Manybody theory of exchange effects in intermolecular interactions.Secondquantization approach and comparison with full configurationinteraction resultsRobert Moszynski, Bogumił Jeziorski, and Krzysztof Szalewicz Citation: J. Chem. Phys. 100, 1312 (1994); doi: 10.1063/1.466661 View online: http://dx.doi.org/10.1063/1.466661 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v100/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Many-body theory of exchange effects in intermolecular interactions. Second-quantization approach and comparison with full configuration interaction results

Robert Moszynski and BogumiJ Jeziorski Department of Chemistry. University of Warsaw. ul. Pasteura 1. 02-093 Warsaw. Poland

Krzysztof Szalewicz Department of Physics and Astronomy. University of Delaware. Newark, Delaware 19716

(Received 2 April 1993, accepted 28 September 1993)

Explicitly connected many-body perturbation expansion for the energy of the first-order exchange interaction between closed-shell atoms or molecules is derived. The influence of the intramonomer electron correlation is accounted for by a perturbation expansion in terms of the M~ller-Plesset fluctuation potentials W A and W B of the monomers or by a nonperturbative coupled-cluster type procedure. Detailed orbital expressions for the intramonomer correlation corrections of the first and second order in WA+ W B are given. Our method leads to novel expressions for the exchange energies in which the exchange and hybrid integrals do not appear. These expressions, involving only the Coulomb and overlap integrals, are structurally similar to the standard many-body perturbation theory expressions for the polarization energies. Thus, the exchange corrections can be easily coded by suitably modifying the existing induction and dispersion energy codes. As a test of our method we have performed calculations of the first-order exchange energy for the He2' (H2h, and Ht'r-H2 complexes. The results of the perturbative calculations are compared with the full configuration interaction data computed using the same basis sets. It is shown that the M~ller-Plesset expansion of the first-order exchange energy converges moderately fast, whereas the nonperturbative coupled-cluster type approximations reproduce the full configuration interaction results very accurately.

I. INTRODUCTION

It is well known that the intermolecular interaction energy is a sum of the long-range component that de­creases for large intermolecular distances R as inverse powers of R and the short-range, repulsive part decreasing exponentially.! The development of analytic models for in­teractions of polyatomic molecules involves obtaining the appropriate analytic form and the numerical values of the adjustable parameters for both components of the poten­tial. The problem has been largely solved for the long­range component when the application of the multipole expansion with damping leads to very satisfactory analytic expressions for the electrostatic, induction, and dispersion energies.2 However, there is no well-established theoretical procedure for obtaining the analytic form of the short­range repulsion energy, although some progress in this di­rection should be noted.3 This difficulty seems to be a major obstacle in developing analytic intermolecular po­tentials. Numerous semiempirical models of the short­range repulsion have been proposed in the literature.4-10 These models cannot be easily justified theoretically and clearly need a reference to accurate ab initio theoretical calculations to verify their validity.

The determination of the short-range part of the po­tential by fitting to experimental data has not been very successful so far since the results of measurements are sen­sitive to the total potential and reliable values of the long­range component at smaller intermolecular separations are

difficult to obtain unambiguously. Moreover, even if a set of experimental data is reproduced accurately, the model repulsion may extrapolate unrealistically to those parts of the configuration space which are not sampled in the fitting (see, e.g., Refs. 11-13).

In view of the above difficulties accurate ab initio cal­culations of the short-range, exchange component of the interaction energy are of particular importance. The symmetry-adapted perturbation theory (SAPT) provides the most natural theoretical framework for performing such calculations. !4-!6 The interaction energy E int is com­puted in SAPT directly as a sum of well-defined, physically interpretable polarization and exchange contributions

E E O) E(1) E(2) E(2) (1) int = po! + exch + pol + exch + . . . .

Expansion (1) is based on the partitioning of the clamped nuclei Hamiltonian H for the complex AB into the unper­turbed operator Ho being the sum of the Hamiltonians for molecules A and B, i.e., Ho=HA+HB' and the interaction operator V defined as the difference V=H-Ho. The po­larization corrections E~l are defined by the conventional Rayleigh-Schrodinger (RS) perturbation expansion based on the same Hamiltonian partitioning as used in SAPT (following Hirschfelder,17 this expansion is referred to as the polarization expansion). The first-order polarization en­ergy E~l is the effect of the classical electrostatic interac­tion of the unperturbed charge distributions in the mono­mers, while E;,l is the sum of the classical induction and

1312 J. Chern. Phys. 100 (2).15 January 1994 0021-9606/94/100(2)/1312/14/$6.00 © 1994 American Institute of Physics

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Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects 1313

quantum mechanical dispersion energies. The exchange corrections ~:~h' n= 1,2, specific to SAPT, provide the short-range repulsion.

The first-order exchange energy constitutes by far the dominant part of the short-range repulsion. I4-16 This cor­rection is defined in the same way in most symmetry­adapted perturbation formalisms, including the simplest one, the symmetrized Rayleigh-Schrodinger (SRS) ap­proach. I8-20 The SRS perturbation corrections E(n) are the coefficients in the Taylor expansion of the function

('1'01 ~V d''I'(~» ('1'0 1 d''I'(~» ,

(2)

where '1'0 is the eigenfunction of Ho, d' is the proper symmetry-projection operator (usually the antisymme­trizer), and 'I'(~) is the solution of the Schrodinger equa­tion (Ho+~V)'I'(~) =E(~)'I'(~). In view of Eq. (2), the first-order SAPT energy is given by

('I' A 'I' B 1 V d''I' A 'I' B)

('I' A 'I' B 1 d''I' A 'I' B) , (3)

where 'l'x, X=A or B is the exact wave function of the monomer X, so that 'l'O='I'A'I' B' The first-order exchange energy ~e!~h is defined by

...(1) _ "'(1) ...tl) ~~xch-..c' -~PoI'

where

('I' A 'I' B 1 V'I' A 'I' B>

('I' A 'I' BI 'I' A 'I' B> •

(4)

(5)

The first-order exchange energy is essentially an effect of the resonance tunneling of electrons between interacting subsystems and falls off exponentially at large intermolec­ular distances. Being of purely quantum-mechanical origin, this energy cannot be easily related to any observable prop­erty of the monomers and is difficult to calculate, even approximately. The exchange energy is also difficult to ob­tain from ab initio variational calculations since it depends strongly on the monomer wave functions in the classically forbidden parts of the configuration space (corresponding to tunneling of electrons into regions of negative kinetic energy). The total electronic energy of monomers is not sensitive to the wave function amplitudes in these regions and, therefore, these amplitudes are not determined accu­rately from variational calculations. Another difficulty arising in calculating exchange energy ab initio is due to fact that the orbitals of the monomer A are not orthogonal to the orbitals of the monomer B. If the interacting systems are described at the Hartree-Fock level, which corre­sponds to the complete neglect of the intramonomer elec­tron correlation, this nonorthogonality can be managed using a suitable density matrix formalism21 and the corre­sponding approximate exchange energy, denoted by ~~~~, can be routinely calculated even for large systems. The few studies performed thus far for small systems using correlated wave functions22

-25 have shown, however, that

the intramonomer electron correlation significantly affects the exchange repulsion energy and cannot be neglected if a quantitative accuracy is required.

Recently, Rijks, Gerritsen, and Wormer26 developed a method to calculate the intramonomer correlation contri­bution to E( I) for larger systems using the configuration interaction singles and doubles (eISD) representation for the monomer wave functions. In this method, the number of matrix elements contributing to E(l) is proportional to ~, where M is the dimension of the configuration inter­action (el) problem, and these matrix elements involve nonorthogonal orbitals. Rijks, Gerristen, and Wormer solved the nonorthogonality problem by neglecting all ma­trix elements involving overlap integrals between occupied orbitals of one monomer and virtual orbitals of the other. However, when a dimer-centered basis set is used to cal­culate the monomer wave functions, this assumption ap­pears to be invalid and may lead to an overestimation of the exchange correlation effect.27 It has also been shown28

that the use of the dimer-centered basis set is essential for a proper representation of the monomer wave functions at large distances from the nuclei (in the tunneling region) and, consequently, for an accurate description of the elec­tron exchange effect.28

A general method of calculating the intramonomer correlation contribution to E~~h can be obtained using the ideas of the many-body SAPT.2

9-31 In this approach the monomer Hamiltonians HA and HB are decomposed as HA=FA+WA and HB=FB+WB, where FA and FB are the Fock operators for monomers A and B, respectively, and WA and W B are the corresponding intramonomer cor­relation operators (or fluctuation potentials) defined as WA=HA-FA and W B=HB-F B (the M0ller-Plesset par­titioning). The total Hamiltonian is parametrized as

H(~,AA,AB)=F+~V+AAWA+ABWB' (6)

where F =F A + F B and the formal expansion parameters ~, AA' and A B have the physical value equal to unity. Apply­ing a suitable triple perturbation theory to Eq. (2), one obtains the many-body expansion of the nth-order SRS energy E(n) in terms of the intramonomer correlation op­erators W A and W B ,

00 00

E(n) = L L E(nij). (7) i=O j=O

The corrections Enij), which are of the nth order in V, of the ith order in WA , and of the jth order in W B, are defined formally as the coefficients in the Taylor expansion of the function

('I'(O,AA,A B) I ~V d''I'(~,AA,AB» ('I'(O,AA,AB) I d''I'(~,AA,AB» (8)

where the wave function 'I'(~,AA,A B) is the solution of the Schrodinger equation

(FA+F B+~V +AAWA+ABW B)'I'(~,AA,AB)

=E(~,AA,A B) 'I'(~,AA,A B) (9)

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1314 Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects

satisfying 'I'(O,AA,A B) = 'I' A (AA)'11 B(A B), where 'I' A (AA) and 'I' B(A B) are the ground-state eigenfunctions of the pa­rametrized monomer Hamiltonians HA=FA+AAWA and HB=FB+ABWB. The SRS corrections E(nij) naturally decompose as

E(nij) =~nij) +~nij) (10) pol exch ,

where the polarization corrections are defined by setting d = 1 and expanding the right-hand side of Eq. (8) in powers of S, AA' and AB. For a fixed n, the exchange contributions E~~{) , defined essentially by Eq. (10), sum up to the total nth-order SRS exchange energy ~:~h ,

00 00

,.,.{n) _ ~ ~ E(nij) ~~xch - £.. £.. exch·

i=O j=O (11 )

The many-body SAPT expansion of the first-order ex­change energy [Eq. (11)], has been introduced in Ref. 32. In this reference and in Ref. 27, such an expansion was applied to the interaction of two helium atoms. For the interaction of arbitrary many-electron systems, the prob­lem of expanding individual terms in Eq. (11) through one- and two-electron integrals (perturbation theory dia­grams) has not been considered thus far.

The aim of the present paper is to introduce a general, explicitly connected expansion of the first-order exchange energy and to derive the detailed orbital expressions for the leading intramonomer correlation corrections. This goal will be achieved by applying a second-quantized represen­tation of the antisymmetrizer. An alternative, first­quantization approach which uses density matrix tech­niques will be presented in Ref. 33. The actual final formulas are strikingly different in these two approaches, but the numerical values of the corrections are identical if a dimer-centered basis set is used.

In our development the coupled-cluster (CC) repre­sentation of the Schrodinger equation3

4--36 is utilized. In this representation the nonphysical, disconnected terms which have to be canceled in the standard perturbational approach do not appear and the perturbation energies are expressed directly in terms of connected contributions (or diagrams). Thus, the many-body SAPT corrections ~~X) are size-extensive (unlike the approximations considered in Ref. 26) and behave correctly with the increasing size of the interacting systems. This is a very important property since the computer codes developed by us are completely general and can be applied to systems with an arbitrary number of electrons.

To simplify derivations, all exchange corrections will be computed by including only those terms which are qua­dratic in the intermolecular overlap densities Pk/(r) =tfJ1(r)tfJf(r), where tfJ1 and tfJf are the orbitals (occupied or virtual) of the monomers A and B, respectively. At the Hartree-Fock level, this approximation has been shown to give accurate results in the region of the van der Waals minimum. 37

The plan of this paper is as follows: In Sec. II, we present a general formulation of the many-body perturba­tion expansion for the first-order exchange energy, based on the coupled-cluster form of the Schrodinger equa-

tion.34--36 The leading intramonomer correlation contribu­tions are considered in Sec. III. Finally, the results of nu­merical calculations for He2' (H2h, and He-H2 complexes are presented and discussed in Sec. IV.

II. MANY-BODY PERTURBATION THEORY OF EXCHANGE INTERACTIONS

The Hilbert space used in SAPT is larger than the conventional Fermion Hilbert space ,7t"F of fully antisym­metric functions since it must include functions like 'I'(O,AA,AB) = 'l'A (AA) 'I' B(AB), which are not antisymmet­ric with respect to exchanges of electrons between interact­ing subsystems. To accommodate such functions, we have to consider the tensor product ,7t" A ® ,7t" B of Hilbert spaces ,7t"A and ,7t" B for molecules A and B, respectively. The Hilbert spaces ,7t"A and ,7t" B are individually of fermion type, i.e., they contain only antisymmetric functions of N A and N B electrons, respectively. The space ,7t" A ® ,7t" B

may be viewed as the linear span of all products <1>'1', <I> E,7t" A, 'I' E,7t" B, provided that the functions <I> and 'I' depend on different arguments, e.g., <I> = <I> (1, ... ,NA ) and '1'= 'I'(NA + 1, ... ,NA+N B). In other words, ,7t"A ®,7t" B is a space of functions of NA+N B variables, which are anti­symmetric separately in variables 1, ... ,NA and N A+ 1, ... ,NA+N B. The conventional fermion space ,7t"F consists of functions which are antisymmetric in all N A + N B variables. Since a function which is antisymmetric in all its variables must also be antisymmetric in subsets of variables, we find that ,7t"F is a subspace of ,7t" A ® ,7t" B, ,7t"F C ,7t" A ® ,7t" B. This means that all physical, Pauli al­lowed solutions of the Schrodinger equation do belong to ,7t" A ® ,7t" B •

The relation,7t"F C ,7t" A ® ,7t" B holds true also for finite­dimensional "full CI" spaces, provided that the spaces ,7t"F, ,7t" A, and ,7t" B are antisymmetric tensor powers of the same finite-dimensional one-electron space 'Y'. This partic­ular case appears when a dimer-centered basis set is used in calculations, i.e., if 'Y' is a linear span of spin orbitals cen­tered on the nuclei of both molecule A and B. The tensor product space ,7t" A ® ,7t" B then contains all products of N A- and N Jrelectron determinants built from the spin or­bitals belonging to 'Y'. Since, in view of Laplace's theorem, each (NA+N B)-electron determinant can be represented as a sum of products of N A-electron and N Jrelectron de­terminants, it follows that the full CI space ,7t"F is a sub­space of the tensor product space ,7t" A ® ,7t" B. In other words, the total antisymmetrizer d for N A + N B electrons does not lead out of ,7t" A ® ,7t" B. On the other hand, if the one-electron spaces used to construct ,7t" A and ,7t" Bare different, e.g., if incomplete monomer-centered basis sets are used, then the space ,7t" A ® ,7t" B is not stable under the action of the antisymmetrizer d. From now on, we shall always assume that a dimer-centered basis set or complete monomer-centered basis sets are used to construct K A and K B, and that d is a well-defined operator acting within KA®KB •

In Ref. 31, the many-body SAPT theory was formu­lated in the generalized Fock space Y AB defined as the tensor product Y A ® Y B of monomer Fock spaces Y A

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Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects 1315

and Y B' The generalized Fock space Y AB , which hap­pens to be isomorphic to the bispinor space of the Clifford algebra approach to the electron correlation problem,38,39 has a more complicated structure than the product Hilbert space K A ® K B' Since all operators used in subsequent developments conserve the number of particles, in the present work we do not need to introduce the generalized Fock space Y AB and can limit ourselves to the Hilbert space KA®KB •

A. Second quantization of exchange operators

The strings of equal number of conventional creation and annihilation operators acting in K A and J¥' B will be denoted by

(12)

and

(13 )

respectively. We will use the same notation for analogous operators actinf.l~. f' A ® J¥' B ~~hough, strictly speaking, the operators aK

1K 2"'K n and u:.1~2 .. . ~m should then be written

1 2 1). ... ). \ 2 m •••

as tensor products aK

1/"

K n ® 1 and 1 ® u:.1~2 ... ~m, respec-

1 2 n 12m tively. One should bear this fact in mind when following fi h .. Al . I h ).1).2"'). urt er denvabons. temabve y, t e operators aK K "'K n

1 2 n

and u:.1~2:::~m may be viewed as acting on different sets of 12m

variables. Consequently, they must commute

(14)

We assume that the spin-orbital basis sets {4>1} and {4>/h- used to define aA. and bl' are orthonormal (note that the spin orbitals 4>1 cannot be orthogonal to spin orbitals 4>: since we assume that these two sets of spin orbitals result from an orthogonalization of the same dimer­centered basis set and, consequently, span the same one-

).). ••• ,1. electron space). The operators a

K IK 2"'K n mUltiply according 1 2 n

to the following rules39-4I

which are very useful in evaluating matrix elements of second-quantized operators. Note that all pairings of the lower indices of the first factor with the upper indices of the second factor are included and that all pairings enter with plus sign (see Sec. 2.3 of Ref. 41). The rules (15)-(17) and similar rules for the operators U:\~2:::~m

\ 2 il2 are a straightforward consequence of the Wick theorem.

The operators Fe and We, C=A or B, can now be represented in the following second-quantized forms:

Fc=(fd~c~ , (18)

(19)

where (/ d~ is the matrix element of the Fock operator for molecule C, wMI is the antisymmetrized two-electron KKI integral

w:'::: = (4)K(1 )c{lK\ (2) I r 121 (1-PI2 )c{I).(1 )c{lA.\ (2», (20)

M'" and the symbol cK

/ •• denotes the operator product of Eq. 1

(12) when C=A and ofEq. (13) when C=B. Summation over repeated lower and upper indices is implied in Eqs. (18) and (19) and throughout the rest of this paper (the Einstein convention). From now on, we shall always as­sume that the indices A,AI, ... K, KI,." (j-L,j-LI> ••• V,VI,"') refer to spin orbitals (both occupied and virtual) of the molecule A(B), while a,a\, ... (/3,/3I"") and p,p\, ... (u,u\> ... ) label occupied and virtual spin orbitals of the molecule A(B), respectively. The indices aa'··· (rr"") are used exclu­sively for the occupied (virtual) orbitals of the molecule A, while the indices bb'··· (ss'···) for the occupied (virtual) orbitals of the molecule B.

To calculate the exchange corrections we have to de­rive the second-quantized form of the exchange operators. In general, the antisymmetrizer d is a complicated, (NA+N B)-electron operator. It can be written as

NAINBI (NA+N B)l dAd B(l+P+P'), (21)

where d x, X=A or B, is the antisymmetrizer of the monomer X, P and P' collect permutation operators inter­changing one and more pairs of electrons, respectively (P and P' include the phase factors corresponding to the par­ity of the permutations), and N x denotes the number of electrons in the monomer X. When all terms quartic and of higher-order in the intermolecular overlap densities Pk/(r) =c{l1(r)c{lf(r) are neglected, the antisymmetrizer d can be replaced by the operator 1 + P, where the single­exchange operator P is given explicitly by

P= - L L Pij . (22) . iEA JEB

In this equation Pij denotes the operator interchanging the spatial and spin coordinates of the ith and jth electrons. Similarly, the operator P' can be written as the sum of operators interchanging simultaneously the coordinates of more than two electrons. Making use of the assumption that the spin-orbital basis sets {4>1} and {4>/h- span the same linear space, one can show that P is a well-defined operator acting within the Hilbert space KA ®K B (this is in spite of the fact that this Hilbert space is not invariant under the action of individual operators Pij ) and can be represented in the following second-quantized form:

P=~:a'J.~, (23)

where the coefficients ~: are expressed through the inter­molecular overlap integrals S~ = < c{l11 c{I:>,

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1316 Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects

(24)

Equation (23) does not hold when a finite monomer­centered basis set is used, i.e., when the orbital basis sets used to construct Yt" A and Yt" B are localized on monomers A and B, respectively. This fact is evident since the right­hand side of Eq. (23) always represents an operator acting within Yt" A ® Yt" B, whereas the operator on the right-hand side of Eq. (22) leads out of the state space Yt" A ® Yt" B if the monomer-centered basis is used. Finite monomer­centered basis sets can, of course, be used to calculate ex­change energies provided that the first-quantized form of P, given by Eq. (22), is employed. A formulation of the theory valid in such a case has been presented in Ref. 33.

Similarly, the intermolecular interaction operator V, defined by

NA NB NA NB

V= I VB(ri) + I VA(r j )+ I I rijl+Vo, (25) ieA jeB ieA jeB

can be represented in the form31

V=~;a1~+ (VA);~+ (vB)!a1+ Vo , (26)

where Vo is the constant nuclear repUlsion term,

I,; = (tPK(1 )tPp. (2) I r1211 tPA (1 )tPv(2», (27)

and (vc>; = (tPp.1 vel tPv) is a matrix element of the electro­static potential of all the nuclei of molecule C.

Equations (18) and (19) are standard expressions of the second-quantization theory.42 To prove Eqs. (23) and (26), it is enough to verify that the matrix elements of P and V in the basis of products of Slater determinants are the same as the matrix elements of the first-quantized op­erators given by Eqs. (22) and (25).

B. Explicitly connected expansion of the first-order exchange energy

The first-order exchange energy has been defined by Eqs. (3)-(5). Neglecting terms of the fourth and higher order in the intermolecular overlap densities we obtain the most important, single-exchange part of E~~~h given by,43

(I) 2 ('II A'll B I VP'I' A'll B) Eexch(S ) = ('II A'll BI 'II A'll B)

(I) ('IIA'II BIP'I'A'II B) - Epal ('II A'll B I'll A'll B)

(28)

The role of the term proportional to E~l is to cancel the unlinked clusters appearing when the expression

('II A'll BI VP'I' A'll B)

('II A'll B I'll A'll B) (29)

is expanded in terms of molecular integrals. The coupled­cluster ansatz for the exact unperturbed wave function '110= 'II A 'II B is 34--36

(30)

where <1>=<1> A<I> B is the product of the Hartree-Fock de­terminants <I> A and <I> B for the monomers A and B, and the

operator T = T A + T B is the sum of cluster operators T A and T B of the isolated molecules A and B. Specifically, T A and T B are given by

(31)

(32)

Thus, the operator T A (T B) is a linear combination of the . • p pp', (bU bUu' ) • h h excitation operators aa' a aa' ••• /3' /3f3" ••• Wit t e co-

efficients t;, ta;;:, ... (~, I!:, ... ), referred to as cluster amplitudes. If we assume that the cluster amplitudes are antisymmetric in their upper and lower indices, then these amplitudes are uniquely defined and can be determined from the Schrodinger equation for the unperturbed mono­mers. By T nm we denote a cluster operator creating the n-fold excitation on monomer A and the m-fold excitation on monomer B.

To derive an explicitly connected expansion of Eq. (28) in terms of intramonomer correlation operators WA

and W B, it is useful to employ the following representa­tion44 of the expectation value of an arbitrary operator X calculated with the coupled-cluster wave function 'IIo=eT <I>

(eTIXeT)

(eT I e1') (33)

where the action of the operator S on the function <I> is given by

(34)

and where we use the following definition of the (degener­ate) scalar product of operators:

(YIX) = (Y<I> IX<I». (35)

Using the fact that T= T A+ T B and that [TA,T B]=O one can easily show that we can set S=SA+SB, where the operator SA satisfies

eT~eTA<I>A (eTA I eTA) (36)

and a similar equation holds for S B' Operators SA and S B are uniquely defined if we require that they assume the following cluster form:

(1)2 al"'a PI"'P S - - S na n na- , p"'P a"'a' n. 1 n 1 n

(1 )2 81"'/3 uI'''u S - - s nb n

On- n! u!'''un /31"'/3n'

As shown in Ref. 44, the operator SA satisfies the following linear equation:

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Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects 1317

(37)

[Y,l1k denotes a k-fold nested multiple commutator, [Y,l1o=Y, [Y,l1n+I=[[Y,l1n,l1, an~ 9110 is a special case of the projection superoperator fJ1 nm defined for an arbitrary operator Z as

9 nm(Z) = (_1_)2 (aal ... anill",PmZ)aPI"'PnbuI",Um. n!m! PI",Pn UI"'Um al···an PI"'Pm

(38) A

One may loosely say that fJ1 nm projects an arbitrary oper-ator Z on the operator manifold of n-tuple excitations on A and m-tuple excitations on B. An equation similar to Eq. (37) holds also for the operator S B. Solving Eq. (37) iteratively, one can express the operator S as a series in powers of the cluster operator T. Up to terms cubic in T, one obtains then44

A tit S=T+fJ1([T ,T]+:iC[T ,T],T]

+H[T,rt],rt]+· .. ), (39) A A

where fJ1 is the sum of the "projectors" fJ1 nm ,

NA NB

9= I I' A

fJ1 nm. (40) n=O m=O

The "prime" at the double sum indicates that the term with n=O and m=O is excluded from the summation. Since the expansion for S given above contains exclusively commutators and since the cluster operator T is con­nected, the operator S must be connected as well.

The expression for E~l can be rewritten as45

(41)

Analogously, the expectation value of the single-exchange operator P in the state IJI 0 = IJI A IJI B= eT <1> can be repre­sented as (eS"le-TpeT). By using Eq. (33) and performing simple manipulations, expression (29) can be transformed in the following way:

(IJI AIJI B 1 VPIJI AIJI B)

(IJI AIJI BIIJI AIJI B) (e-SeTt Ve-TteS"1 eS"te-TPeTe-S\

(42)

The "ket" on the right-hand side of Eq. (42) can be fur­ther transformed using the following identity valid for an arbitrary operator X acting in JY'A ® JY' B:

X<1>= (X) <1> + 9 (X) <1>, (43)

where (X) denotes the average value of an operator X calculated with the unperturbed wave function <1>, i.e., (X) == (<1> 1 X<1». Using Eqs. (43), the right-hand side of Eq. (42) can be written as

(eS"te-TVeTe-st) (eS"te-TPeTe-st)

+ (e-SeTt Ve-TteS" 1 9 leS"te-TpeTe-S\ A A

where we use the notation (X 1 9 1 Y) == (X 1 9 ( Y) ). The first, disconnected term in the above sum is equal to the product

R 1) (IJI AIJI B 1 PIJI AIJI B) pol (IJI AIJI B IIJI AIJI B)

and cancels out when the difference in Eq. (28) is taken. Thus, the first-order exchange energy can be expressed in the form:

E~~~h(S2) = <e-seTt Ve-TteS"1 9 1 eS"te-TPeTe-S\ (44)

Equation (44) is the main result of this section. Together with Eq. (39), giving S as a function of T, this equation expresses the exchange energy E~~~h(S2) in terms of the exact cluster operators T A and T B of the monomers. Equa­tion (44) can be expanded in powers of Sand T using the (finite) nested commutator expansion

-T TIl e Xe =X + [X,T] +21 [X,Th+3! [X,Th+ ... ,

(45)

which shows that the exchange energy E~~~h(S2) can be expressed entirely through commutators. Since a commu­tator of two second-quantized operators contains only con­tracted (connected) terms (diagrams), the first-order ex­change energy of Eq. (44) must be connected. This means that the individual terms resulting from the expansion of Eq. (44) (into a sum of products of cluster amplitUdes

t~~;::: and I!::::, a two-electron integral v1~~" and a ma­trix element P;~v') cannot be written as a product of two factors having no common index, i.e., cannot be repre­sented by disconnected diagrams. Actually, a somewhat stronger result holds. If ~v, is expressed as a (discon­nected) product of overlap integrals ~v, = -S~,S;, , then the considered term remains connected, i.e., cannot be written as a product of two factors with no common index. To see why this cannot happen, we note that an arbitrary term resulting from the expansion of Eq. (44) can be

written as A~'~v,B~', where the factors A~' and B~' are themselves connected. Substituting the overlap integrals for ~v" we find that this general term is equal

to - A~'~,S;,B~', i.e., remains connected. Thus, discon­nected (size non extensive ) objects do not appear at the intermediate levels of the theory and do not have to be eliminated. This feature of our approach not only simplifies an order-by-order perturbation theory expansion, but also allows a nonperturbative treatment of intramonomer cor­relation effects via size-extensive infinite-order (in WA and W B) selective summation techniques.

In Sec. III, we consider in detail the intramonomer correlation corrections through the second order in W= WA+ W B to the first-order exchange energy, as well as some infinite order coupled-cluster type approximations.

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1318 Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects

We show how these corrections can be expressed in terms of the cluster operators T A and T B of the isolated mono­mers and present final formulas in terms of molecular in­tegrals and cluster amplitudes. Since according to Eq. (26) the operator V is expressed in terms of Coulomb integrals, and the operator P in terms of overlap integrals, Eq. (44) gives E~~~h(S2) in terms of only Coulomb and overlap in­tegrals (the terms Coulomb, exchange, and hybrid inte­grals refer here to the two-electron integrals of the type abba d aa '1) Vab' Vab' an Vab' respective y .

III. MANY-BODY PERTURBATION THEORY EXPANSION OF THE FIRST-ORDER EXCHANGE ENERGY

To derive the many-body perturbation expansion of the first-order exchange energy, it is useful to introduce the following parametrization of the cluster operators T and S,

(46)

and

(47)

The operators TA(AA) and T B(AB) are the exact cluster operators for the molecules A and B, respectively. They can be obtained by solving the coupled-cluster equations for the monomers A and B with Hamiltonians FA +AAWA and F B+ABW B, respectively. Since the cluster operators are now functions of A A or A B, these operators can be expanded as series in AA or AB,

(48)

SA(AA) = L A~S~k), (49) k=1

where T~k) , k= 1,2, ... , are the consecutive MBPT correc­tions to the exact cluster operator of the monomer A.44,46

Note that AA and AB are dummy parameters introduced only to define the perturbation expansion, and their phys­ical values are equal to 1. It follows from Eq. (39) that the operator S(AA,AB) is uniquely defined by T(AA,AB)' It has been shown in Ref. 44 that the difference between Sand T appears only in the third and higher orders in W A or W o, i.e., S~I)=T~I) and S~2)=T~2). Inserting Eqs. (48) and (49) and similar expansions of the operators T Band So into Eq. (44), one obtains explicit formulas for the corrections E( lij) exch •

In the following, we shall also use double-superscript energy corrections ~~~h' defined generally by

They represent the intramonomer correlation effect of the lth order in the total correlation operator W = W A + W o.

Since all the corrections appearing in the following devel­opments are defined in the S2 approximation, we will drop from now on the symbol indicating this approximation.

A. Hartree-Fock exchange interaction energy ~~OJh

When the intramonomer correlation is completely ne­glected, the exchange energy is approximated by the cor­rection of the zeroth order in WA and W 0, i.e., by E~~~( This correction represents the exchange interaction of mol­ecules described at the Hartree-Fock level of approxima­tion. From Eq. (44), we immediately find that

",,(10) h

.eexch = (VI P), (50)

where

A A A A.

P= f!JJ 10(P) + f!JJ 01 (P) + f!JJ II (P). (51 )

Using Eq. (23), the operators f) nm(P), denoted from now on by Pnm , can be expressed through overlap integrals, i.e.,

(52)

where

(53)

(54)

(55)

Using these definitions, the Hartree-Fock exchange inter­action energy E~~6 can be written as

(10) h h h h

Eexch = (VP) = (VPIO ) + (VPOI ) + (VPl1 ). (56)

To express the expectation values ofEq. (56) in terms of one- and two-electron integrals, one can apply subse­quently Eqs. (15)-(17), the factorization rule

A.IA. """A. and the fact that the density matrices (aK /"K n) and I 2 n

(u:,1~2"",,~m) are equal to antisymmetrized products of Kro-12m

necker deltas when all indices refer to occupied spin orbit-als and are equal to zero otherwise. After performing the necessary spin integrations, this procedure leads to the fol­lowing expression for E~!~6:

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Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects 1319

E~~~ = - 2 (a> B) ~~ - 2 (a> A )~S~ - 2v~uS'~ , (58)

where (a> B)7= (v B)7+2v7i denotes a matrix element of the complete electrostatic potential a> B of molecule B calcu­lated using the HF density matrix. Note that expression (58) for the Hartree-Fock exchange energy E~~~~ is for­mally very similar to the formula for the polarization cor­rection If:.20) =E.20 ) +E(~O)

pol md dlSP'

where €k denotes the orbital energy corresponding to the orbital labeled by the index k, and differs considerably. from the standard expression (see, e.g., Ref. 31)

",(10)_ [~rJJ c:rza' c:rz'a c..a :-:b'b :-:bb' .nexch - -2 vab+,)a,(2vab -vab ) +.Jb,(2vab -vab )

(60)

where

k k' +S/S/' Vol (NAN B)'

Our formula for the correction ~~~~ shows that the ex­change energies can be expressed exclusively through the Coulomb integrals, monomer cluster amplitudes, and over­lap integrals. The summation ranges are longer (since these summations now include the virtual orbitals) than in the standard expression of Eq. (60), but the use of Eq. (58) does not require the time-consuming four-index transformation of exchange and hybrid integrals. In fact, the correction E~~~ can be obtained, practically at little extra cost, as a byproduct in a calculation of the Hartree­Fock induction and dispersion energies.31

B. First-order Intramonomer correlation correction E11)

exch

The leading intramonomer correlation correction to ~~~~ is of the first order in Wand is given by E~~6 ' i.e., by the sum of If:.~~) and ~~~~) . Since the appropriate expres­sion for E~~tj is obtained from the expression for E~~~~) by interchanging symbols pertaining to molecules A and B, below we consider only the correction E~~~~) .

Keeping in Eq. (44) only linear terms in T and in S one can easily find that

~~~~)= (V19 ([P,Ti6'] »+ ([ v,TM)] 19(P», (61)

where we used the fact that S(k) = T(k) for k<.2. By T<JJ)

we denote the n-particle part of the T~k) operator of Eq. (48), or equivalently, the kth order part of the n-particle cluster operator T nO defined by Eq. (31). Note that in the triple perturbation theory approach of Ref. 31, the opera­tors T(k) and T(k) were denoted by T(OkO) and T(OOk) nO ~ nO ~,

respectively. The first term ofEq. (61) can be transformed using the "Hermiticity" of the projection superoperator

A A

(X I&'( Y) ) = (&' (X) I Y) (62)

A (I) A

and rewritten as (VI [P,T20 D, where the operator V is defined as

V=91O(V)+901(V)+911( V)= VIO+ VOl + VII, (63)

and given explicitly by

(64)

(65)

(66)

with (a> B) ~ = (VB) ~ + v~. Thus, the final expression for E~~~~) now reads

(67)

Employing now the explicit form of the first-order cluster operator46

(68)

evaluating the commutators using the multiplication and factorization rules described in Sec. III A, and performing the spin integration, one obtains

E ( 1 10) £laa' ( ) r f;'r' rJJ £lrr' a's f;'O b exch = -4 Re Urr' a>B tt-'b')·a,-2uaa,vr'tJ-' sSr

2£laa' rs f;'r'Sb - 0", Vab"-'s a"

where

(69)

(notice a factor of 2 difference compared to the definition of Ref. 31), and the first-order spin-free double excitation

amplitUde t':;.: is given by

Expression (69) is structurally similar to the formula for E(~IO) 47 so the correction E(1IO) can be coded by modify-dlsp , exch ing the program for the calculation of E(?IO) 31,47 dlSp •

C. Second-order intramonomer correlation correction E!,~~h

The intramonomer correlation correction of the second-order in W= WA+ WB is given by E~~~~

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1320 Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects

=E(111)+E(120)+E(102) Since the expression for E(l02) exch exch exch' exch

can be obtained from the expression for E~~;~) by inter~ changing symbols pertaining to molecules A and B, it is sufficient to consider only the bilinear term E~!~~) and the

d . . W . E(120) term qua rabc III A, 1.e., exch'

1. Bilinear term Ei~~~

Collecting the terms of the first order in W A and W B in the expansion of Eq. (44), we can write

E(111)- ([ [V T(I)] T(1)] I & (P» exch- '20' 02

+ ([ v,Ti~)] I & ([p,TM)]»

(70)

In view of Eqs. (52), (62), and (63), the final expression for the correction E~~~~) can be rewritten as

Other terms in Eq. (44) do not contribute since they gen­erate a different rank excitation in bra vs keto In Eq. (73), we have replaced all the S operators by the T operators since, as shown in Ref. 44, S(I)=T(I) and S(2)=T(2). As in Ref. 27 we introduce the following partitioning of E I20) exch ,

(74)

where K{(A) and IG.(A) collect terms linear in TW and Ti~) , respectively, while K;' I CA) is quadratic in Tio) . The final commutator expressions for the components IG (A) and K{(A) can be found from Eq. (73) using Eqs. (52), (62), and (63). They have the form

A (2) (2) A

K2(A)=(VI [P,T20 ])+([V,Tzo ] IP), (75)

(76)

The formula for K'{ I can be written as

(71)

Evaluating the commutators and performing the integra­tion over spin coordinates, one finds that

Expression (72) for the correction E~~~~) is similar to the formula for E(211) so the bilinear term E(lll) can be ob-

disp , ex~~

tained as a by-product in a calculation of E~lis~1) .31,47

2. Quadratic term ~~cOJ

Again using Eq. (44), the expression for the correction E( 120) can be written as exch

(73)

(1) A A A A (1) KfI(A)=([V,T20 ] 1&10+&11+&20+&211 [P,T20 ])

- (VI [[P,T~)],Ti~)t])

(77)

The last two terms in this equation can be further trans­formed to a computationally convenient form to obtain the following alternative expression for K'{ 1 :

(I) A A A A (I) Kfl (A) = ([ V,T20 ] 1& 10+ &' 11 + &' 20+ &' 211 [P,T20 ])

+ (Ti~) I [( v,Ti~)],P] > + (V01Ti~) I [p,Ti~)] >

(78) Using the rules described in Sec. III A, it can be shown that the orbital expressions for the components IG(A), K{(A), and KfI(A) can be written as

(2) (Z)

IG(A) = -4 Re ():::' (wB):;S~S!,-2 () :'V~:PsS~ (2)

aa' rs f;!r'Sb - 2 () IT' Valt-' s a"

(2) (2)

K{CA) = -2 t rr-2 t ~G~,

(79)

(80)

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Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects 1321

where the intermediate quantities appearing in Eqs. (79)­( 81) are defined by the equations

(2) (2) (2) (2) e aa' = ( err' )*=2 t (2)aa' _ t (2)a'a

IT' aa' IT' rr"

(82)

(83)

(84)

(86)

aa' b r" ,..b a af I' baa' , baa" b a" a' TIlT's =Ss ~;., tIT" +Ss Sr tr"r'-S<; Sa" tTT' -S<;Sa" trr, ,

(87)

An a'sO"' ab=Vr'b aa" (88)

b Ish £ra' ~ =Ss a' IT' ' (89)

Ish ~a' X<:=Sb a' TT' ' (90)

a' rr' ~=«(llO)r'eaa" (91)

The second-order spin-free mono- and double-excitation (2) a (2) aa'

monomer cluster amplitudes t r and t rr" respectively, needed for the evaluation of /G(A) and K{(A) are given by

and

a'r" fi a" -Va"r lr"r' ]/(Ea+Ea,-Er-Ert).

The expressions for the components /G(A), K{(A), and K~'t (A) are similar to the formula for the correction ~f5~)' Note, that some of the intermediate quantities de-

(81)

fined by Eqs. (82)-(91) are needed to compute the E~~O) correction, while others may be obtained in a calcu­lation of this correction by performing additional summa­tions using already available integrals.31 Specifically, the component K{ (A) can be computed as a byproduct in a calculation of the single-excitation part of the correction E~~O). Similarly, the sum /G(A) +~l (A) can be obtained as an additional quantity in a calculation of the double-, triple-, and quadruple-excitation parts of E~t~O).

D. Coupled-cluster approximations

Numerical results through the second order in WA and W B obtained thus far13,27,33 show that the convergence of the many-body perturbation expansion of the exchange en­ergy is only moderately fast. Therefore, to obtain very ac­curate results, it may be necessary to resort to nonpertur­bative techniques, i.e., to a summation of certain classes of diagrams through infinite order in the correlation opera­tors WA and Wo.

The simplest way of performing such infinite-order summations in W is to approximate the energy E~~~h by a first few terms of the commutator expansion of Eq. (44) in which the exact cluster operators T A and To are replaced by the converged operators of the coupled-cluster singles and doubles (CCSD) theory. 48 Keeping the linear and the most important quadratic terms in T A and Toone can define in this way the following approximation for E~~~h :

E~~~h(CCSD) =E~~:) +/G(CCSD) +K{(CCSD)

+KlI (CCSD) +~t(CCSD), (92)

where the last four successive terms on the right-hand side denote the expressions ofEqs. (75), (76), (78), and (71), respectively, in which the first- and second-order cluster operators are replaced by the converged CCSD operators. The expressions for /G (CCSD), K{ (CCSD), and Kll (CCSD) include the total correlation effect, i.e., the sum of the components coming from the correlation in the monomers A and B, e.g., the expression for K{(CCSD) is the sum of the expressions for K{(A) and K{(B).

For cluster operators involving two-particle and higher excitations an inclusion of higher multiple commutators in Eq. ( 44) is computationally difficult, and was not at­tempted. However, when T is replaced by the sum of one­particle cluster operators T= T lO+ TOI , then all multiple commutators in Eq. (44) are summed up to infinity by the following simple expression resulting directly from Eq. (28):

-(I) - - - - - - --Eexch= (<I> A<t> B I VP<I> A<I> B) - (<I> A<t> 01 V<I> A<t> 0)

X (cI> AcI> 0 I P<I> AcI> 0) , (93)

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1322 Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects

TABLE I. Geometries of complexes and basis sets used in this paper. All the y and z coordinates are zero.

Complex

"Reference 50. ~eference 51. cReference 52.

Atom

Hel He2

HI HZ H3 H4

HI H2 He

where <I> A is defined as

eT\O<I> A

x

0.000 R

0.000 1.4 R

1.4+R

0.000 1.4

0.7+R

Basis set

7s5p .. b

5s2pC

(Hh 5s2pc (He) 7s5p .. b

(94)

and a similar formula holds for <I> B' If the cluster operators TIO and TOl are obtained from the CCSD theory, the cor­relation part of E~~~h will be denoted by i{ (CCSD), i.e.,

-, _ ~1) (10) K2 (CCSD) -I!,exch-Eexch'

This notation is motivated bt the fact that the leading, linear in TIO+Tol , term in K{(CCSD) is then equal to K{(CCSD). Expression (93Us very easy to cal£ulate in practice since the function <I> A (and similarly <I> B) is a Slater determinant for monomer A in which the occupied Hartree-Fock spin orbitals ¢A are replaced by the corre­lated spin orbitals (1 + T IO )¢! defined by the TIO operator. Since correlated orbitals of this form are identical up to the second order in the fluctuation potential with the exact Brueckner orbitals,49 the infinite-order contribution of the single-particle cluster operators to ~~~h can also be ob­tained by evaluating formula (93) with the Brueckner de­terminants.

IV. NUMERICAL RESULTS AND DISCUSSION

Numerical calculations have been performed on He2, (H2h, and He-H2 systems. Geometries and basis sets used in this study are specified in Table I. Since the aim of the calculations was a comparison with the reference full CI

TABLE II. Decomposition of the first-order exchange energy for the He dimer.

4.0

0.1753( -Z)" 0.Z713(-4)

-0.1310( -6) 0.3406 ( -4) 0.6106( -4) 0.1110(-3) 0.1115(-3)

5.6

0.3559( -4) 0.5318( -6)

-0.1339( -8) 0.1449( -5) 0.1980( -5) 0.3845( -5) 0.3914( -5)

"The expression (-N) denotes the factor of IO-N•

7.0

0.1072(-5) 0.1436( -7)

-0.1645( -10) 0.6605(-7) 0.8039(-7) 0.1594(-6) 0.1640( -7)

TABLE III. Decomposition of the first-order exchange energy for the Hz dimer.

4.0

0.ZZ51( -1)" 0.1112( -2)

-D.Z762( -4) -0.2898( -3)

0.7947 ( -3) 0.7699(-3) 0.8184(-3)

"See footnote a of Table II.

5.0

0.3982( -2) 0.Z497 ( -3)

-0.3772( -5) -0.7573(-4)

0.1702( -3) 0.1596(-3) 0.1668( -3)

6.5

0.2493 ( -3) 0.1786(-4)

-0.1335(-6) -0.7235( -5)

0.1049(-4) 0.8234( -5) 0.8354( -5)

data, and since the full CI calculations with large basis sets are not possible at the moment, we employed basis sets of relatively modest size. All calculations have been done us­ing the dimer-centered basis sets. Atomic units are used throughout this paper (the units of distance and energy are bohrs and hartrees, respectively).

The computer codess3 for the correlation corrections to the first-order exchange energy have been written using either the expressions derived in this paper or the alterna­tive ones, based on the interaction density matrix formal­ism given in Ref. 33. In the density matrix approach the final expressions for the corrections ~~~{) involve not only Coulomb integrals, but also exchange and hybrid integrals. The alternative expressions for the exchange corrections were coded independently and gave results identical to those obtained using the expressions derived in the present paper. This gives us confidence in the correctness of our codes for these rather complicated expressions.

The computer codes developed by us are completely general and can be applied to study interactions of systems with arbitrary number of electrons. In the present paper, we present results for four-electron systems, for which the performance of our method can be tested by comparison with accurate reference· data obtained using the full CI method. Applications to larger systems have already been performed and are being published separately.33,s4,ss

In Tables II-IV we present numerical results of per­turbative calculations through the second order in WA+ W B, as well as the results of coupled-cluster calcu­lations using Eq. (92). To demonstrate the convergence properties of the perturbation series for various dimers, we have tabulated consecutive corrections E!~:6, the sum of

TABLE IV. Decomposition of the first-order exchange energy for the He-Hz system.

4.0

0.8272( -Z)" 0.2348( -3)

-0.Z092( -5) -0.2707( -4)

0.2056 ( -3) 0.2905(-3) 0.2991( -3)

"See footnote a of Table II.

5.0

0.1111(-2) 0.3378( -4)

-0.2130( -6) -0.7333( -5)

0.2623 ( -4) 0.3791(-4) 0.3962 ( -4)

6.5

0.4988( -4) 0.1434(-5)

-0.4863 ( -8) -0.5722 ( -6)

0.8573(-6) 0.1172( -5) 0.1269(-5)

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Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects 1323

the series truncated after the kth order, the correlation part of the CCSD exchange energy

E~~h(CCSD) =~~~h(CCSD) -E~~~6, (95)

and the full CI (FCI) exchange-correlation energy

E~~~h(FCI) =~~~h(FCI) -~~~6. (96)

The importance of the correlation corrections to E~~~6 grows with the increase of the intermolecular distance R. For all the complexes studied in this paper, the intramono­mer correlation effect is repulsive and represents approxi­mately 5% of E~~6. For interaction of many-electron sys­tems, the importance of the exchange-correlation energy varies with the system. For the Ar-H2 (Ref. 54) and He-HF (Ref. 55) interactions, this effect is quite impor­tant and amounts to 15% and 22% of the total interaction energy at the van der Waals minimum, respectively. For the He-F- (Ref. 13) system, the exchange-correlation en­ergy is of the same order of magnitude as the total inter­action energy. A more detailed discussion of the exchange­correlation effects in the interactions of many-electron systems is presented in Ref. 33.

Our FCI results for the He dimer are in very good agreement with those of Ref. 23. The agreement with the data of Refs. 24 and 25 is somewhat worse (our exchange energies could not be compared with the results of the calculations of Rijks et al. 26 since these authors did not separate the exchange and electrostatic contributions of their first-order energy). The results of Ref. 25 obtained using a correlated Gaussian geminal basis agree very well with the results of recent calculations24 utilizing large or­bital basis sets. These comparisons show, as has been also demonstrated in Ref. 24, that the correlation part of ~!~h converges rather slowly with the increase of the basis set. One should also mention that it is possible that for the largest distance considered here R = 7, our result is more accurate than that of Ref. 25. The results of Ref. 25 have been computed using a very large basis set of correlated Gaussian geminals and, at least at the van der Waals min-

TABLE V. The structure of E.~~ and E~~h(CCSD) corrections for the He dimer.

4.0 5.6 7.0

E.~~~ 0.2713( -4)" 0.5318( -6) 0.1436(-7)

~ 0.3609 ( -5) 0.6683( -7) 0.1832( -8)

E.~~+~ 0.3074( -4) 0.5987( -6) 0.1619( -7) ~(CCSD) 0.2994( -4) 0.573l( -6) 0.1517( -7) K{ 0.3304( -4) 0.1323( -5) 0.6018(-7) ~(CCSD) 0.8336( -4) 0.3137( -5) 0.1367(-6) K{(CCSD) 0.8455 ( -4) 0.321l( -5) 0.1594( -6)

Kf, -0.2592 ( -5) 0.5970(-7) 0.4036(-8) Kf,(CCSD) -O.2180( -5) 0.1366( -6) 0.7520(-8) ~II')

exch -0.1310(-6) -0.1339( -8) -0.1645(-10) K:f(CCSD) -0.1735( -6) -0.1723( -8) -0.2036( -10)

E.~~~ + E.~;~ 0.6106(-4) 0.1980( -5) 0.8039(-7) E~(CCSD) 0.1110(-3) 0.3845( -5) 0.1594( -6) E~(FCI) 0.1115( -3) 0.3914( -5) 0.1640(-7)

"See footnote a of Table II.

TABLE VI. The structure of F~;~ and E~~h (CCSD) corrections for the H2 dimer.

4.0 5.0 6.5

E.~~~ 0.1112( _2)" 0.2497 ( -3) 0.1786( -4)

K2 0.2537( -3) 0.6303 ( -4) 0,4820( -5)

E.~l~+K2 0.1366(-2) 0.3128( -3) 0.2268(-4) ~(CCSD) 0.1487( -2) 0.3473 ( -3) O.2550( -4) Kf

2 -O.2846( -3) -O.9330( -4) -0.9209 ( -5) ~(CCSD) -0.2672 ( -3) -0. 1073 ( -3) -0. 1196( -4) K{(CCSD) -0.2793 ( -3) -O.1078( -3) -O.1185( -4)

K't, -O.2590( -3) -O.4546( -4) -0.2845 ( - 5) K't,(CCSD) -O.3885( -3) -O.7202( -4) -0.5012( -5) ~II')

exch -0.2762( -4) -0.3772( -5) -0.1335( -6) K:f(CCSD) -0.6100( -4) -0.8300( -5) -0.2939( -6)

F~~~+E.~~~ 0.7947(-3) 0.1702( -3) 0.1049(-4)

E~~~h (CCSD) O.7699( -3) O.1596( -3) O.8234( -5)

E~~~h(FCI) 0.8184( -3) 0.1668( -3) 0.8354( -5)

"See footnote a of Table II.

imum, are expected to be very close to the basis set satu­ration limit. However, since only monomer-centered gem­inal basis sets have been used, for sufficiently large R the accuracy must deteriorate. It is clear that the exchange energy is very sensitive to the tails of the wave functions in the region far from the nucleus, and that basis sets giving very accurate monomer energies may not be sufficient to obtain accurate values of E~!~h at sufficiently large inter­monomer distances.

Our FCI exchange-correlation energy for the H2 dimer is in good agreement with the results of Chalasinski52 ob­tained using various monomer-centered basis sets with bond functions, i.e., basis functions localized at the center of mass of the H2 molecule. At larger intermolecular dis­tances, where this agreement deteriorates, our results are probably more accurate, again due to the use of the dimer­centered basis set vs the monomer-centered one.

The results reported in Tables II-IV show that the convergence of the many-body perturbation expansion of the first-order exchange energy is only moderately fast. For He2 and He-H2 systems, the sum of the first two terms reproduces only 50% and 70% of the FCI result, respec­tively. Note, that the leading (first-order) intramonomer correlation correction E~!~6 is smaller than the next term, i.e., the second-order correction E~!~6. The smallness of E~~~~ may be related to the fact that the corresponding polarization correction E~:) vanishes on account of the Brillouin theorem, but is difficult to explain by inspection of Eq. (67). In contrast to these results, the convergence properties of the many-body perturbation theory expansion of the exchange energy for the H2 dimer are quite satisfac­tory. The sum of corrections through the second order in WA + W B reproduces the FCI exchange-correlation energy with an error of 3% and -2% for R=4 and 5 bohr, respectively. Only for R = 6.5 bohr this rapid convergence deteriorates and the sum of perturbation series overesti­mates the FCI result by 25%. The inclusion of the higher­order terms by means of the CCSD approximation [Eq. (92)] improves the situation considerably for the slowly convergent cases and f!;~h ( CCSD) recovers the full CI

J. Chern. Phys., Vol. 100, No.2, 15 January 1994

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1324 Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects

TABLE VII. The structure of ~~;~ and €~~~h (CCSD) corrections for He-H2 system.

~~~6 ~ ~~~~+~ ~(CCSD)

K{ ~(CCSD) K{(CCSD)

Kfl KfI(CCSD) E III )

exch K'tt(CCSD)

~~~~+~~;~ €~~b(CCSD) €~~~h(FCI)

4.0

0.2348 ( _3)" 0.5030( -4) 0.2851( -3) 0.3038( -3)

-0.2748( -4) 0.5416( -4) 0.5343( -4)

-0.4989( -4) -0.6382( -4) -0.2092( -5) -0.3635( -5)

0.2056( -3) 0.2905(-3) 0.2991( -3)

'See footnote a of Table II.

5.0

0.3378( -4) 0.7417( -5) 0.4119( -4) 0.4417(-4)

-0.8988( -5) 0.2183( -5) 0.1906( -5)

-0.5761( -5) -0.8071( -5) -0.2130( -6) -0.3692( -6)

0.2623(-4) 0.3791( -4) 0.3962( -4)

6.5

0.1434( -5) 0.3110( -6) 0.1745( -5) 0.1862( -5)

-0.6837( -6) -O.3306( -6)

-0.3552( -6) -0.1994(-7) -0.3505( -7) -0.4863 ( -8) -0.8166( -8)

0.8573( -6) 0.1172(-5) 0.1269( -5)

exchange-correlation energy to within few percent for all the systems considered at all the distances. This result is very gratifying since the inclusion of multiple commutators involving single- and double-excitation cluster operators in the explicitly connected expansion of E~~~h would be very difficult.

In order to get still more insight into the convergence properties of the many-body perturbation series for E~~~h' in Tables V -VII we display results showing the structure of the E~~;6 and €~~~h C CCSD) corrections. For the He dimer, the largest contributions to these corrections are given by the K{ and K{CCCSD) components, respectively. These terms are at least one order of magnitude larger than the remaining ones. For the He-H2 system, the largest part of the exchange-correlation energy comes from the .A1CCCSD) component, while for the H2 dimer, the ex­change correlation energy results from a delicate balance of all terms entering the expressions for E~~6 + ~~;6 or €~~~hCCCSD). Note, that E~~~6 and .A1 are the first two terms in the many-body perturbation theory expansion of .A1 C CCSD ). Similarly, K {, IG l' and E~~~~) are the first terms in the many-body perturbation series of K{CCCSD), IGICCCSD), and K'ttCCCSD), respectively. An inspection of Tables V-VII shows that the sum ~~J~ +.A1 reproduces quite accurately the converged result .A1 C CCSD). The perturbative approximations to K{CCCSD), IGICCCSD), and K'ttCCCSD) are, however, less accurate. For the He dimer at R = 5.6 bohr, the com­ponents K{ C CCSD), IG 1 C CCSD), and K'tt C CCSD) are underestimated by approximately 58%, 56%, and 22%. For the He-H2 complex, the K{ approximation to K{ (CCSD) does not reproduce correctly even the sign of this component for some distances. These results can be rationalized by relating them to the number of terms in the many-body perturbation expansion of each component dis­cussed above. K{CCCSD), IGICCCSD), and K'ttCCCSD) are only the first terms in the corresponding many-body perturbation expansions, while .A1 C CCSD) includes two perturbation corrections calculated by us.

In Tables V-VII, we present also the results of calcu-

lations of the quantity i{CCCSD) ofEq. (93), containing the contributions of all multiple commutators involving the one-particle CCSD operators T 10 and T 01' These re­sults show that the total contribution of the single­excitation operators to the first-order exchange energy is very close to the linear term K{ C CCSD) and, in general, €(1)hCFCl) -€!~~CCCSD) is not dominated by K7c CCSD) - Ki C CCSD). Thus, the small differences be­tween the FCI and CCSD exchange-correlation energies are probably due to the neglect of the commutators involv­ing both single- and double-excitation operators. The effect of this coupling of single- and double-excitation operators can be investigated in practice by evaluating Eq. C 44 ) within the framework of the Brueckner coupled-cluster theory.56,57 Work in this direction is in progress in our group.

ACKNOWLEDGMENTS

This work was supported by grants KBN-2 0556 9101 and NSF CHE-9220295. We thank Dr. Paul E. S. Wormer for valuable discussions.

IR. Ahlrichs, Theor. Chirn. Acta 41,7 (1976). 2 A. van der Avoird, P. E. S. Wormer, F. Mulder, and R. M. Berns, Topics Current Chern. 93, I (1980).

3K. T. Tang, J. P. Toennies, M. Wanschura, and C. L. Yiu, Phys. Rev. A 46, 3746 (1992).

4C. Nyeland and J. P. Toennies, Chern. Phys. 122, 337 (1988). sc. Nyeland and J. P. Toennies, Chern. Phys. Lett. 127, 172 (1988). 6S. H. Patil, J. Chern. Phys. 89, 6357 (1988). 7S. Kita, K. Noda, and H. Inouye, J. Chern. Phys. 64, 3446 (1976). By. S. Kim, S. K. Kim, and W. D. Lee, Chern. Phys. Lett. 80, 574

(1981). 9K. Ch. Ng, W. J. Meath, and A. R. Allnatt, Chern. Phys. 32, 175

( 1978) . 10K. Ch. Ng, W. J. Meath, and A. R. Allnatt, Mol. Phys. 37, 237 (1979). II c. C. Kirkpartick and L. A. Viehland, Chern. Phys. 98, 221 (1985). 12 A. D. Koutselos, E. A. Mason, and L. A. Viehland, J. Chern. Phys.93,

7125 (1990). 13R. Moszynski, B. Jeziorski, and K. Szalewicz, Chern. Phys. 166, 329

(1992) . 14p. Arrighini, Intermolecular Forces and their Evaluation by Perturba­

tion Theory (Springer, New York, 1981). 15B. Jeziorski and W. Kolos, Int. J. Quantum Chern. 12, 91 (1977). 16B. Jeziorski and W. Kolos, in Molecular Interactions, edited by H.

Ratajczak and W. J. Orville-Thomas (Wiley, New York 1982), Vol. 3, p. 1.

17J. O. Hirschfelder, Chern. Phys. Lett. 1, 325 (1967) . IBB. Jeziorski, K. Szalewicz, and G. ChaIasinski, Int. J. Quantum Chern.

14,271 (1978). 19D. M. Chipman and J. O. Hirschfelder, J. Chern. Phys. 73, 5164

(1980). 2oT. Cwiok, B. Jeziorski, W. Kolos, R. Moszynski, and K. Szalewicz, J.

Chern. Phys. 97, 7555 (1992); J. Mol. Struct. (Theochern.) (in press ). 21B. Jeziorski, M. Bulski, and L. Piela, Int. J. Quantum Chern. 10,281

(1976); M. van Hemert and A. van der Avoird, J. Chern. Phys. 71, 5310 (1979).

22G. ChaIasinski, S. van Srnaalen, and F. B. van Duijneveldt, Mol. Phys. 45, 1113 (1982).

23G. ChaIasinski and M. Gutowski, Mol. Phys. 54, 1173 (1985). 24M. Gutowski, J. G. C. M. van Duijneveldt-van de Rijdt, J. H. van

Lenthe, and F. B. van Duineveldt, J. Chern. Phys. 98, 4728 (1993). 25S. Rybak, K. Szalewicz, and B. Jeziorski, J. Chern. Phys. 91, 4779

(1989). 26W. Rijks, M. Gerritsen, and P. E. S. Wormer, Mol. Phys. 66, 929

(1989).

J. Chern. Phys., Vol. 100, No.2, 15 January 1994

Downloaded 18 Oct 2012 to 128.143.23.241. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

Moszynski, Jeziorski, and Szalewicz: Many-body theory of exchange effects 1325

27p. Jankowski, B. Jeziorski, S. Rybak, and K. Szalewicz, J. Chern. Phys. 92, 7441 (1990).

21M. Gutowski, G. Chalasinski, and J. van Duijneveldt-van der Rijdt, Int. J. Quanturn Chern. 26, 971 (1984).

29S. Rybak, K. Szalewicz, B. Jeziorski, and M. Jaszunski, J. Chern. Phys. 86, 5652 (1987).

JOB. Jeziorski, R. Moszynski, S. Rybak, and K. Szalewicz, in Lecture Notes in Chemistry, edited by U. Kaldor (Springer, New York, 1989) Vol. 52, p. 65.

31 S. Rybak, B. Jeziorski, and K. Szalewicz, J. Chern. Phys. 95, 6576 (1991 ).

32K. Szalewicz and B. Jeziorski, Mol. Phys. 38, 191 (1979). HR. Moszynski, B. Jeziorski, S. Rybak, K. Szalewicz, and H. L.

Williams, J. Chern. Phys. (to be published). 34F. Coester, Nucl. Phys. 7, 421 (1958); J. Cizek, J. Chern. Phys. 45,

4256 (1966); Adv. Chern. Phys. 14, 35 (1969); J. Cizek and J. Paldus, Int. J. Quanturn Chern. 5, 359 (1971); J. Paldus, J. Cizek, and I. Shav­itt, Phys. Rev. A 5, 50 (1972).

35R. J. Bartlett, Annu. Rev. Phys. Chern. 32, 359 (1981); J. Phys. Chern. 93, 1697 (1989).

36 J. Paldus, in NA. TO A.S! Series B, Vol. 293, edited by S. Wilson and G. H. F. Diercksen (Plenum, New York, 1992), p. 99.

37M. Bulski, G. Chalasinski, and B. Jeziorski, Theor. Chim. Acta 52, 93 (1979).

38 J. Paldus and C. R. Sarma, J. Chern. Phys. 83, 5135 (1985); J. Paldus, M. J. Gao, and J. Q. Chen, Phys. Rev. A 35, 3197 (1987).

39J. Paldus and B. Jeziorski, Theor. Chirn. Acta 73,81 (1988). 4OW. Kutzelnigg, Chern. Phys. Lett. 83, 156 (1981).

41 J. Paldus, in NATO AS! Series, edited by G. L. Ma1Ii (Plenurn, New York, in press).

42 J. Paldus and J. Cizek, Adv. Quanturn Chern. 9, 105 (1975). 43D. R. Williams, L. J. Schaad, and J. N. Murrell, J. Chern. Phys. 47,

4916 (1967). 44B. Jeziorski and R. Moszynski, Int. J. Quanturn Chern. 48,161 (1993). 45R. Moszynski, B. Jeziorski, A. Ratkiewicz, and S. Rybak, J. Chern.

Phys. 99, 8856 (1993). 46H. J. Monkhorst, B. Jeziorski, and F. E. Harris, Phys. Rev. A 23, 1639

(1981). 47R. Moszynski, B. Jeziorski, and K. Szalewicz, Int. J. Quanturn Chern.

45,409 (1993). 48G. D. Purvis III and R. J. Bartlett, J. Chern. Phys. 76, 1910 (1982). 49L. Z. Stolarczyk and H. J. Monkhorst, Int. J. Quanturn Chern. Syrnp.

18, 267 (1984). 5°F. B. van Duijneveldt, IBM Research Report RJ945, 1971. 51 G. Chalasinski, S. van Srnaalen, and F. B. van Duineveldt, Mol. Phys.

45, 1271 (1982). 52G. Chalasinski, Mol. Phys. 57, 427 (1986). 53B. Jeziorski, R. Moszynski, A. Ratkiewicz, S. Rybak, K. Szalewicz, and

H. L. Williams, in Modern Techniques in Computational Chemistry: MOTECC-93, edited by E. Clernenti (Escorn, Leiden, in press).

54H. L. Williams, K. Szalewicz, B. Jeziorski, R. Moszynski, and S. Ry-bak, J. Chern. Phys. 98, 1279 (1993).

sSR. Moszynski et al. (to be published). s6R. A. Chiles and C. E. Dykstra, J. Chern. Phys. 74, 4544 (1981). s7N. C. Handy, J. A. Pople, M. Head-Gordon, K. Ragavachari, and G.

W. Trucks, Chern. Phys. Lett. 164, 185 (1989).

J. Chern. Phys., Vol. 100, No.2, 15 January 1994

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