An analytical investigation into the bsse problem

Journal of Molecular Structure (Theochem), 227 (1991) 43-65 Elsevier Science Publishers B.V., Amsterdam

43

AN ANALYTICAL INVESTIGATION INTO THE BSSE PROBLEM*

I. MAYER

Central Research Institute for Chemistry of the Hungarian Academy of Sciences, H-1525 Budapest, P.O. Box 17 (Hungary)

L. TURI

Department of Chemists, Laboratory of Theoretical Chemistry, Roland Eiitviis University, H-1518 Budapest 112, P.O. Box 32 (Hungary)

(Received 2 January 1990)

ABSTRACT

Analytical and simple numerical model studies are performed to clarify some aspects of the basis-set-superposition-error (BSSE) problem in the theory of intermolecular interactions. It is deduced from the requirement of the energy to be real and have appropriate numerical values that whatever wavefunction is used, the energy should be determined as the conventional expectation value of the total Hamiltonian. In other words, one must not utilize expressions of interaction energy from which terms vanishing in the case of an exact description of the individual monomers have been dropped. This conclusion may have an important impact on the “exchange perturbation theory” and can serve as the firm theoretical justification of the chemical Hamiltonian approach with conventional energy (CHA/CE) scheme introduced recently, as giving the conceptually best interaction energy for the given monomer basis sets.

The interaction energy given by the usual Bosy-Bernardi (BB) scheme contains two types of (minor) extraneous terms, having opposite signs. Accordingly, the BB scheme can give both “un- dercorrected” and “overcorrected” interaction energies; it is also expected to overestimate slightly the equilibrium intermolecular distance.

INTRODUCTION

It is well known that “straightforward” quantum-chemical calculations of intermolecular interactions suffer from the so-called basis-set-superposition error (BSSE), leading to overestimated stabilization energies for the weakly bonded complexes. This purely mathematical “phenomenon” is due to the approximate wavefunctions (finite basis sets) used for the individual constituent molecules (“monomers”). When the complex (“supermolecule”) is calculated, the basis functions of one molecule are utilized to some extent for im-

*Dedicated to Professor Rudolph Zahradnfk.

0166-1280/91/$03.50 0 1991- Elsevier Science Publishers B.V.

44

proving (in the energetic sense) the description of another monomer, as well. Therefore the supermolecule wavefunction corresponds to lower intramolecular monomer energies than those calculated for the free monomers in their own monomer basis. The BSSE is due to this imbalance between the approximations used for the supermolecule and for the free monomer. Accordingly, the BSSE could be avoided in the standard scheme of calculations only by using practically saturated basis sets; this is usually not practicable, however.

As the BSSE distorts the estimated intermolecular energies, some correction must be introduced when the basis is not large enough to be saturated. This correction may be either of a posteriori character, when one tries to estimate how large the influence of the BSSE is on the actual supermolecule energy, or it may be of an a priori character, when one introduces some modifications in the supermolecule calculation in order to prevent the appearance of BSSE effects. The previous scheme is realized in the “counterpoise correction” or Boys-Bernardi (BB) method [ 1,2] while the latter is attempted in the so- called chemical Hamiltonian approach (CHA) introduced recently [ 3-71. (Both schemes may have some variants.)

The problem which one encounters when dealing with any BSSE correction scheme is that the “BSSE-free intermolecular interaction energy corresponding to a given basis set” does not represent a well-defined concept. Our physical intuition may give a qualitative idea about it, but there is no strict mathematical formulation behind this quantity. The aim of the present paper is to discuss this problem in some detail by using very simple analytical models and numerical examples. The two-electron models we consider are the simplest ones in which effects like overlap repulsion and delocalization caused by the BSSE and true interactions appear simultaneously. Accordingly, these models are somewhat more complicated than the one-electron-two-orbitals model discussed previously [ 5,8]. The models permit a deeper understanding of the BSSE problem to be obtained. As will be seen, the results of this analysis give a fur- ther confirmation of the idea on which the most recent “CHA/CE” and “CHA/ F” schemes [ 5-71 are based: the best BSSE-free interaction energy is obtained if one determines the supermolecule wavefunction by preventing the delocalizations caused by the BSSE effects and then calculates the conventional expectation value of the energy. (CHA/CE= CHA with conventional energy; CHA/F = CHA applied to the Fockian. There is only a slight difference between these two methods: the same philosophy is used either at the level of the whole Hamiltonian or directly to the Fockian; in the SCF case they give practically coinciding results. Both schemes apply the conventional energy expectation value, instead of the non-symmetric CHA energy formula used previously (in refs. 3 and 4). In refs. 5-7 different examples are shown, indicating how the CHA/CE or CHA/F methods solve the BSSE problem at the SCF level of theory. )

45

ANALYTICAL AND NUMERICAL MODEL STUDIES

3 T X2 T A B

Fig. 1. The two-electrons-two-orbitals model.

x2 viz-t.

‘Y4 uirt.

3 T - occ. X3 T- occ.

A B

Fig. 2. The two-electrons-four-orbit& model.

We consider here models in which one or two basis orbitals Xi are centred on each “molecule” A and B (Figs. 1 and 2). The orbitals are assumed normalized but, in general, not orthogonal; their overlap integrals are Sij= (xi 1 a). In both models there are two electrons of the same (a) spin, so we are dealing with a triplet state. For the sake of simplicity we consider explicitly one-electron interactions only in our analytical models; the Hamiltonian

~=~+~A+~JB,~A+~JB,~B+~A (1)

can be considered, for instance, as some effective one-electron Hamiltonian. In eqn. (1) 2? is the operator of kinetic energy, 0’ and uB are those of the potential energy due to the nuclei in A and B, respectively, LA= $+ oA and hB= p+ 0’ are the intramolecular Hamiltonians of the individual monomers. The matrix elements of these operators are denoted hij = (xi 1 h 1% ) , h$ = (XiIhAl*)9 u$=tXi16Al&),e~-

Interaction energy for unperturbed monomer orbital

In the case of weak interactions the constituent molecules conserve their individuality to a great extent and some important effects can be accounted for by considering a supermolecule wavefunction obtained as a properly antisymmetrized product of the free monomer wavefunctions: using this wavefunction one should recover some of the leading terms of the true interaction energy. In particular, the very important phenomenon of the overlap repulsion should be described qualitatively well at this level corresponding essentially to the first order of a usual perturbation theory.

For the simplest model shown in Fig. 1, the wavefunction obtained by anti- symmetrizing the unperturbed monomer ones is

(2)

where A denotes the antisymmetrizer, and ri and oi are the spatial and spin coordinates of the ith electron, respectively. As the orbitals x1 and x2 are not orthogonal, the determinantal wavefunction, Y, is unnormalized.

The energy of the system [expectation value of the Hamiltonian hfl, =

i h(i)] is given by i=l

(3)

Using the different possible decompositions (eqn. (1) ) of the Hamiltonian, one obtains from eqn. (3) the energy change, AE, with respect to the sum htl + hF2 of the unperturbed monomer energies

hE=(1-)s,2(~)-‘[u~~-s,2u~~+u~2-s2,~~2

-s,z(G -s,,w -S21M2 -s,,m 1 (4)

An interesting aspect of this expression is that the terms containing matrix elements h$ and h; vanish, if x1 and x2 are exact monomer orbitals, i.e. exact eigenfunctions of the respective monomer Hamiltonians: hAxl (r ) = h$xl (r ) and hBx2(r) =hF2x2(r). Then, as it is easy to see, the equalities h& =szlh$ and hB 12 =s12 hF2 hold, so eqn. (4) reduces to

A&= UB 11 -s12 El + G2 -s21 G2

l- Is12 I” (5)

We shall note that this expression does not contain matrix elements of the kinetic-energy operator p. This result is not in contradiction with the state- ment made in Ref. 9, connecting overlap repulsion with the increase of the kinetic energy, but certainly shows a different aspect of the problem.

We can introduce some operators B * and B B, to which the BSSE effects can be attributed [ 51. Operator B * is defined as the difference between the exact intramolecular Hamiltonian, hA, and its representation in the finite (one-dimensional) molecular basis, /&,,. = h& Ix1 ) (xl ( .

~A=hA-li~,,$iA-h~l 1x1) (xl ( @a)

Analogously, the operator BB is defined as

P=P-h&.=P-@2 1x2) <x2 I (6b)

The matrix elements B & = (x2 ( B * I x1 ) = h& - hfl s21 = B $ vanish if x1 is an exact eigenfunction of h*: hAxl (r ) = h&x1 (r ) . If this is not the case, one can write

%= <X2 I (I- lx1 > <XI I )hAX1> (7)

47

where it is easy to identify the projector 1 - 1 x1 ) (x1 1 onto the orthogonal com- plement of x1. According to eqn. (7)) the matrix element 23$ is originated from that component of the intramolecular function hAxl, which cannot be expanded in the molecular basis; this is the component which gives rise to the BSSE [5]. (Similar considerations apply to the matrix element BF2 = (x1 1 (1 - ]xZ ) (x2 ) )hBx2), of course.) Using these operators, eqn. (4) can be written as

Now, the question arises, whether eqn. (5) or eqn. (4), i.e. eqn. (8), represents the best estimate of the interaction energy in this simple model, or - in other words -whether it is of meaning to separate out the BSSE contributions in the expectation value of the energy calculated over the antisymmetrized (but otherwise unperturbed) monomer wavefunctions. The importance of this question extends, of course, far beyond this particular model, as the terms of this type enter any actual problem in which overlap repulsion takes place.

The point is that those terms of the energy expression (eqn. (8) ) which contain the matrix elements B& and By2 reflect the imperfect description given for the free monomers, as it follows from the definition (eqn. (6) ) of the BSSE operators l?* and BB. As we are not interested in the intramolecular problems, a possible choice would be to consider eqn. (5) as the best possible “BSSE-free” estimation of the interaction energy which is accessible in the given model, even if x1 and x2 are not really exact eigenfunctions of GA and hB, respectively. Such a point of view was expressed in our recent papers [ 581 treating the simplest one-electron-two-orbitals model, and it was assumed that this view can be generalized to more complex problems as well [ 51. (Note that standard versions of the “exchange” PT of intermolecular interactions correspond to the use of eqn. (5)) so our considerations have an immediate impact in that field, as well.) In the meantime, however, it became clear, that this approach is not adequate in the cases that also involve overlap repulsion. This revision resulted from both theoretical and numerical considerations, some of which are considered below. It is important to point out, however, that the new conclusion does not question the adequacy of the CHA/CE or CHA/F methods proposed in refs. 5-7; on the contrary, there are some convincing arguments in their favour, because now all the terms of the energy fit the same conceptual frame.

From a purely theoretical point of view, the most serious argument against the use of eqn. (5) is that there is no guarantee for A&, to be real if complex orbitals x1 and x2 are admitted. The energy is necessarily real only if the terms which vanish in the case of exact monomer orbitals are explicitly conserved. This problem does not arise in the analogous energy formula of the one-electron-two-orbitals model considered in refs. 5-8, but seems to be characteristic of all systems with two or more electrons.

48

At first sight the problem of complex energy does not seem to be a serious one, as in practice we usually do not use complex wavefunctions; nevertheless, it calls attention to the fact that energy expressions of this type can give completely inconsistent results. There is also an element of arbitrariness in the way of dropping the terms which vanish if the monomers are described exactly; for instance, instead of eqn. (5) one could obtain its complex conjugate as well.

One might consider the above model as being a rather artificial one. This is not the case, however. For instance, a formula, closely related to our eqn. (5)) is quoted in the Appendix of ref. 10, where the three-electron problem of the He- - l H colliding system was considered: the third electron of the opposite spin does not introduce any essential complications. (In ref. 10 the two-electron part of the interaction was also considered, of course, and the possible complex character of the different quantities was simply neglected.)

Figure 3 shows an illustration that expressions like eqn. (5) are problematic not only in a conceptual sense but also give numerically inadequate results. The model considered is that of the triplet state of the H, molecule, the interaction energy for which can be obtained by adding the nuclear-nuclear and electron-electron repulsions (including exchange) to the right-hand sides of eqns. (4) or (5). One adds the same terms to eqns. (4) and (5), so the differences in the results reflect the behaviour of the one-electron terms only. The minimal basis set considered was that of the 1s Slater-type orbitals with three different exponents czO.9, 1 and 1.1. The expressions of different integrals given in Slater’s book [ 111 have been used. For [=l we have the exact 1s hydrogen orbitals, so the results given by eqns. (4) and (5) coincide. The curve obtained in this case is taken as the reference one, as it describes the overlap repulsion for the unperturbed exact monomer wavefunctions. If the monomer orbital is not the exact one (<# 1 ), the interaction energy formula (within the framework of unperturbed monomer orbit&) which gives results closer to the reference curve {= 1 is the preferred one. The results indicate unambiguously that the conventional energy expectation value, eqn. (4) or (8), gives much better approximations. It is therefore inappropriate to drop the terms SLOB& and szlBFz from eqn. (8)) attributing them to some intramolecular effects which are irrelevant from the point of view of the true “intermolecular” interactions. Although these terms can indeed be formally related to the deficiency of the monomer description, they appear important both in providing the real energy and in providing appropriate numerical values. This conclusion forces us to modify to a great extent the basic assumptions on which the original development of CHA [ 121 was based and stresses the basic correctness of the CHA/ CE formalism. (We shall return to this point. )

The use of energy expressions derived by utilizing explicitly the assumption about the exact description of the individual monomers may make question- able many results of the so-called “exchange perturbation theory” (for a sur- vey see ref. 18) of the intermolecular interactions, as well. (It might also be

49

(a) 0.20

0.05

0.00 1

(b)

0.07

i 0.06

t

& w" zw 0.05 z 0 E 2 g 0.04

0.03

0.02

0.01 2

SO

F

\

T-

'.50

2.50 3.50 4 50 a 5.50 R /au

2.;5 3.60 3.k 3 5 R /mu.

0

Fig. 3. (a) Overlap repulsion for the triplet state of two hydrogen atoms described by one Slater- type orbital each. ( 1) Exact hydrogen orbit& (the exponent C= 1); (2 and 3 ) results corresponding to eqn. (4) or (8) for {=a.9 and 1.1, respectively; (4 and 5) results correspondingto eqn. (5) for [=0.9 and 1.1, respectively. (b) Enlarged part of (a).

50

worthwhile to check what versions of the exchange PT guarantee the energy to be real. The equation in ref. 15 mentioned above was also obtained in the framework of a “label-free” exchange PT.)

The above numerical comparisons have an analogue in the field of exchange PT: Conway and Murrell [ 141 have calculated the repulsion between rare-gas atoms by introducing a correction term, A, to the first-order exchange PT energy formula, whichvanishes for the exact monomer eigenfunctions. This means that they have calculated the actual expectation value of the energy for the unperturbed (but antisymmetrized) monomer wavefunctions, instead of using a PT formula analogous to our eqn. (5). The results given in ref. 14 indicate the great importance of the correction term A. (The large sensitivity of A is also worth mentioning: when the overall interaction energy of two helium atoms at 5.5 a.u. is already stabilized up to ca. 3 x 10m7 a.u., the correction energy component is still changing by ca. 2 x 10m6 a.u. [ 141. ) This also calls attention to the fact that, when developing perturbation theories of intermolecular interactions, it is imperative to carry out considerations similar to those discussed above. One has to keep in mind that it can be dangerous to substitute approximate wavefunctions into expressions derived for exact ones, in much the same way as Hellmann-Feynman or virial theorems must not be used without introducing special requirements.

Closely related problems have been met in the first developments based on CHA [ 3,4]. The general non-symmetric CHA energy formula - as in refs. 5 and 6 we denote this “CHA/NE” (CHA with non-symmetric energy expression) for distinction - [3,4,12] also gives results which are necessarily real only in the case of real wavefunctions (or if some additional conditions are fulfilled providing the “density matrix” to be at least real [ 41) . The different problems encountered in both perturbational- [ 31 and SCF-type [ 41 applica- tions can be more or less directly related to this fact. At the SCF level these difficulties have been solved by the recently introduced CHA/CE scheme [ 5,6], in which the energy is calculated as a conventional expectation value, i.e. by using an analogue of eqn. (4) instead of one of eqn. (5). (The CHA/CE scheme was, however, introduced on the basis of much less strong arguments [5]: it was related to the difference between the quality of description given for the occupied and virtual monomer orbitals. )

The formal, but conceptually important requirement that the energy must be real rules out eqn. (5) derived for exact monomer wavefunctions and forces us to consider the conventional expectation value of the energy (eqn. (4) or (8) ) as the only correct energy expression for the model considered, even if eqn. (8) contains terms which might really be considered to have BSSE origin. This conclusion also means that the exclusion of the BSSE at the level of the Hamiltonian, as was assumed when the development of CHA was started [3,4,12], cannot be done in such a simple manner: it appears that the CHA Hamiltonian can be used to obtain the wavefunction, but not the energy. Such

51

a combined procedure is just the essence of the CHA/CE scheme. Its success indicates that BSSE is an effect which can be attributed to the wavefunction, rather than to the Hamiltonian.

As the terms present in the simple model discussed above also enter any realistic problem, one arrives at the general conclusion that whatever wavefunction is used, the energy should be determined as the expectation value of the conventional (Born-Oppenheimer) Hamiltonian.

Two-electrons-four-orbitals model

Let us consider the model in which both “molecules” A and B bear two orbitals: x1 and xz are centred on “molecule” A and orbitals x3 and x4 on “molecule” B. Let x1 and x3 be occupied by one electron each and let both electrons be of the same spin, cy (Fig. 2). As in the previous model, we only consider explicitly one-electron interactions. We assume that the orbitals x1 to x4 represent solutions of the “canonic” eigenvalue problems of the individual monomers to which they belong, i.e. s12 = s34 = h$ = hF4 = 0.

In what follows we shall assume that both the physical interactions and the BSSE effects are weak enough to justify the use of some simplifying assumptions. For this reason we consider the diagonal matrix elements hii, hi, h:, U$ and Uz as zeroth-order quantities and all the non-zero off-diagonal matrix elements aLi. hij, h$, h$, U$ and Uz (i#j) as well as the polarization-delocalization parameters qij as first-order ones.

The system is described by the determinant Y built up of the molecular orbitals #1 and $g2:

(9)

Similarly to the previous case, the molecular orbit& (MOs) & and $J~ are not assumed orthonormalized, the determinantal wavefunction Y is, in general, unnormalized.

In our analysis we utilize the following theorem.

Theorem Any “not singularly wrong” (see below) determinantal wavefunction, Y, of

the model studied can be built up by using (unnormalized and generally non- orthogonal) orbitals of the special structure

$1 =x1 + %2X2 +%4x4

and (10)

$2 =x3 + q32x2 + v34%4

Equations (10) mean that the expansion of the MO & does not contain the basis orbital x3 and the expansion of g2 is lackingx,. In other words, any wave-

52

function of the supermolecule can be described without admitting delocalization to the occupied orbitals; accordingly, any mixing which may be observed in a supermolecule between the occupied monomer orbitals is merely an ortho- gonalization effect. Note that this theorem is not restricted to the case of weak interactions, i.e. it is valid even if strong delocalizations take place between the constituent parts of a compound system.

Proof The determinantal wavefunction, Y, is uniquely defined (up to its normal-

ization) by the subspace of the occupied orbitals; therefore, in order to build up Y, it is sufficient to possess any pair of linearly independent orbitals d1 and flz lying in this subspace. For this reason, we prove the existence of orbitals & and & which have the required structure (eqn. ( 10) ) and lie entirely in the occupied subspace. (The theorem represents an application of a more general result [ 151 obtained in a different context; we present the proof here for the sake of completeness.)

The orbitals of the general formula v/1 =pxI + qx2 + r-x4, with arbitrary coef- ficientsp, Q and r, span a subspace of dimension three. The occupied subspace has the dimension two. As the sum of these dimensions is five then, while the dimension of the whole basis is only four, the two subspaces must have a com- mon vector. In other words, there exists an orbital w1 with some values of p, q and r which also lies in the subspace of the orbitals occupied in Y. We call “not singularly wrong” the case when coefficient p#O for this orbital; then by di- viding by p # 0 we obtain the orbital & = (l/p) v1 having the required structure. (Note that the requirement p # 0 can also be formulated in the following way: there is no occupied supermolecule orbital which could be expanded exactly in terms of virtual monomer MOs.)

The existence of the orbital & can be proved in the same way as that of @1; their linear independence follows from the independence of x1 and x3; therefore, the determinant \Y can be built up of these orbitals of structure (10). Q.e.d.

Note that the above theorem also holds in the case of any number of electrons; for the proof in the general case and for an algorithm of actual deter- mination of the orbitals of this special structure we refer to ref. 15, where the problem was originally discussed in the framework of the localized MO analysis.

In the ref. 16 (also dealing with localized MOs) we gave the series expansion of the delocalization and polarization parameters, q, for the orbitals of the special structure (10). In this case the first-order expressions are simply

53

h q32 = -

23 -h33s23

h22-h33 (11) h

2114 = - 41-hs41

h 44 -h,

h 2734 = - h 44 :h3,

These expressions are quite analogous to eqn. (5) of ref. 5, related to the case of a one-electron model; for v12 and q34 it is also taken into account that the equality s12 = s34 = 0 has been assumed in the present case.

The energy of the system (the expectation value of the Hamiltonian

HC1, =i$(i)) can easily be calculated by using Lowdin’s general formulae for

non-orthogonal orbitals:

E= <YI%, I ‘y>

(YlY)

+~~21~l~2>~~~I~~~)~~~~I~~>~~21192>-I~~~I~2z12)-' (12)

Evaluating the matrix elements in eqn. (12) and utilizing eqns. (11) for the q values, after a simple but somewhat lengthy algebra one obtains the following expression, which is valid up to second order,

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

+~34~34+~114(h4-s14hd+t132(h32-s32h33) (13)

We now consider some special cases. In ref. 5 a similar analysis was performed for the one-electron-two-orbitals model; therefore we concentrate here mainly on the differences between the one-electron and two-electron systems and, in particular, on the consequences of the conclusion drawn in the previous section.

The ‘t’nteraction only” case In this case xl and x3 are assumed to be exact monomer eigenfunctions:

hAxl(r)=h~lxl(r) and hBx3(r)=hf3x3(r). In this case no BSSE appears in the model, because the extension of the basis cannot lower the energy of A with

54

respect to the exact eigenfunction x1; similar considerations apply to monomer B, of course.

The delocalization-polarization parameters qij and the interaction energy become

%2 = -

Gl h-22 - hll

q32 = - G3 - Gs23

h22 - h33

2714 = -

cl - G541

h 44 -hll

%4=-h _h

44 33

and

~~=~~l+~~-~l3~~~l-s3lu~l~-s31~u~3-sl3u~~- m212

h2 -hll

vc4l” IUQ-s4,Ub12_lU~-S23hLi~312

- h44 - h33 - h44 - hu 22 33

(14)

(15)

where the subscript “0” again indicates that eqn. (15) refers to the case of exact monomer orbitals.

Equation (15) is analogous to eqn. (5)) but terms reflecting the polarization and delocalization (charge transfer) caused by the true physical interactions are also accounted for, and all terms are expanded up to second order. The overlap repulsion terms, quite similar to those in eqn. (5)) mean that eqn. (15 ) guarantees a real value of the interaction energy for complex orbitals only if one utilizes explicitly that x1 and x3 are actual eigenfunctions of fi* and hB, respectively. For this reason (contrary to that we have assumed previously [ 51)) expansions of this type should not be used with approximate monomer wavefunctions.

The ‘BSSE only” case Let us consider monomer A in the whole supermolecule basis; this means

that uB = 0 and the Hamiltonian is that of “molecule” A (&= h*); orbitals x3 and x4 serve as “ghost orbitals”. The orbital x1 delocalizes into

Vl =x1 +&3x3 +x14x4 (16)

Orbital x2 is absent in this expansion because the monomer problem of A for the basis of orbitals x1 and x2 and Hamiltonian hA is assumed solved (s12 = hf2 = 0) , so no first-order polarization can take place. The operator I?* respon-

55

sible for the BSSE can again be defined as the difference of the exact monomer Hamiltonian, h*, and its finite-basis representation in the (two-dimensional) monomer basis, t&,,. =h:: IX1 > tx* I +@2 Ix2 > (x2 I9

B*&*-li& =i*- vel Ix1 > (XI I +ez Ix2 > (x2 I ) (17)

(Note that eqn. (17) contains only diagonal matrix elements h: owing to the assumed equality h t2 = 0. )

In terms of operator B* the delocalization and energy lowering due to the pure BSSE effects in the “ghost orbital” calculation of monomer A become

and

I% I’ 1% I” m*(AB)=-hA _hA -hA _hA

33 11 44 11

(18)

(19)

respectively. The matrix elements B& and B& (and thus the BSSE energy lowering) vanish trivially, if xl is an exact eigenfunction of h*. Quite similar expressions hold if monomer B is considered in the whole supermolecule basis; operator BB is defined analogously to eqn. (17).

Note that contrary to the supermolecule case described by eqn. (lo), here we must also consider a delocalization to the orbital x3 which is occupied in monomer B. As indicated by eqn. (19), this delocalization (obviously) also contributes to the energy lowering in the “BSSE only” (ghost orbitals) case.

Standard supermolecule calculations In this case both the BSSE and the true interactions are present. Using eqns.

(11) and (13) and the definition (17), the energy change obtained in the standard supermolecule calculation can be presented as

m

sup. -

_m

0- 2Re B~l(U~4-s,,U::)+B~(U~-s,,U~) I h44-h,, h22 - h33 >

-s,,&% -s3&3 - 1% I” I@3 I2

h44 -h -hm -h33 (20)

In addition to the right-hand side of eqn. (15)) this expression contains the terms +13%1 - ssl BF3, providing the. formula of overlap repulsion is real (analogously to the case of eqn. (8) ), as well as pure BSSE terms proportional to ) B$ ( 2 and I BF3 I 2. Furthermore, similarly to the case of the one-electron model discussed in ref. 5, there are also some “cross-terms” containing matrix

56

elements of both the BSSE operators and the physical interactions. We discuss these terms in a later section of this paper.

The Boys-Bernurdi aposteriori correction scheme Taking the difference A&+ - AEA ( AB ) - ABn (AB ) we obtain

+hf3-hfI+h&hf3-2Re h44-hll I B%J?4 -s,4u~~,+B~(U~-s32U~) h22-h33

(21)

Here we have introduced the coefficients K& and KF3 which are analogous to the parameter K in the one-electron model discussed in refs. 5 and 8:

K:I =hA hh -hi’1 1

44 - h& + Uf4 - UFI = 1 + Uy4 - UyI

h& -hf’l

K?3 =hB hk -h& 1 I

22 - hf3 + Uf2 - Ut3 = 1 + Uf2 - Uf3

h% -h%

(22)

It can be difficult to give realistic estimations of the values of these coefficients. Inspection of Eqn. (21) leads to the following conclusions. (i) The BB scheme

excludes pure BSSE effects on the delocalizations from the occupied orbital of one monomer to the virtual orbital of another only if the parameters KfI and KF3 are both close to unity, which is not necessarily the case. (ii) The BB correction contains spurious terms originating from the delocalization from the occupied orbital of one monomer to the occupied orbital of another, taking place when the monomer calculation is performed in the supermolecule basis. The energy lowering due to such delocalizations is also ascribed to BSSE in the BB scheme, and this may be the source of some “overcompensation” of the BSSE effects. (We return to these points later.)

The “chemical Hamiltonian approach” We follow a procedure quite similar to that described in ref. 5 for the one-

electron case. By writing, for instance, hB1 = h$ + U& = (x3 I pAhAxl ) + (x31 (l-~A>hAx,)+U~I=s3,h~I+U:I+B~I (where PA= 1x1) (Xl I + )x2 ) (x2 1 is the projector onto the (two-dimensional) subspace of the orbitals of monomer A ) , and dropping terms like B fI which cause the BSSE, we obtain the (non-Hermitian) CHA Hamiltonian of the model as

57

h-IA= I (23)

By solving the generalized eigenvalue equation of this matrix we obtain a pair of orbitals which can be presented as

$; =x1 +v;2x2 +hG3x3 +z7114x4

and (24)

$I =x3 + 9~lxl+yl~2x2 + h4x4

Restricting again the calculations to the terms of first order in q, we get vi2 =

21127 h4 =‘?14, vi2 =?I32 and 5’;4 =‘i’34, i.e. these coefficients are the same as in the “interaction only” case. The coefficients r& and qkl are obtained as

Gl -s31 U?l

v’3=- h,,-hll

and (25)

21[71= - ut - s13 G3

h 11 - h33

However, it can be shown that, to a first approximation, the “occupied-occupied” mixing described by these quantities influences only the normalization of the determinantal wavefunction. This means that the wavefunction built up of the orbitals given in eqns. (24) differs from that built up of the orbitals given by eqns. (10) only by a normalization factor and some higher-order terms. This conclusion is also confirmed by explicit calculation of the energetic quantities: after a lengthy algebra one finds that all terms containing vi3 and vi1 cancel up the second order. (This is a quite general result and may be considered an independent justification for using the orbitals of form given by eqn. (lo).) It can be concluded, therefore, that the CHA method gives correct orbitals for which the delocalization effects caused by the BSSE are absent.

Concerning the CHA energy, similarly to ref. 5, we investigated the CHA/ NE and the CHA/CE cases, i.e. the energy formulae obtained as the expectation value of the non-symmetric CHA Hamiltonian and of the conventional Hamiltonian, respectively. We do not quote here the rather complicated CHA/ NE energy formula, but only mention that (similar to the one-electron-two- orbitals model [ 51) it reduces to eqn. (15) corresponding to the “interaction only” case if, and only if, both the occupied and the virtual orbitals are exact monomer orbitals. This is, obviously, a too strict condition, as no BSSE ap-

58

pears in the present model if the occupied monomer orbitals are exact, irre- spective of the properties of the virtual orbitals. Accordingly, the CHA/NE energy formula does not fulfil the natural criterion [ 51, according to which an acceptable theory must reproduce A& as given by eqn. (15) in the special case when x1 and x3 are eigenfunctions of & and fin, respectively. This fault, to- gether with the fact that the CHA/NE energy formula does not guarantee the energy to be real if complex orbitals are admitted, indicates that the CHA/NE scheme used in our first papers [ 3,4] on the subject can give quite unpredictable results. The CHA/CE scheme introduced in ref. 5 is, however, free of these disadvantages.

The CHA/CE energy is again obtained as the expectation value of the original supermolecule Hamiltonian (i.e. as the conventional energy value) calculated over the CHA wavefunction

&HA/~ = ~IJ -%,%I -S31%3

-2R G - U::yp + G3 - Uf3s23BB

h 44 -hll l4 h22-h33 32

(26)

As noted above, the CHA method permits orbitals to be obtained which are free of the BSSE delocalizations but take into account properly the physical interactions (the coefficients r] reproduce those orbitals obtained in the “interaction only” case). Also, it was shown above that the proper description of the overlap repulsion forces us to calculate the energy as a conventional expectation value of the total Hamiltonian. Therefore, the CHA/CE energy given in eqn. (26) should be considered as the best possible estimate of the interaction energy which can be obtained within the given model.

Inspection of eqn. (26) indicates that (contrary to the BB case) the CHA/ CE energy is completely free of any pure BSSE terms, and no spurious occupied-to-occupied delocalizations appear either. This equation contains the terms s13B& and s3,By3 which originate from the overlap repulsion of the unperturbed orbitals and, quite analogously to the similar terms in eqn. (8)) they are necessary to guarantee the energy to be real. There are also some terms containing matrix elements of both the physical interactions (potential energy oA or oB) and the BSSE operators B* or BB. These terms represent some “cross-terms” (or “interferences”) [ 5,8] between the physical interaction and the BSSE, but their appearance seems to be inevitable according to our conclusion about the use of the conventional energy expectation value. The existence of these terms shows the inherent non-additivity of the BSSE problem and may indicate that the “BSSE-free interaction energy calculated in a given basis” is a concept which can be defined with some limited accuracy only. (The terms s~~B$ and s31B73 can also be considered as “cross-terms” between the overlap and the BSSE effects. Their more fundamental importance is, how-

59

ever, indicated by the fact that they are necessary to guarantee that the energy is real. )

Comparison of the BB and CHA/CE methods

We discuss here in more detail the comparison of the results predicted by the BB and CH/CE schemes, representing different (a posteriori and a priori, respectively) approaches to the same problem. Inspection of eqns. (21) and (26) indicates that the BB energy formula contains two types of additional terms which are absent in the CHA/CE case. Both types are of “pure BSSE” character in the sense that their numerators contain matrix elements of the BSSE operators only. The first type of these additional terms corresponds to the BSSE-caused delocalizations from the occupied orbitals of one molecule to the virtual orbitals of another. These terms of the BB energy formula (eqn. (21) ) vanish only if the parameters K$ and K& are equal to unity. This requirement is fulfilled if the BSSE-caused occupied-to-virtual delocalizations in the “ghost orbitals” calculation are equal to those in the actual supermolecule (parameter K 14 = q14, etc. ). For this to be the case, it is necessary that the energy denominators determined for the “ghost orbitals” problem of one molecule are not influenced by the appearance of the potential due to the nuclei and electrons of another one. This influence is expected to be small for non- polar molecules and to be rather large in problems of ion-molecule interactions.

Note that orbital x4 is situated near to “molecule” B, while orbital x1 is fur- ther away. Therefore, the effect of the attractive potential, i&, should be greater for x4 than for x1, i.e. Uy4 - Uyl < 0 (and, similarly, U& - U& < 0). This means that the parameters of type K,4 and K& are expected to be greater than one, especially if the corresponding monomer molecule has a considerable electron affinity. Consequently, the terms of the given type lead to the energy lowering in the BB scheme. This is in contradiction with the usual belief that BB can only “overcorrect” for the BSSE, but is confirmed in some actual calculations (see Fig. 4 (a) ) . There are cases in which this effect is even more pronounced, and a number of others in which it does not occur or is observed for some (e.g. intermediate) distances only, while at smaller distances it is overweighted by the spurious occupied-to-occupied delocalizations (see below). Examples of this type can be found in refs. 5 and 6; see also Fig. 4 (b) .

The terms IB& I”/(h&-h&) and IBfl 12/(h~l-h&,) in eqn. (21) are due to those delocalizations which take place from the occupied orbital of one monomer to the occupied orbital of another in the “ghost orbitals” calculations. As noted above, the mixing between the occupied orbitals is a higher-order effect in the actual supermolecule, so it does not influence the energy formula (eqn. (20) ) in which terms up to second order only are conserved.

It has been the subject of extensive debates in the last decades (e.g. refs. 17- 25) whether the corrections corresponding to such “occupied-to-occupied” de-

-12

-16

1.5 2.0 2.5 3.0 3.5 40 4.5 NH.. Li) /A

HF dime,. 3-ZIG

1.5 2.0 2.5 30 3.5

R@’ F) /A

Fig. 4. Potential curves of the (a) LiH and (b) HF dimers calculated in the 3-21G basis set. SCF, standard supermolecule SCF calculations; BB, the SCF results corrected by the Boys-Bernardi “full counterpoise” method; CHA/CE, the results given by the method consisting in calculating BSSE-free orbit& by the “chemical Hamiltonian approach” and then the conventional energy expectation value.

61

localizations should be included when the interaction energy is calculated, or whether the “ghost-orbitals” calculation should only involve the virtual orbitals of the partner molecule. The latter “virtuals only” correction scheme is, however, less practical and is used less commonly, so we shall not discuss it in detail. It seems sufficient to mention that the results obtained with our analytical model would differ from those predicted by the CHA/CE scheme only by terms connected with the deviation of parameters K from unity. So the “virtuals only” scheme is expected to “undercorrect” BSSE due to the fact that K> 1.

The BB scheme was originally introduced on the basis of the observation that “the calculation of the energy of the separate molecules as if they were just a sub-case of the composite system of two molecules appears to introduce a strong effect of error cancellation” [ 21. This is undoubtedly the case, and the BB scheme surely represents a rather useful practical recipe for this purpose. However, we cannot agree with the assumption that the BB results are also conceptually “exact” and represent the most correct interaction energies which can be obtained when a finite basis set is used for the monomers.

We shall not repeat here our objections [ 5-71 to the “theoretical arguments” used to attribute such a privileged role to the BB scheme, but mention only the following. According to the BB philosophy, this method should be applicable also to the two-electrons-two-orbitals problem of two hydrogen atoms forming a triplet, which has been discussed above: one uses the same basis for treating the monomers and the “supermolecule”. Now, the BB correction can only increase the supermolecule energy whereas the use of approximate monomer wavefunctions ([# 1) can both increase and decrease the overlap repulsion with respect to the reference value corresponding to the exact monomer orbitals ([= l), which by definition lack a BSSE. Therefore, the BB correction scheme does not necessarily improve the results predicted by the standard supermolecule calculations as far as pure overlap repulsion is concerned. (As there is only one triplet wavefunction in this model, there is no actual BSSE- caused delocalization in the supermolecule, but of course correction terms appear in the “ghost orbitals” calculations.) This is illustrated by the energy- level schemes shown on Fig. 5. It also follows from the comparison of the analytical models considered above that the effects observed for the overlap repulsion in this simplest model also enter any realistic problem. In other words, the “tails” of the monomer wavefunctions generated by the “ghost orbitals” calculations are not the adequate tools to correct the basis set deficiency in describing overlap repulsion. The assumption that these tails can improve the description of the overlap repulsion is (at least implicitly) implied in some arguments in favour of the BB scheme. We think, however, that overlap repulsion between molecules treated in a given basis has a clear physical or mathematical meaning only if it is calculated by using just the unperturbed monomer wavefunctions. There can be basis-set defects, but no arbitrary and

62

R = 2

< = 1.1

A!p, T 0.1580

= AET = 0.1568 c 0.9

AETCC=l, = 0.1540 -------_--

Ap= T 0.1509

AET = 0.1495

R = 3

5 = 1.1 c = 0.9

AEBB= T 0.0377

AET(C=l, = 0.0374 ----------

AEp= 0.0368 =T = 0.0369

AET = 0.0363

Fig. 5. Level schemes illustrating the repulsion between two hydrogen atoms forming a triplet state. Values (in atomic units) for two different distances R and three different Slater exponents [ are shown. The case of c= 1 corresponds to the exact description of the individual hydrogen atoms and represents the reference value. (Note that for the minimal basis applied, there exists only a single triplet wavefunction, so that the Heitler-London and MO results coincide.) AET, uncorrected interaction energy for the given exponent; AEeB, interaction energy corrected using the Boys-Bernardi method (the individual atoms are calculated in the basis of both atomic orbitals; AET( c= 1) , interaction energy calculated for the exact hydrogen atomic orbitals (reference value).

unpredictable corrections are introduced. This is of special relevance if correlated wavefunctions are used. We recall here that for atoms or other symmetric systems the BB scheme corresponds to the overlap repulsion between monomers of improper (broken) symmetry.

The error introduced by the “occupied-to-occupied” delocalizations in the BB scheme may be relatively small in many cases, as compared with the “true” (occupied-to-virtual) BSSE terms in the supermolecule energy, explaining the successes of this a posteriori correction scheme. It can also be assumed that there can be some compensation between the two effects of opposite direction discussed above, i.e. the energy lowering due to parameters of type K& being greater than one and the energy increase due to the occupied-to-occupied delocalization terms in the “ghost orbitals” calculations. This can be of importance in understanding the known stability of the BB results.

63

In cases where significant occupied-to-occupied delocalizations occur in the “ghost orbitals” calculations, the BB method gives somewhat too high energies (“overcompensation”). This effect obviously increases as the intermolecular distance decreases. As a consequence, the minimum on the BB curve can be slightly shifted to larger intermolecular distances. Examples of this type can be found in refs. 5-7; see also Fig. 4(b). This effect is not restricted to the one- electron level; the same conclusion was drawn in ref. 18 for correlated wavefunctions. (In ref. 26, among others, the He*- -He potential curve was calculated in an enormous basis set of 100 contracted functions, taking into account electron correlation via the CEPAB method. It was found that the interatomic distance was larger in the BB case by ca. 0.1 a.u. than in the uncorrected case; the latter practically coincided with the experimental value.)

CONCLUSIONS

The analytical and numerical model studies presented here permit the following main conclusions to be drawn.

(i ) The requirement of the energy to be real indicates that, whatever wavefunction is used, the energy should be determined as the expectation value of the total Hamiltonian. In other words, one must not utilize expressions from which terms that vanish in the case of an exact description of the individual monomers have been dropped. These terms are necessary both to guarantee that the energy is real and to obtain appropriate numerical values when the effects of overlap repulsion are involved. This conclusion has an important impact on both the “exchange perturbation theory” of intermolecular interactions and the “chemical Hamiltonian approach” to the problem.

(ii) The above conclusion serves as a firm theoretical justification of the CHA/CE scheme introduced recently. In this scheme one uses the CHA method to determine wavefunctions which include delocalizations due to the true physical interaction but not those caused by the BSSE, and then calculates the conventional energy over these BSSE-free CHA wavefunctions. Therefore BSSE can be excluded from the very start and we have a truly a priori treat- ment of the BSSE problem. Accordingly, the CHA/CE scheme can be considered to give the best possible BSSE-free interaction-energy values.

(iii) The analytical model discussed indicates that the interaction energies given by the a posteriori Boys-Bernardi correction scheme differ from the CHA/ CE ones by two types of (minor) terms which have opposite effects. First, the presence of the nuclei and electrons of the partner molecule is expected to favour BSSE-type delocalizations in the actual supermolecule (through its influence on the energy denominators), as compared with the “ghost orbitals” case. This effect leads to a lowering of the resulting BB energy (“undercorrec- tion” ). Second, the spurious occupied-to-occupied delocalizations taking place in the “ghost orbitals” calculations leads to an increase in the resulting BB

64

energy (“overcorrection”). The numerical example indicates that these occupied-to-occupied delocalizations cannot be considered as factors correcting the insufficiency in the description of the overlap repulsion, contrary to some assumptions implied in the arguments in favour of the BB scheme.

For different systems either of the two effects can be more important; this can also be changed for different intervals of the same potential curve. Ac- cordingly, the BB scheme can give both overestimated and underestimated interaction energies. One can expect that for most (but not all) systems the “overcompensation” due to the occupied-to-occupied delocalizations domi- nates the BB curve at smaller distances. In these cases the BB method slightly overestimates the equilibrium intermolecular distance of the complex. (The energy lowering due to the energy denominators is expected to be more or less irrelevant for the position of the minimum.)

The presence of two effects of opposite sign, which compensate each other to some extent, can be an important factor in the observed numerical stability of the BB results. However, the numerical data obtained up to now [5,6] indicate that the CHA/CE results also have a similar, or even better, stability; the CHA/CE and BB energies are usually close to each other. Finally, we should mention that the absence of the non-physical BSSE-caused delocalizations in the CHA wavefunctions should be advantageous if physical quantities other than energy are required for the complex.

ACKNOWLEDGEMENT

This work was supported in part by the Hungarian Research Fund (OTKA No. 785). Also the numerous discussions with Dr. P.R. Surj4n are gratefully acknowledged.

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An analytical investigation into the bsse problem

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