenergy is invariant with respect to rotations among occupiedor virtual orbitals, the elements R q

ij and R qab are redundant and

can be taken to be zero. The non-redundant elements Rqai are

collected in a vector r q, which can be obtained by solving thecoupled-perturbed Hartree±Fock (CPHF) equations

r q � ÿH ÿ1 f �q� � Bs qÿ �

: �12�These equations can be derived from the requirements that theBrillouin condition is ful®lled for all geometries, i.e., f

qai� 0.

The vector f (q) contains the Fock matrix elements fai(q) eval-

uated with the integral derivatives h qmn and (mnjrs)q, Bai, mn are

the prefactors of the overlap derivatives S qmn which are col-

lected in the vector s q, and Hai, bj� (q2ESCF=qRaiqRbj)0 is the(symmetric) Hessian matrix.Using eqn. (10), (11), and (12) the gradient expression (9)

can be rearranged to

Eq �Xij

Z jiG ij ;�q� �� T ji �K ij ;�q� �� �� zyf �q� � xy � zyBÿ �

s q;

�13�where z is the solution of the Hartree±Fock z-vector equations

y�Hz � 0 �14�with

xmn � ÿ 12 �C �YCy�mn; �15�

yai � �Yÿ �Yyÿ �

ai; �16�

�Y � CyY: �17�

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Since the quantities f (q), K� ji, (q), and Gij, (q) depend only on thederivatives of the AO-integrals, the gradient expression canfinally be rewritten in the AO basis and evaluated in an inte-gral direct way (cf. section 3.3).

3. Local coupled-cluster gradient theory

3.1. Transformation properties of amplitudes and residuals

In most local correlation methods1±23 the occupied canonicalorbitals are transformed into orthogonal localized orbitals(LMOs)

jfloci i �

Xm

jwmiLmi

L � CoW with W yW � 1;

�18�

where Co and L represent the occupied canonical andlocalized orbitals, respectively, in the AO basis {wm}. In thecase that uncorrelated core orbitals are present, only thevalence space is localized, in order to avoid arti®cal mixingsof core and valence orbitals which a�ect the correlationenergy. In the current work the unitary transformation Whas been obtained by the method of Pipek and Mezey,34

since this is particulary e�cient and simple in the context ofcomputing energy gradients (cf. Section 3.3). However, inprinciple other localization procedures, e.g. the Boys locali-zation,35 can also be used. The localization a�ects neitherthe SCF wavefunction nor the general formalism of the localcorrelation methods. Only the explicit form of the coupledperturbed localization equations depends on the localizationcondition (see below).In conventional calculations the virtual orbital space, into

which excitations are made, is usually spanned by the ortho-normal canonical orbitals Cv . Due to the orthonormalityrestriction these orbitals cannot be properly localized. In orderto exploit the local character of electron correlation it istherefore necessary to use non-orthogonal functions. In prin-ciple, any set of functions {fr} that can be represented in theAO basis by a matrix A

jfri �Xm

jwmiAmr �19�

can be chosen. In order to guarantee strong orthogonalitybetween the virtual and occupied spaces these functions mustbe projected against the occupied orbitals yielding projected

functions j ~fri

j ~fri � 1ÿXi

jfiihfij !

jfri �Xm

jwmi ~Pmr : �20�

In theAObasis the projected coe�cientmatrix takes the form

~P � PA where P � 1ÿ LLyS � CvCyvS �21�

is the projection matrix. Here and in the following the tilde (�)indicates quantities in the projected basis. The projected func-tions are in general non-orthogonal among themselves, butstrongly orthogonal to the occupied space, i.e., ~PySL� 0.One particular choice of A is the unit matrix, i.e., the indi-

vidual basis functions are projected. This is the case in ourpresent program. The resulting projected atomic orbitals

(PAOs) are then represented by the projector itself, i.e., ~P�P.Another choice would be to use atomic SCF or natural orbitalsas a basis; in this case A would be block-diagonal, where eachblock represents the orbitals of one atom.The PAOs de®ned in this way are neither normalized nor

orthogonal. For numerical reasons it may be convenient to nor-malize them; furthermore, it is sometimes useful to reorder thePAOs, which corresponds to a permutation of the columns of A.

In the local correlation methods one exploits the fact that innon-metallic systems with a su�ciently large HOMO±LUMOgap the PAOs are inherently local. As outlined in the Intro-duction, this allows one to introduce two di�erent approx-imations: ®rstly, excitations from a given localized orbital pair(ij) are only allowed into an associated subset of PAOs, whichis denoted as [ij]. These subsets are called pair domains, and arerepresented in the AO basis by the rectangular coe�cientmatrices ~P�ij�mr , r 2 [ij]. Consequently, only the amplitudes ~T ij

rs

with rs 2 [ij] are nonzero. Secondly, the small correlationcontributions of weak pairs (ij) may be treated at a lower level,e.g., MP2. This means that only a subset of strong pairs(ij) 2 {Ps}, where {Ps} is a prede®ned pair list, needs to beincluded in the coupled-cluster calculation. Since the numberof strong pairs increases only linearly with the molecular size,and since the number of PAOs in each pair domain [ij] isindependent of the molecular size,14,17,19 the total number ofamplitudes and corresponding residual equations scales onlylinearly. This forms the basis for local correlation methodswith linear cost scaling, as recently developed for the wholerange of single reference correlation methods.17±23

In general, any choice of projected functions ~P can beexpressed in the basis of orthonormal virtual MOs Cv , i.e.

~P � CvQ with Q � CyvSA: �22�The transformation matrix Q is rectangular (Nvirt�NAO). Theoverlap matrix of the projected functions can be expressed interms of Q

~S � ~PyS ~P � QyQ: �23�Due to the projection ~S has Nocc zero eigenvalues. Later theright inverse transformation

�Q � A ÿ1Cv �24�with the property QQ� � 1v will be needed. Since we use PAOs,A is diagonal so that its inversion is trivial. For simplicity inderiving the gradient expressions, we will furthermore assumethat A is geometry independent. This is of course the case ifpure PAOs are used, i.e., A� 1.It is now straightforward to derive the CCSD equations on

the basis of PAOs from the canonical equations by simpletransformations (the resulting equations are valid for any setof projected orbitals, but for convenience we denote the non-orthogonal functions PAOs). Provided that all pairs i5 j areincluded in the calculation, the form of the CCSD equationsas given in ref. 32 is independent of unitary transformationsamongst the occupied orbitals (but note that this is not the casefor a truncated pair list). As will be shown in the following, thetransformation of the canonical virtual orbitals (MOs) to thenon-orthogonal PAO basis leads merely to the occurrence ofthe overlap matrix of the PAOs in the LCCSD equations.As has been discussed in more detail earlier,14 the residuals

and amplitudes transform as

~V ij � QyV ijQ and T ij � Q ~T ijQy; �25�respectively. As an example, consider the transformation of theMP2 residuals in the orbital invariant form,1 which will beneeded in section 4

V ijMP2 � K ij � FT ij � T ijFÿ

Xk

FikTkj � T ikFkj

ÿ �: �26�

Using eqn. (23)±(25) one obtains in the PAO basis

~V ijLMP2 � ~K ij � ~F ~T ij ~S� ~S ~T ij ~Fÿ

Xk

~S Fik~T kj � ~T ikFkj

� �~S:

�27�The local CCSD residual equations14 can be derived in anentirely analogous way.

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The gradient theory for local correlation methods is sig-ni®cantly complicated by the fact that excitations are restrictedto the domains [ij]. In this case the amplitudes ~T ij

rs are deter-mined by the conditions

~V ijrs � �QyV ijQ�rs � 0 for r; s 2 �ij�; ij 2 fPsg: �28�

As a consequence of the fact that a di�erent subspace(domain) [ij] of PAOs is used for each pair, and a sparse pairlist may be used, the correlation energy is no longer invariantwith respect to the unitary localization matrix W, andtherefore the geometry dependence of W must be taken intoaccount. This will be considered in section 3.3. Furthermore,the orbital subspaces spanned by the domains are geometrydependent, and, as will be shown below, this gives rise toadditonal terms in the gradient expression which only vanishif the domains span the full virtual space. In order to eluci-date these problems, we will derive the local gradientexpression from the canonical equations using similar trans-formations to those discussed above.

3.2. General gradient expression

The local correlation energy, ~Ecorr , can be obtained byinserting eqn. (25) into eqn. (1).

~Ecorr �Xij2fPsg

�K ijQ ~T jiQyD E

�Xij2fPsg

~�K ij ~T jiD E

�29�

Di�erentiating with respect to a nuclear coordinate q yields

~E qcorr �

Xij2fPsg

Q ~T ijQy �K ji ;q �� ~�K ij ~T ji ;q

D E� 2 Qqy �K ijQ ~T ji

D Eh i:

�30�Note that only quantities indicated by the tilde are in the PAObasis, while at this stage some matrices are still in the canonicalMO basis. The amplitude derivatives ~T ji,q are formally deter-mined by the coupled perturbed local coupled-cluster(CP-LCC) equations.

~Vij ;qrs � �QyV ijQ�qrs � QqyV ijQ�QyV ijQ q �QyV ij ;qQ

ÿ �rs� 0

for rs 2 �ij�; ij 2 fPsg: �31�Substituting eqn. (4) for V ij,q with T ij ;q� (Qy ~T ijQ)q, and col-lecting all prefactors of the derivatives Q q in a matrix Xv oneobtains after some straightforward algebra

~Eqcorr �

Xij2fPsg

Q ~T ijQy �K ji ;qD E

� Q ~Z ijQyG ji ;qD Eh i

� Q qXv

�; �32�

where the matrices ~Z ij are obtained by solving the LCC-Zequations in the PAO basis

~Rijrs � QyR ijQ

� �rs� ~�K ij �

Xkl2fPsg

~hkl;ijy ~Z kl

" #rs

� 0

for rs 2 �ij�; ij 2 fPsg; �33�with ~h ij ;kl

rs;tu�P

abcdQarQbsQctQduhij ;klab ;cd . Explicit expressions

for the residuals ~Rij in a computationally convenient formare given in Appendix A for LQCISD. The matrices ~Z ij canbe transformed into the MO basis similar to the amplitudes,see eqn. (25). The matrix Xv in the last term of eqn. (32) is spe-cific to the local theory. It is defined as

Xv �Xij2fPsg

~Z ij ~V ji � ~T ij ~R ji� �

�Q � ~X �Q; �34�

where we have used QQ� � 1v (cf. eqn. (24)). The subscript v

indicates that Xv is a rectangular (NAO�Nvirt) matrixin which the columns correspond to virtual orbitals.

~V ij� ~V ij( ~T ) and ~R ij� ~R ij( ~Z ) are the residuals of the LCCand LCC-Z equations in the projected basis, eqn. (25) and(33). The residuals vanish only in the domain [ij] [cf. eqn.(28) and (33)], and therefore Xv is nonzero unless all domainsspan the whole virtual space, as is the case in conventionalcalculations. From eqn. (34) it can be seen that the residuals~V ij and ~R ij are multiplied from the left by the local quantites~T ji and ~Z ji, and therefore only the rows corresponding to thedomain [ij] are needed. However, there is no such projectionfrom the right, and consequently all columns are required ineach matrix. Hence, the number of residual elements to beevaluated scales formally with o(n2), where n is a measureof the molecular size. Therefore, linear scaling cannot beachieved as straightforwardly as for coupled-cluster energy cal-culations.19

As in the canonical case, we can ®nally separate the con-tributions of AO-integral derivatives and orbital derivatives tothe matrices ~�K ij,(q) and ~G ij,(q). This yields

~E qcorr �

Xij2fPsg

~Z ij ~G ji ;�q�D E

� ~T ij ~�K ji ;�q�D Eh i

� S qAXvCyv

�� C qyv SAXv � Yv� � �� LqyYo

�; �35�

where the subscript o and v indicate the blocks correspondingto localized occupied and canonical virtual orbitals, respec-tively. The quantities

~�K ij ;�q� � Qy �K ij ;�q�Q and ~Gij ;�q� � QyG ij ;�q�Q �36�depend only on the derivatives of the AO integrals hqmn and(mnjrs)q. The treatment of the orbital derivatives is describedin the next section.

3.3. Coupled perturbed localization

In the conventional CCSD case the energy is invariant withrespect to rotations amoung occupied orbitals, and conse-quently yij� ( �Yijÿ �Yji)� 0. As already mentioned in section3.1, this is no longer the case if the excitations are restricted topair-speci®c domains of PAOs or if only a subset of pairs isconsidered. One must therefore take into account the geometrydependence of the localization matrix W, which we express asW(q)�W(0)V(q). Since the localization matrix W is unitary atall geometries, V(q) is unitary as well, and therefore V q must beantisymmetric. The derivatives of the coe�cients of the loca-lized orbitals can be expressed as26

L q � CU qW� LV q: �37�Considering again the elements of the matrices S q

mv , Rqai , and

Vqij , (i > j ) as elements of vectors s q, R q, and V q, respectively,

and substituting the expressions for C qv and L q into the gra-

dient expression (35) yields

~E qcorr �

Xij2fPsg

~T ij ~�K ji ;�q�D E

� ~Z ij ~G ji ;�q�D Eh i

� ~xysq � ~yyr q � ~wyvq;

�38�with

~xmn � 12

ÿ�XvCyv ÿ �C �Y �Cy

�mn �39�

~yai � �Yÿ �Yy ÿ �XyvSLÿ �

Wy� �

ai�40�

~wij � � �Yÿ �Yy�ij; �41�where C� � (L jCv), Y� �C� yY, and X� v�AXv . These equationsare very similar to the corresponding eqn. (15) and (16). Theadditional contributions in the local case are the terms invol-ving X� v and ~wij .

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In order to exploit the local character, the external blocksof the quantities �Xv and Y� can be expressed by the corre-sponding quantities in the PAO basis ~X (cf. eqn. (34)) and ~Y(cf. Appendix B), respectively. The matrices ~X and ~Y areneeded in the full PAO space and are not restricted todomains. As mentioned before, it might therefore be di�cultto achieve linear scaling even in a full local implementation.This will be further investigated in future work.The elements v

qij �i > j� are the solutions of the coupled

perturbed localization (CPL) equations, which follow fromthe condition that the localization criterion is ful®lled at anygeometry. For the Pipek±Mezey localization the CPL equa-tions can be written as26

0 �as q �br q � cvq: �42�Explicit expressions for the matricesaij; uv , bij; ai and cij; kl canbe found in ref. 26. Note that the form of these equations isindependent of the correlation method used, but speci®c forthe type of localization. Inserting r q from eqn. (12) yields forv q the formal solution

vq � ÿcÿ1 as q ÿbH ÿ1 f �q� � Bsqÿ �� �

: �43�After inserting eqn. (12) and (43) into eqn. (38) the gradientexpression can be rearranged to

~E qcorr �

Xij2fPsg

~T ij ~�K ji ;�q�D E

� ~Z ij ~G ji ;�q�D Eh i

� ~xy � uya� zyBÿ �

s q � zyf �q�; �44�where

~w� cu � 0; �45�

~y�byu�Hz � 0: �46�Hence, only one set of linear equations of each type has to besolved, independent of the number of degrees of freedom.First, the localization z-vector equations (45) are solved for u,which is then used on the lhs of the HF-Z equations (46) to besolved for z. If the domains for all pairs span the whole virtualspace, Xv � 0, as explained earlier. If in addition all pairs (for agiven orbital subspace) are used, the energy is invariant withrespect to unitary transformations of these orbitals, andtherefore w� 0 and u� 0. In this case the gradient expressionreduces to the one given in section 2.2. In the case thatuncorrelated core orbitals are present, additional terms areneeded to determine the z-vector elements zic corresponding tocore-valence orbital rotations Ric . Details can be found inref. 26.Since the quantities f �q�, ~�K ji ;�q� anf �G ij ;�q� depend only on the

derivatives of the AO-integrals, the gradient expression can®nally be rewritten in the AO basis as

~E q �Xmn

d�1�mn hqmn �

Xmnrs

d�2�mn; rs�mnjrs�q �Xmn

lmnSqmn �47�

The e�ective density matrices d�1�mn , d�2�mn; rs as well as lmv depend

on the solutions of the z-vector equations and the amplitudes.These quantities can be generated on the ¯y and the gradientexpression can be evaluated in an integral-direct mannerwithout storage of the integral derivatives. The Hartree±Fockcontribution to the energy gradient can be included simply byadding the appropriate terms to d(1) and d(2) and l. Theexplicit expressions for these quantities, given in Appendix D,include both the LQCISD and the Hartree±Fock contributions.

4. Gradient of the LMP2 weak pair energy

Section 3 describes the additional terms arising in local gra-dient theory due to the restriction of the excitations to pair

domains, which is one of the essential approximations made inlocal correlation theories. As explained in the Introduction,further computational savings can be gained by a hierarchicaltreatment of the pair correlations, depending on the distanceof the correlated electrons in the orbitals i and j. A mixedscheme, in which strong pairs are computed at the LCCSDlevel, while weak pairs are treated at the LMP2 level (which isdenoted LCCSDjLMP2) has recently successfully been usedfor calculating reaction energies,36 and it has been found thatthe overall accuracy is hardly a�ected by the weak pairapproximation.In the LCCSDjLMP2 method the weak pair correlation

energy is computed prior to the LCCSD iterations, by per-forming a LMP2 calculation for all (strong�weak) pairs andsubsequently gathering the weak pair energies. Within theLCCSD iterations couplings between strong and weak pairsare entirely neglected, i.e., the LCCSD equations are solved forthe strong pairs ij2 {Ps} only. Consequently, the LCCSDjLMP2 gradient is simply a sum of the CCSD gradient asoutlined in the previous sections, and the LMP2 gradient forthe weak pairs. In this section we brie¯y discuss the evaluationof the latter contribution. A minor complication arises fromthe fact that the weak pair energy does not minimize theHylleraas functional,37 as is the case for the total LMP2energy. Therefore, the weak pair energy is not stationary withrespect to small variations of the pair amplitudes, and conse-quently a set of LMP2-Z equations has to be solved.The weak pair energy is de®ned as

~Eweak �X

ij2fPwg

~�K ij ~T jiLMP2

D E; �48�

where fPwg is the list of weak pairs. The amplitudes ~T jiLMP2

are determined from the condition that the LMP2 residuals,eqn. (27), vanish for all pairs. (In the following, we will omitthe subscript LMP2 with the understanding that all quan-tities in the current section refer to LMP2.) Consequently,the amplitude derivatives are determined by the coupled-per-turbed LMP2 equations for all pairs. Applying the sameprocedure as in sections 2 and 3 the energy gradient canbe expressed as

~Eqweak �

Xij2fPwg

�Kij ;qQy ~T jiQD E

� Qy ~Z ijQG ji ;qD Eh i

� Q qXv

�;

�49�where the residuals Rij and V ij in Xv (eqn. 34) are defined byeqn. (7) and (8), respectively, evaluated with projected LMP2amplitudes T ij�Qy ~T ijQ and Z-matrices Z ij�Qy ~Z ijQ. Ofcourse, the quantities Gij ;q and hij,kl appropriate for MP2have to be used here. The Z-vector is obtained by solving thelinear equations

~Rijrs � lij

~�K ij � ~F ~Z ij ~S� ~S ~Z ij ~FÿXk

~S Fik~Z kj � ~Z ikFkj

� �~S

" #rs

� 0 for rs 2 �ij�; ij 2 fPsg � fPwg: �50�

These equations di�er from the LMP2 residual eqn. (27) by

replacing ~K ij by lij~�K ij, where the factor lij is one if ij is a weak

pair and zero if ij is a strong pair. Note that solution ofeqn. (50) yields the matrices �Zij for all (strong�weak) pairs,but only the ones for the weak pairs are needed in eqn. (49).The treatment of the orbital derivative contributions followssimilar lines to those described in section 3.3. The corre-sponding matrices X� v and Y� for the LCCSD and LMP2 havesimply to be added. Thus, the orbital contributions of bothparts can be treated together and only one set of CPL-Z andHF-Z equations has to be solved.

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In summary, a gradient calculation for LCCSDjLMP2requires the following steps:1. Solve the LCC-Z equations for the strong pairs

(cf. Appendix A).2. Solve the LMP2-Z equations, eqn. (50).3. Construct X�v (eqn. (34)) and �Y (cf. Appendix B)

using the LCCSD and LMP2 amplitudes, Z-vectors, andresiduals.4. Solve the CPL-Z and HF-Z eqn. (45) and (46),

respectively.5. Construct the e�ective densities (cf. Appendix C) and

assemble the gradient according to eqn. (47).

5. Test calculations

All methods described above were implemented in the MOLPRO

package of ab initio programs.38 Although the e�ciency of thelocal correlation methods is not yet fully exploited in ourcurrent implementation, geometry optimizations with morethan 200 basis functions can easily be performed as shown inthe test calculations provided below. In this section we com-pare structural parameters for a small set of moleculesobtained from QCISD, LQCISD, LQCISDjLMP2, and LMP2calculations.The e�ect of the local approximations on structural para-

meters and harmonic vibrational frequencies of standardmolecules has been investigated before at the LMP2 level26,39

and was found to be very small. In general, excellent agree-ment between the conventional and local results was observed.Typically, the local treatment leads to a slight bond elonga-tion. This e�ect decreases as the basis set is increased, and isprobably due to a reduction of the intramolecular basis setsuperposition error (BSSE).In the current study we have selected a few molecules from

our recent LMP2 test set for investigating the e�ect of the localapproximations on computed equilibrium structures at higherlevel, namely LQCISD, with and without weak pair approx-imation. Computed bond lengths are presented in Tables 1 to 3.

A complete set of geometrical parameters is provided asElectronic Supplementary Information.y The 1s core orbitalsof the carbon, oxygen, and nitrogen atoms were not correlated.Orbital domains were generated automatically as proposed byBoughton and Pulay.7 Within this procedure a threshold of0.02 was used in combination with Dunning 's cc-pVDZ basis40

and a value of 0.015 for the cc-pVTZ basis set. The domainsand pair lists were kept ®xed in order to ensure smoothpotential energy surfaces during the optimizations. Tables 1to 3 show that the local correlation treatment has little e�ecton the bond lengths and angles. For the cc-pVDZ basis theabsolute mean deviations between LQCISD and QCISD are inthe same range as for (L)MP2 i.e. 0.002 AÊ for bond lengths and0.3� for bond angles (see ESIy). With the larger cc-pVTZ basisthe deviations are even smaller, i.e. 0.001 AÊ and 0.2�. The mostsensitive parameter appears to be the N � � �H5 distance inglycine, which is a hydrogen bond. At the QCISD=cc-pVTZlevel this distance was computed to be 1.942 AÊ , while it is1.968 AÊ at the LQCISD=cc-pVTZ level (without weak pairapproximation). It is likely that this e�ect is due to a reductionof BSSE e�ects in the local treatment.14,26,41,42 For thecc-pVDZ basis the discrepancy is even slightly larger.The treatment of weak pairs at the LMP2 level has in gen-

eral little impact on the results. For the standard moleculesconsidered here the absolute mean deviations from conven-tional results are even slightly smaller than without weak pairapproximation, which is probably due to some error com-pensation. However, the deviations appear to be less sys-tematic and the LQCISDjLMP2 bond lengths and angles arenot necessarily in the range between the pure LMP2 andLQCISD values, as is observed, for instance, for the N±C1

bond or the N±C1±C2±O2 dihedral angle in glycine (see ESIy).In order to test the e�ect of the approximate weak pair

treatment more thoroughly we studied two molecules which arewell known for their sensitivity to dynamical electron correla-tion e�ects, namely 5H-tetrazole43 and oxadiazole-2-oxide(furoxan)44,45 (cf. Tables 4 and 5). MP2/cc-pVDZ calculationslead to bond lengths that are far from experimental results46 or

Table 1 Geometrical parameters of glycine (C1)a

cc-pVDZ cc-pVTZ

Parameter LMP2 QCISD LQCISDb LQCISDc LMP2 QCISD LQCISDb LQCISDc

N±C1 1.473 1.472 1.476 1.472 1.467 1.465 1.468 1.464C1±C2 1.541 1.538 1.543 1.540 1.531 1.528 1.532 1.529C2±O1 1.212 1.207 1.207 1.207 1.206 1.200 1.200 1.200C2±O2 1.348 1.346 1.348 1.348 1.343 1.338 1.341 1.340N±H1 1.021 1.021 1.022 1.023 1.010 1.009 1.009 1.010N±H2 1.023 1.024 1.024 1.025 1.011 1.010 1.011 1.012C1±H3 1.104 1.105 1.106 1.107 1.089 1.090 1.091 1.092C1±H4 1.104 1.105 1.106 1.107 1.089 1.090 1.090 1.092O2±H5 0.985 0.980 0.978 0.979 0.981 0.973 0.972 0.974

a Bond lengths in AÊ . b Local QCISD values without speci®c weak pair treatment. c Local QCISD values with weak pairs treated at the LMP2 level.

y Electronic Supplementary Information available. See http:==www.rsc.org=suppdata=cp=b1=b105126c=

4858 Phys. Chem. Chem. Phys., 2001, 3, 4853±4862

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from more accurate calculations. The most critical parametersare the N2±N2

0 bond length in 5H-tetrazole and the N1±O2 bondlength in furoxan. At the cc-pVDZ level LQCISD leads to anN2±N2

0 bond elongation relative to QCISD of 0.033 AÊ , whichshrinks to 0.011 AÊ for the cc-pVTZ basis set. However, at thecoupled LQCISDjLMP2 level the bond is stretched even by0.129 AÊ with respect to the conventional QCISD=cc-pVDZvalue, indicating that the very large error of LMP2 (0.50 AÊ ) isonly partly eliminated. Using the larger cc-pVTZ basis thiserror is reduced to 0.029 AÊ , even though the error of the LMP2value is increased (0.54 AÊ ). For the critical N1±O2 bond lengthin furoxan the weak pair approximation has no larger in¯uencethan found for the standard molecules discussed above.From these preliminary results it appears that the approx-

imate weak pair treatment at the LMP2 level can be safely usedfor most molecules, even when dynamical electron correlatione�ects are important. Only in very extreme cases when theMP2 approximation breaks down entirely, as for the 5H-tetrazole bond lengths, this approach may lead to questionableresults, in particular when small basis sets are used. The

reliability of LQCISD and LCQISDjMP2 vibrational fre-quencies will be reported in a separate paper.47

6. Conclusions

The theory for local coupled-cluster energy gradients has beenpresented in a general way, and the relation between local andconventional coupled-cluster gradient theory has been eluci-dated. The canonical and local theories di�er mainly by twocontributions: ®rstly, additional terms arise from the geometrydependences of the excitation domains. These terms(cf. eqn. (34)) involve the residuals of the CC and CC-Z equa-tions, which do not fully vanish in the local case, unless domainsspanning the whole virtual space are used. Secondly, the geo-metry dependence of the unitary localization matrix must betaken into account, which requires the solution of a set ofcoupled perturbed localization z-vector equations (cf. eqn. (45)).The theory has been implemented for QCISD and LQCISD,

andanumberof test calculationswerepresented inorder to studythe e�ectof the local approximations at this level for twodi�erent

Table 2 Geometrical parameters of oxalic acid (C2h)a

cc-pVDZ cc-pVTZ

Parameter LMP2 QCISD LQCISDb LQCISDc LMP2 QCISD LQCISDb LQCISDc

O1±C 1.217 1.210 1.210 1.209 1.210 1.201 1.201 1.201O2±C 1.330 1.330 1.332 1.332 1.327 1.323 1.325 1.325C±C0 1.540 1.545 1.553 1.549 1.539 1.538 1.543 1.540O2±H 0.980 0.977 0.976 0.976 0.975 0.970 0.970 0.970

a Bond lengths in AÊ . b Local QCISD values without speci®c weak pair treatment. c Local QCISD values with weak pairs treated at the LMP2 level.

Table 3 Geometrical parameters of ethanol (Cs)a

cc-pVDZ cc-pVTZ

Parameter LMP2 QCISD LQCISDb LQCISDc LMP2 QCISD LQCISDb LQCISDc

C1±C2 1.523 1.523 1.527 1.524 1.514 1.514 1.517 1.514C1±O 1.426 1.423 1.427 1.425 1.426 1.422 1.424 1.421O±H1 0.967 0.965 0.966 0.967 0.961 0.957 0.958 0.959C1±H2 1.110 1.111 1.112 1.113 1.094 1.095 1.096 1.097C2±H4 1.103 1.105 1.105 1.106 1.089 1.090 1.091 1.092C2±H5 1.102 1.104 1.105 1.105 1.088 1.089 1.090 1.091

a Bond lengths in AÊ . b Local QCISD values without speci®c weak pair treatment. c Local QCISD values with weak pairs treated at the LMP2 level.

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basis sets. In general, geometry parameters obtained fromLQCISDgeometry optimizations are in excellent agreementwithconventional QCISD results and show a similar relative beha-viour as found when comparing LMP2 with MP2 results.Since weakly correlated electron pairs can be treated at a

lower correlation level without introducing signi®cant errors inrelative energies, we have performed some LQCISD bench-mark calculations with a weak pair treatment at the LMP2level. The approximation appears to be safe for geometryoptimizations of standard molecules and even compensatespartly the e�ects (e.g. bond elongation) caused by the local

correlation treatment. In summary, both the local correlatione�ects as well as e�ects due to an approximate weak pairtreatment are very small and can be neglected in most cases.Exceptions were found only for extremely correlation sensitivegeometry parameters.

Acknowledgements

This work has been supported by the Deutsche For-schungsgemeinschaft and the Fonds der Chemischen Industrie.

Table 4 Geometrical parameters of 5H-tetrazole (C2v)a

cc-pVDZ cc-pVTZ

Parameter LMP2 QCISD LQCISDb LQCISDc LMP2 QCISD LQCISDb LQCISDc

C±N1 1.510 1.474 1.479 1.480 1.500 1.461 1.465 1.456N1±N2 1.197 1.221 1.218 1.210 1.184 1.214 1.214 1.222N2±N2

0 2.178 1.676 1.709 1.805 2.157 1.615 1.626 1.586C±H 1.102 1.102 1.103 1.105 1.088 1.089 1.089 1.092

a Bond lengths in AÊ . b Local QCISD values without speci®c weak pair treatment. c Local QCISD values with weak pairs treated at the LMP2 level.

Table 5 Geometrical parameters of oxadiazole-2-oxide (Cs)a

cc-pVDZ cc-pVTZ

Parameter Exp. LMP2 QCISD LQCISDb LQCISDc LMP2 QCISD LQCISDb LQCISDc

C1±C2 1.401(2) 1.405 1.428 1.430 1.428 1.393 1.415 1.418 1.416C1±N1 1.302(2) 1.359 1.332 1.333 1.334 1.349 1.319 1.321 1.322C2±N2 1.292(2) 1.341 1.323 1.313 1.315 1.329 1.300 1.301 1.302N1±O2 1.441(2) 1.564 1.442 1.441 1.445 1.486 1.416 1.416 1.418N2±O2 1.379(1) 1.330 1.370 1.373 1.371 1.343 1.368 1.371 1.369N1±O1 1.240(1) 1.201 1.219 1.220 1.219 1.207 1.218 1.219 1.218C1±H1 0.92(2) 1.087 1.086 1.087 1.088 1.074 1.072 1.073 1.074C2±H2 0.97(2) 1.089 1.090 1.091 1.091 1.076 1.075 1.076 1.077

a Bond lengths in AÊ . b Local QCISD values without speci®c weak pair treatment. c Local QCISD values with weak pairs treated at the LMP2 level.

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Appendix A: The LQCISD-Z residuals

The pair residuals of the LQCISD-Z equations as discussed ineqn. (33) are given as

~R ij � ~K� ~Z ij � ~�E ij � ~�E jiy� �Xkl

akl;ij ~S ~Z kl ~S� ~Z ij ~S ~T lk ~SD E

~K lkh i

� ~N ij � ~N jiy �51�with [ ~�E ij]rs� ~E ij ÿ 1

2~E ji

h irs� ~z i

rdjsÿ 12~z jrdis , and

~K� ~Z ij�rs �Xtu

~Zijtu�rtjsu�; �52�

where (rtjsu) are the two-electron integrals in the PAO basis.~K( ~E ij)rs is de®ned analogously using 3-external integrals (rtjsj).The corresponding singles contributions take the form

~ri � ~Fÿ�X

kl

~�K kl ~T lk

�~S

" #~zi � 2 ~K� ~Z ij�� j � ÿ 2 ~S

Xkl

~Z lk ~k kli

�Xj

2 ~S ~Z ij ~f j � 2 ~K ij ÿ ~J ij� �

~z j ÿ bij ~S ~z jh i

�Xl

�Xk

~�K ik ~�T kl

�~S ~z l �53�

with the de®nitions

aij; kl � K klij � h ~T ij ~K lki ÿ dijbki ÿ dkiblj ; �54�

bki � Fki �Xl

h ~�K kl ~T lii: �55�

For quantities not de®ned in the previous sections weadopted the nomenclature used in ref. 14. The local matrices~N ij can be expressed as

~N ij��

~Fÿ�X

kl

~�Kkl ~T lk

�~S

�~Z ij ~S�

�1ÿ 1

2tij��

~f i�Xl

~�K il ~t l�

~z jy ~S

ÿ 12

Xk

�~�kijk ~zky

�~Sÿ ~S

�Xkl

�~Zkl ~S ~T lk� 1

2dkl ~zl ~t ly��

~�K ij

ÿXkl

�h ~S ~T lk ~S ~Zkii� 1

2dklh ~ziy ~S ~tli�

~�K lj

��2ÿ tij� ~SXk

~Z ik

�~Kkjÿ 1

2~Jkj� 1

2~SXl

~�Tkl ~�K lj

��Xk

��Xl

~K lj ~Tkl

�~Sÿ ~J jk

��~Z ik� 1

2~Zki

�~S �56�

Appendix B: The orbital derivative prefactors

The prefactors of the orbital derivatives are collected in thematrix Y� (cf. section 3.3). In order to use the local character,the external parts of this matrix can be evaluated in theprojected basis. For a compact matrix notation of the explicitequations, it is convenient to augment the local matrices by theoccupied parts

~S �1 0

0 ~PySAO~P

!; ~F �

LyFAOL LyFAO~P

~PyFAOL ~PyFAO~P

!; �57�

and similarly for other quantites like ~J kl, ~K kl etc. The rectan-gular transformation matrices Q and Q� (cf. eqn. (22) and

(24)) are augmented by a unit matrix, similar to ~S. The blocks

of the amplitudes ~T ij, ~t i and Z-vectors ~Zij, ~zi which involve

occupied orbital indices are zero, e.g., ~T ijkl � ~T ij

kr � ~T ijrk � 0.

Thus, the matrix Y� in the MO basis can be obtained from

the corresponding local quantity ~Y as follows

�Y � �Qy ~Y �Q: �58�with

~Y � ~F ~A� ~f ~t yh i

~S� ~F ~t� 2 ~G� ~A� ~t �1o�Xij

~J ij ~B ji � ~K ij ~C ji � ~K� ~D ij� ~Z ji � ~K� ~Z ij� ~D jih

� 12

~K� �T ij�1j ~ziy � 12

~K� ~E ij � ~E jiy� ~�T ji � ~K ij ~�T jii

~S

�Xij

~J� ~B ij� � ~K� ~C ij � ~T ij�h i

1 ji � 12

~K� ~�T ij� ~z j1iyn o

�59�

where (1 k)r� drk , (1kl)rs� drk dsl , and 1o is a unit matrix in the

occupied space and zero otherwise. Furthermore, ~f � ~F1o . Simi-larly, ( ~t )ri� ~t ir , (~t)rs� 0, and ~D ij

rs � ~T ijrs � ~t irdsj � ~t jsdir . Finally,

~J� ~B ij�rs �Xtu

~Bijtu�rsjtu�; �60�

and ~G( ~A) is the two-electron part of the closed-shell Fockmatrix evaluated with the density A. The e�ective densitymarices ~A, ~B ij, and ~C ij are de®ned in Appendix C.

Appendix C: E�ective LQCISD densities

The expression for ~Y given in Appendix B depends one�ective density matrices, arising as prefactors of di�erentintegral types. These density matrices will also contribute tothe explicit gradient expression given in Appendix D. In thefollowing, the indices r; s correspond to PAOs, and i; j; k; l toLMOs. The density arising from one-electron integrals isde®ned as

~Aij � ÿXk

h ~T ik ~S ~Z kj ~S� ~T jk ~S ~Z ki ~S i ÿ 12� ~z iy ~S ~t j � ~t iy ~S ~z j�;

~Ari � ~Air �Xj

� ~Z ij ~S ~t j � ~�T ij ~S ~z j�r � 12~zir;

~Ars �Xij

� ~Z ij ~S ~T ji � ~T ij ~S ~Z ji�rs � 12

Xi

� ~z i ~t iy � ~t i ~z iy�rs : �61�

The e�ective density arising from the prefactors of theCoulomb integrals (ijjrs) is given by

~Bijrs �

hÿ2

Xk

� ~Z ik ~S ~T kj � ~T kj ~S ~Z ik� ÿ ~zi ~t jyirs

�62�

This matrix is nonzero in the external±external part only, i.e.,~B ijrk � ~B ij

kr � 0 for any index r. Only the symmetric part~B ijsym � 1

4 � ~B ij � ~B ji � ~B ijy � ~B jiy� is needed. The correspond-ing e�ective density for the exchange integrals (rijsj) is givenby

~Cijkl � h ~Z ij ~S ~T lk ~Si;

~Cijrs �

�~Z ij �

Xkl

h ~S ~T ij ~S ~Z lki ~T kl ÿXk

~�T ikÿh ~z ky ~S ~t ji

� 2Xl

h ~Z kl ~S ~T lj ~S i�ÿ ~�T ij ~S

�2Xkl

~Z kl ~S ~T lk �Xk

~zk ~t ky�

� 2Xk

~�T ik ~Sÿ

~Z kj � ~z k ~t jy��X

kl

h~�T ik ~S ~Zkl ~S ~�T lj

ÿ 12

~�T jk ~S ~Zkl ~S ~�T li � ~T kj ~S�

~Z lk � 12

~Z kl�

~S ~Tili

ÿXk

ÿ ~�T jk ~S ~z k�~tiy � 2 ~t i ~z jy

�rs

;

~Cijrk � ÿ

�~Z ij ~S ~t k � 1

2~�T ij ~S ~z k

�r

�63�

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These matrices can be symmetrized according to~C ijsym � 1

2 �C ij � ~C jiy�. For weak pairs treated at the LMP2level one simply has �C ij

rs � �Z ijrs and all other elements are

zero.

Appendix D: Explicit gradient expressions

The total SCF plus LQCISD one-electron density matrix inMO/PAO basis is

~D�1� � 21o � ~A� ~t� ~t y � 12� ~z� ~zy� �64�

which can be transformed into the AO basis using

d�1� � L j ~P� � ~D�1�oo ~D�1�op

~D�1�po ~D�1�pp

0@ 1A Ly~Py

� ��65�

where the indices o and p denote the occupied and PAOspaces, respectively. The prefactors of the derivatives of theoverlap integrals are

lmn � ~x� uya� zyBÿ 2CoeCyoÿ �

mn �66�where e is a diagonal matrix holding the canonical orbitalenergies of the occupied orbitals. The product zyB can becomputed as

�zyB�mn � ÿ 14

Xri

�CmiCnr � CmrCni� zF� Fz� 4G�Z�� �ri �67�

where zai is the solution of the HF-Z equations (zij� zra� 0),F is the Fock matrix in the MO basis, and G(Z) is the two-electron part of the closed-shell Fock matrix in the MO basisevaluated with the density Z� 1

2(CvzC yo�C ozyC yv). Finally, the

e�ective second-order density matrix in the AO basis is givenby

d�2�mn;rs � DSCFrs

ÿd�1�mn ÿ 1

2DSCFmn

�ÿ 12D

SCFmr

ÿd�1�ns ÿ 1

2DSCFns

��Xij2fPsg

ÿ�T ijmn � Cij

mn

�LniLsj � �Tij

mrzinLsj

h� Bij

mnLriLsj � ZijmrD

jinr

i�68�

In the case of a speci®c weak pair treatmentP

ij2fPwg�T ijmrLviLsj

has to be added with pair amplitudes computed at the LMP2level.

Appendix E: Erratum to ref. 26

In ref. 26 there is an error in eqn. (52). The two-particle densitymatrix G depends on the SCF density and the one-particledensity matrix g rather than the MP2 density d. The correctequation in the notation of ref. 26 is

Gmn;rs � �gmn ÿ 12Dmn�Drs ÿ �gmn ÿ 1

2Dmn�Dsn

� 2Xij

Lmi �T ijnrLsj �69�

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