Electron correlations in molecules. I Bond orbital approximation

11

Chemical Physics 106 (1986) 11-26 North-Holland, Amsterdam

ELECTRON CORRELATIONS IN MOLECULES. I. BOND ORBITAL APPROXIMATION Werner

BORRMANN,

Max-Planck-Institut

Andrzej

M. OLES ‘, Frank

fiir Festkiirperforschung,

PFIRSCH,

Peter FULDE

D-7GOO Stuttgart 80, FRG

and Michael

C. BfiHM

Institut fiir Physikalische Chemie, Physikalische D-6100 Darmstadt. FRG

Chemie III, Technische Hochschule Darmstadt,

Received 27 September 1985; in final form 17 March 1986

A simple and fairly accurate approach to electron correlations in the ground state of molecules is presented. The two parts of electron correlations, inter- and intra-atomic correlations, are treated separately and by different methods. The interatomic correlations are studied by means of a variational local ansatz. It starts from the self-consistent field (SCF) ground state and optimizes the energy by the appropriate reduction of charge fluctuations. This procedure is qualitatively illustrated in the bond orbital approximation (BOA). It is found that the correlation energy decreases with increasing bond polarity at,, being ’ J’2. It is also shown that the contributions to the correlation energy due to one-particle proportional to the factor (l- an) excitations become important only in more strongly correlated polar bonds. The intraatomic correlations are obtained from an atoms-in-molecule-type of approach. It makes use of a population analysis of the correlated ground state wavefunction. The used probability distribution P(n), where n is the valence electron number, is found to be well approximated by a normal distribution. The advantages and limitations of the method are shortly discussed.

1. Introduction The phenomenon of electron correlation remains one of the most challenging problems in solid state physics and quantum chemistry. A good understanding of correlations in molecules should be the first step in understanding those in corresponding solids so that both fields, i.e. molecules and solids, should be considered as intimately connected. In solids some of the long-range correlations are well understood. We mention only the pair correlations leading to the phenomenon of superconductivity [l] and the long-range part of the screening cloud in metals [2]. But it is the short-range part of the correlation hole which, ’ On leave of absence from the Institute of Physics, Jagellonian University.

Cracow, Poland.

0301-0104/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

despite of its great importance, has hardly been studied for electrons in different solids. For example, it contributes to the cohesive energy, band gap and other band properties as well as to, e.g., magnetism in transition metals. One of the problems is that the classical methods of treating correlations in molecules cannot be carried over to infinite systems, i.e. solids. They are almost all based on configuration interaction (CI) (see, e.g., ref. [3]). In order to treat solids and in particular delocalized excited states one would need an infinite number of interacting configurations in order to describe the correlation hole around a moving electron. There is an alternative approach to the correlation problem which avoids calculating wavefunctions but determines ground state properties (i.e. energies and densities) directly. This is the density B.V.

12

W. Borrmann et 01. / Electron correlations in molecules. I

functional method. It requires, however, making important simplifying assumptions in order to become applicable to real systems. The most popular one is the local density approximation (LDA) [4]. It has been successfully used to improve the Hartree-Fock (HF) results for the binding energies and the low-lying excited states of various simple molecules [5]. Also such intramolecular properties as bond length an vibrational frequencies have been systematically reproduced with high accuracy [6]. When the method works it is by far the simplest and most efficient way to incorporate electron correlations into electronic structure calculations. Its disadvantages are that it is very difficult to improve on its insufficiencies and furthermore that one does not really obtain a clear and simple physical understanding of the effects of correlations. For these reasons we shall concentrate here on a method for calculating correlated wavefunctions and correlation energies which emphasizes the local nature of electron correlation in molecules, as well as solids. This approach requires, like others do, a SCF calculation as a starting point to which correlation calculations are added. It has been applied before to small molecules for which the SCF and correlation energy calculations were done on an ab initio level [7,8]. There is no difficulty to perform similar calculations also for large molecules. However, it has turned out [9], and in fact is intuitively obvious, that correlation energies are rather insensitive to fine details of a SCF wavefunction. Therefore, we devote this paper and two following ones [lO,ll] (referred to hereafter as part II and III, respectively) to the development of a simple and efficient scheme which resigns from the ab initio accuracy of the SCF part of the calculations when one is interested in a study of correlation effects only. However, simplifying the SCF part of the calculations simplifies tremendously the correlation energy calculations. This enables one to investigate large numbers of molecules and also large single molecules in order to find chemical trends in the behavior of various correlation energy contributions. The SCF part will be generally calculated within a semi-empirical INDO (intermediate neglect of differential overlap) approximation [ 121. The parameters con-

tained in such a scheme are chosen to reproduce ab initio SCF results as closely as possible. Since one is working within a minimal basis set when such semi-empirical schemes are used, one can supplement a SCF calculation only by a treatment of interatomic correlations. The remaining intruatomic correlations can be taken into account only by a different procedure. Thereby use is made of earlier findings that intraatomic correlations of different atoms are almost independent of each other [7,13]. This suggests applying an “atom-in-molecule” approach (for a review see MaksiC et al. [14]) in a version proposed by Lievin and co-workers [15]. It involves a population analysis of the wavefunction obtained within a minimal basis set and uses available atomic data [16]. The entire scheme relies therefore on an earlier finding that the correlation energy of a molecule may be broken up into different contributions which may be calculated separately by different methods [17]. It turns out that treating electron correlations as indicated above supplies us with a rather simple and surprisingly accurate method of calculation. As will be shown in part III, it reproduces very well various experimental quantities. At the same time it provides an appealing, simple interpretation of the different correlation energy contributions. In order to gain as much insight as possible into the different effects which influence the size of a correlation energy contribution we have used in this paper one further simplification when doing the SCF calculations, namely we have adopted the bond orbital approximation (BOA). It was originally introduced for solids [18]. In that case analytical expressions can be obtained for the interatomic correlation energies. This enables one to study directly the influence of various physical quantities on their values. The present paper is organized as follows. In section 2 we discuss a variational approach to interatomic correlations, i.e. a local ansatz (LA) developed in refs. [7,19]. In section 3 the BOA is introduced and applied to the study of correlations in different bonds. Their dependences on bond polarities and on one-particle excitations are analyzed. This analysis is extended in section 4 to a study of correlations between different bonds.

13

W. Borrmann et al. / Electron correlations in molecules. I

The method of calculating intraatomic correlation energies is presented in section 5. The probability distribution of finding a certain electron configuration at a given atomic site and its dependence on the interatomic correlations is also discussed in that section. The limitations of the present formalism are pointed out in section 6. Section 7 contains the conclusions and some final remarks. 2. Interatomic correlations The correlation calculations require the SCF ground state I&.-.) as a starting point. We consider valence electrons only and start out from a hamiltonian

one-half of the bond-order matrix as defined in closed-shell SCF procedures. Here and in the following the abbreviation ( ) = (&-r 1 1GSCF) is used. The ground state 1&-F) is Written as IdkF)

= l-b;0 PO

(2.5)

IO>,

where 10) is the vacuum state and c,‘a creates an electron with spin (I in the CM0 CL. The Fock matrix is computed within the INDO approach and we refer to refs. [20,21] for a review and further details of the application of this method. The remaining part of the hamiltonian (2.1) is H,,, = i c

Yjk,a,:akfa,a,/jo

H = Ho + Hint, Ho= Ct,,aLaj,, (2.6)

(2.1)

The a,:( a,,) are creation (annihilation) operators for electrons in atomic s- or p-orbitals with spin u. We denote them by fi(r - R,) where R, is the position of the atom at which they are centered. The zero-differential overlap (ZDO) approximation is made which implies that the overlap matrix between different orbitals is diagonal (exception: kinetic hopping integrals). This approximation leads to a considerable simplification of the interaction elements cjk, and thus enables efficient approach calculations, as, e.g., in the INDO [20,21]. The canonical molecular orbital (CMO) energies and wavefunctions follow from the diagonalization of P-2)

f,, is the Fock matrix

where hj

=

tt~

+

c

(2y,kl

-

r/;lk,)

Dkl

(2.3)

and describes the residual interactions. It results in the correlation effects which we want to consider. As has been extensively discussed before [7,9,19], electron correlations are predominantly local in nature because they describe how electrons avoid each other and thereby minimize the effect of their Coulomb repulsion. For their description one must introduce local functions g,(r) because the extended CMOS are an inappropriate set of functions for that purpose. We choose for them functions which are limited to one atomic site only so that they can be expressed as gi(r,

R,)

= CY;j(R,)f,(r-R,)* i

(2.7)

In the case of a H atom gi(r, R,) is identified with the atomic 1s function. For the second- or third-row atoms there are four functions g,(r, R,) per atom which are chosen as follows. First one constructs from the occupied CMOS x,(r) a set of localized orbitals X,(r). They are related to the former by a unitary transformation

kf

and

X,(r)

is the one-particle

density

matrix

or, equivalently,

= Y VjXj(‘)

(2.8)

i

(2.4)

D,i = (&a,,>

and

correspond

to the classical

chemical

bonds.

To achieve this we have used the method of Foster and Boys [22]. The A,(r) are obtained from the requirement that the quadratic repulsions of the different h,(r) are minimized, i.e. occ

This condition is equivalent to

The local functions g;(r, R,) are obtained from the localized orbitals X,(r) as follows. First the h,(r) are projected onto the different atoms characterized by the functions f,(r - R,). For each atom of the second or third row there are four functions X,(r) with the largest projections. We denote these projections with g:(r). For example, for the C atom in CH, the g,‘(r) are the four sp3 hybrids. The g,!(r) are generally not orthogonal and therefore we apply next a Liiwdin (S- i”) orthogonalization for each atom. The functions which are obtained in this way are identified with the functions g,(r). The correlated ground state wavefunction 1So> is written in the form (2.11)

I+,> =eSl#scF),

which are contained in 0,, we require that all contractions within the O,] are forbidden when expectation values of operators containing the 0,, are evaluated. (By a contraction we understand the expectation value with respect to 1‘jbStF) of the product of one electron creation and annihilation operator as it appears when Wick’s theorem is applied.) The effect on 1tjapscF)of operators of the form exp( --‘lijO,,) has been discussed at length in refs. [7,9,19]. In short, they project a correlation hole around each electron and reduce the charge fluctuations at the different atoms. One could also suppress spin fluctuations by supplementing the 0,, with operators of the form S’S,, where S, is the spin operator for electrons m region gi(r). However, their contribution to the correlation energy is very small 171.Therefore they are neglected here. This does not imply though, that they cannot modify considerably special configurations contained in 1(P~,-.~). The new feature as compared with refs. [7,9] is the additional inclusion of one-particle excitations in S with independent variational parameters [,. They re-optimize the CMOS in the presence of the two-particle operators Oi,. The variational parameters .$,, VI,,are found by minimizing E = (&I I H I ~~)/(~~ = (eSH eS), .

where S= -C&n,i

C17,j"ij

(2.12)

rj

describes local correlations. The 5; and 9ij are variational parameters. The operators 4, and Oij describe one-particle and two-particle excitations. The n, are density operators referring to the local regions g,(r). Thereby n, = Cani and ni,, = b&,. The bz(b,,) create (destroy) electrons with spin u in states g,(r). The operators Oij are defined as 0l.i = 0,; - ( 0,; >9 O;j=n,,n,i, = Rifil,

(2.13a)

if i=j, otherwise.

(2.13b)

In order to eliminate the one-particle excitations

I h>

(2.14a) (2.14b)

The last expression is obtained by making use of the linked cluster theorem 1231.The notation ( ), implies that only connected diagrams are taken when the expectation value is calculated. The evaluation of (2.14b) is not possible without further approximations. We shall replace in the following exp S L 1 + S when evaluating (2.14b). This is well justified as long as the correlations are not too strong as it is the case for most of the stable molecular species. On the other hand this approximation excludes us from treating strongly correlated systems. This point will be the subject of some later discussion (see section 6). With this simplification the energy E becomes

where EsCF = (H) and J?,!~~~is the energy due to interatomic correlations. In what follows we shall use the convention that j?Jiiy is calculated with the full ansatz (2.11) and (2.12), and E&y under the neglection of one-particle excitations. By minimizing E one obtains the following relations C (Oi,HOk,),Vk/ + C (OijHn/),t, i

~n,HO~,~~~ij + ~~n~Hn,~~i

= (‘,,H)c

= 0.

*

(2.16)

i

ij

Since ]&-r) is optimized with respect to the one-particle densities it follows that (2.17)

(n/H}, = 0,

which has been used in eqs. (2.15) and (2.16). It is convenient to define a matrix loi*

HOk,)c

(n,HOk,),Sjj

(“ijHn,)cskl

(n,Hn,)c6ij6kl i .

(2.18)

The solution of the set of equations (2.16) can then be written as follows (2.19) and the correlation energy takes the simple form ijkl

(2.20) ij The correlation energy consists of additive contributions introduced by the different operators Ojj. This allows for interpreting EC,,, as a result of different correlation processes. Of course, the inversion of the matrix T is possible only if it is non-singular. Otherwise the different operators Oij and n, are not independent of each other and describe the same processes. In that case the respective nij(&) are set to zero and the ap=

-

propriately reduced matrix is inverted. Eqs. (2.18)-(2.20) constitute the essential results of the LA with respect to interatomic correlations. They are the starting point of the investigation of different types of molecules. Their main advantage is the simplicity and transparent physical interpretation as will be seen below. The method has been used before in ab initio calculations for molecules as well as solids [7,13,24] and by considering the effects of electron correlations in terms of model hamiltonians 125-273. The present scheme for calculating ground state correlations has similarities to the PCILO (perturbative configuration interaction using localized orbitals) method [28]. In both cases the correlation energy contributions are due to two-particle intrabond and interbond excitations. Both the approaches are, however, quite different. First, the PCILO method starts out from a product of localized bonds while here the starting point is ) +SCF). Second, one uses a perturbation approach in PCILO method while here we are dealing with a variational calculation. Third, certain excitations which are present in KILO, are neglected in our variational ansatz. The first difference disappears when we make the bond orbital approximation for 1(pscF). It should also be mentioned that the LA can be easily extended to the calculation of excited states of finite as well as infinite systems while this is not the case with the PCILO method. In order to characterize the state of a given local region gi( r) we introduce the probabilities of finding there zero, one and two electrons, to be defined as P,,, Pli and Pzi, respectively. They fulfill the following conditions

$ Pni”l,

CVij(OjH)c*

C nP,, = iii, ?I=1

5 n’P,,=Tf

(2.21)

?I=1

and may be determined if the average densities 7i, and average square densities ;;f are known. The averages are calculated either in 1+sPSCF)or in I$,). The charge fluctuations s;f - iif decrease

16

W. Borrmann

et ul. / Electron

due to electron correlations, while the density 2, remains almost unchanged, except for strongly polar bonds. Thus, typically the probabilities Pa, and Pz, decrease while P,; increases with increasing electron correlation. Therefore, the pair distribution function is changed. It is convenient to introduce the following measure of the strength of correlations in a given region g,(r)

correlatrons

in molecules. I

Let us consider a single bond I, for instance between an atom of the first row in the periodic system, e.g., C, and a H atom, constructed from the localized hybrid gi(r) and 1s function g,(r), respectively. They may form a bonding and antibonding linear combination B:, = ab:,,

+ (1 - a2)“2b:,,

A:, = (1 - a2)“2b;o

,

- cub,+,, ,

(3.2)

where if (n,,) =l_

< Q,

(~,l(l-n,t)(l-n,1)1~,) ((l-niT)(l-niL)>

otherwise.

’ (2.22)

We will call A(i) in the following the correlation strength parameter. Globally the correlation strength is a measure of the reduction of charge fluctuations at a particular atomic site X

(n,,)

= (Yz= :(l

+ a,),

(n,,)

= 1 - (Y2= :(l

-a,)

(3.3)

It describes the and (or is the bond polarity. charge transfer taking place between the hybrids 1 and 5 forming the bond, and is determined by the parameters contained in the Fock matrix. If T3 measures the energy difference between the SCF levels 27. =fss -_fi,

(3.4)

and To is the SCF hopping where nx is the total number of electrons at site X. We illustrate the above procedure for treating interatomic correlations and its application in section 3.

To = t, + :(l

parameter

- CX;)“~V,,~~,

the bond polarity

(3.5)

(or is given by

(or = T,/( T3’ + T;)“2.

(3.6)

3. Bond orbital approximation The simplest way to approximate the SCF state of a molecule is by means of the BOA [18]. It assumes that two electrons participate in each chemical bond. Thereby the localized functions X,(r) which are constructed as described above, are identified with chemical bonds and the local regions defined by g,(r) with the corresponding atomic hybrids which participate in the chemical bond. The simplification of the BOA is that different bonds do not mix with each other, i.e. the one-particle density matrix is block-diagonal with blocks of size 2 x 2. Then the SCF state takes the simple form (3.1) where B,+, is a creation operator the bonding state of bond Z.

for an electron

in

Here t, is the bare hopping (i.e. kinetic energy) integral. The simplified form (3.1) for I c#+.~) makes it possible to derive analytical expressions for E,!ijy. We want to use them in order to study how the interatomic correlation energy depends on the bond polarity (or and furthermore how important the one-particle excitations may become. As an illustrative example consider again a single bond only for which we want to determine the correlation energy. It is realized, for instance, in a diatomic XH molecule where X stands for any element of the first or second full row in the periodic system. In that case the ansatz (2.11) reduces to

I kl> = (1 - 5% - vo,) I %.CF)?

(3.7)

where we have introduced

0, =

the abbreviation

17

W. Borrmann et al. / Electron correlations in molecules. I

Furthermore ties will be useful

01, =nlT%l* v, =t[t(K,,,

+ v,,,,)

the following

- l&l

quanti1.0

3

(3.8) (3.9)

F,-‘(a,)

= 1 + a;(1

- a;:)“2VO/t0,

G,‘(a,)

= 1 + :(1-

a;)“2V0/t0.

(3.10)

0.8 0.6 S Gji 0.4 t

As before, the indices 1 and 5 refer to the two hybrids which form the bond. In the INDO scheme there are no other matrix elements yjk, except for the ones which appear in eq. (3.8). For &!ii: the following expression is found - inter E con-

_ -

02

0.4

0.6

08

1.0

aP Fig. 1. Factor (1 - I$)‘/~ which modifies the correlation ergy in a polar bond as a function of bond polarity a,,.

en-

>c

where (O,H),=t(l

(O,Hn,),

0

mm

(3.11)

(n,Hn,),

0

(OIHO,

(O,HO,),

0.2

(3.12)

-a;)‘& = a(1 - a;)3’2~,,F;1(a&

(3.13)

= (1 - c~;)“~t,,G,&x,),

(3.14)

= - &r(l

(3.15)

- a;)‘&.

-a;)5’2&(ap)V;/t0.

(3.17) where Y = a;(1

When one neglects the one-particle excitations, the small density changes due to correlations, finds E corr inter= -a(1

Let us compare the correlation energy in the presence and absence of one-particle excitations. For that purpose we calculate

i.e. one

- a;)2Fo(a,)G,(a,)(

Vo/to)2.

(3.18)

A E;$; is displayed in fig. 2 for various ratios of &/to. In a homopolar bond, i.e. if ai, = 0, there is no contribution to the correlation energy from one-particle excitations since the electron distribution has already been optimized in the SCF wave-

(3.16) 0.25 ,I

The obtained correlation energy E&i:’ (3.16) is the same as the one which follows from the coupled electron pair approximation (CEPA-0) [29] applied to a single bond. Eq. (3.16) makes transparent the dependence of the correlation energy on the bond polarity a,,. It decreases with increasing ap and its behavior is mainly determined by the factor (1 - ai)‘j2 which, for convenience, is plotted in fig. 1. The factor F,( ap) is only weakly dependent on ap. For ap --j 1 the interatomic correlation energy vanishes because the charge fluctuations are going to zero. For ap f 0 the result goes beyond second-order perturbation theory because of the denominator of

Fig. 2. Correction of correlation energy Al&,, due to one-particle excitations as a function of ap for different values of

Ma,,).

Vo/kJ.

’

0.20

I

v0/to =1.5

0.15 ,l.O

0.10 5 0.05 Y

0.5

hl

0 IL3

0

0.2

0.4

0.6

0.8

1.0

aP

18

W. Borrmann

et al.

/ Electron

function (i.e. it is completely determined by symmetry). On the other hand, in a polar bond one obtains energetic corrections due to the coupling between two-particle and one-particle excitations. The curves show a maximum at (or = 0.58. For example in hydrocarbons &/to < 1. For Vo/to > 1 the LA becomes unreliable due to the expansion of exp S (see section 6). Because of the correlations the probabilities are reduced to find two electrons simultaneously in hybrid 1 or 5. For the SCF ground state one finds (3.19a) (3.19b)

0.25

0.5

0.20

0.4

I t

correlutions tn molecules. I

while for the correlated

state

I$,j)

it is

(~o)=~(l+n,)2-$(1-a;)2TJ,

(W%%l

(3.20a) ($0 111511151I Ji”) = $(l - (xJ2 - +(1 - a$)2TJ, (3.20b) with (3.21)

?j = (1 -tX;)“‘F,(a,)V,/r,.

A plot of (3.19b) and (3.20b) is shown in fig. 3 for &/to = 0.4. Although the difference between the two curves decreases with increasing bond polarity (Ye, the relative reduction of double occupancy is larger in polar bonds. This feature is illustrated in fig. 3a by the increasing interatomic correlation strength A [see eq. (2.23)]. In the ionic limit (at aP = 1) the electrons are perfectly localized. A decreases rapidly when this limit is approached. The suppression of charge fluctuations is described by

0.3 02 0.1 a

= (A&[1

-:(l

-~;)3’2~0(~,,)VU/~U], (3.22)

0 where (AH’,)

denotes the charge fluctuations in the independent electron approximation. For a comparison we have also shown in fig. 3b by a dashed line the minimal charge fluctuation

0.3 0.2 N$ z

(3.23)

= +(l - CX;)

0.1 (CAnZC>)min

=

ap(1

-

ap>e

(3.24)

0 0

0.2

0.4

0.6

0.8

1.0

UP Fig. 3. Properties of correlated single bond in BOA. (a) Double occupancy (n,+ JIM,) in I+scF) and IqO) (solid lines) and electron localization parameter A (dashed line). (b) Charge fluctuation (An&) on X atom in 1+scF) and in IIJJ~) (solid lines) and in the perfectly correlated limit (dashed line), as function of the bond polarity ap.

The latter is obtained when the empty (if 0~~> 0), or the doubly occupied (if (or < 0) configuration is totally suppressed. The results for a single bond are little changed when we consider several bonds. There are, of course, new types of correlations, i.e. interbond correlations, appearing as will be discussed in section 4.

W. Borrmann et al. / Electron correlations in molecules. I

4. Application of the BOA to simple molecules In the following we want to consider the qualitative changes which take place when there are several bonds present. For that purpose we consider isoelectronic molecules of the form XH; where X = Li, Be, B, C, N, 0 and F. They are schematically shown in fig. 4 where we have also numbered the different functions forming the bonds. The variational parameters q,i and 6, reduce to three. One parameter n,, corresponds to the intrabond correlations, 9, to interbond correlations and 5 to the one-particle excitations. The corresponding non-equivalent correlation operators are 0, = O,, = n, t n, 1, 0,2 = nlnZ and n,. The interatomic correlation energy takes the form inter E- corr

_ -

-

hd0,H)c

+

411;(O,H&,

- I211(O,*H), +W(O,,HO,,),

+ 4(0,&O,,),)

+2490%(O,HO,*), +4t2((@n,), +h,~(O,H~,),

(1- ~y’F,b,)w~0)

171=

+ 4(Wn2),) + 24wi(012Hnl>c. (4.1)

After evaluating the different expectation values and minimization of the energy one obtains for the variational parameters the coupled equations

X[1-f(l-a~)vo+2a,t], - a~)“‘Go(a,)(Wfo)

E= &,(I

x [170+ W:/w?J. Here the following duced 6 =

WI,22

Fr’(a,)

-

=

(4.2)

new quantities

t v,2,2>

+

have been intro-

2h.61~

(4.3)

- ai)“‘V,/t,.

(4.4)

v,%,

-

1 + +(l - az)3’2Vo/lo +2(1

+ +a;)(1

V, is usually by one order of magnitude smaller than V, because it describes the interaction of electrons in different bonds while V. describes the one within the same bond. The simplest approximation is to neglect the one-particle excitations as well as the coupling term (0, HO,,), between intrabond and interbond correlations. In that case eqs. (4.2) have the solutions T#‘=

(1 - a~)1’2Fo(a,)Vo/to,

vi’)=

(1 - a~)“‘F,(a,)V,/lo.

(4.9

Those expressions can be used as input on the right-hand side of eqs. (4.2) so that one obtains no = #‘[l

- 3q\O’(l - at)V,/Vo

+ 2a,[‘“‘],

170= (I - ~;)“2F,(~,)W~o) ?J,= Tp [ 1 - iTjo(O)(1 - a;) + 2a,E’O’] , .$“O’= iar(l ‘X

- ai)2Go(a,)

[ &(a,)

+

~F,(~,)(I/,/V,)‘](V~/~O)‘-

(4.6) The resulting

correlation

energy is written

as

i”ter= 4c, + 6q, E- corr

(4.7)

/H

where co is the intrabond one bond

correlation

H -._X H’

\ H

IZo = - +(l - ai)5’2Fo(ap)(

Vz/to)

Fig. 4. Schematic structure of XH; molecules described in section 4. Each bond consists of one hybrid (X) and 1 s A0 (H) which are labelled as in the text.

X [l - 3(1 - a~)(V,/Vo)q$+

energy

of

2a,5’0’] (4.8)

and f, is the correlation Cl

=

-(l x

energy between

two bonds

a;)“‘q~,p~/~o)

[l - I(1 - cx’,)?#‘+24*q.

(4.9)

The prefactors 4 and 6 in eq. (4.7) result from the number of bonds and bond pairs, respectively. This can be compared with the corresponding expressions of perturbation theory

(4.10) The main effect of the bond polarity is a reduction of the correlation energy by the prefactor (1 LY’)“~. In a subsequent paper (II) we shall calcul$e the correlation energy for a number of molecules without making use of the BOA. Here we want to estimate these energies within the BOA by using the INDO parameter values I$, Y,, t, and aP .for CH,, BH; and NH:. In all three cases aP = 1. In order to have (Ye appreciably different from unity we have extended the calculation to include the other (unphysical) molecules of the series XH: where X is an atom of the second row. The atomic distances are kept the same as in CH,. The parameter values are listed in table 1 and the results of the interatomic correlation energies f0 and f, are shown in figs. 5a and 5b, respectively. It may be seen that the intrabond correlation energy Q, has a maximum at NH:, whereas the

interbond one E, increases rather fast between CH, and FH:+. The maximum in the former case is a consequence of the suppression of Em by the factor (1 - (Y:)“~ in polar systems. On the other hand, the ratio P’,/tO increases much faster than &/iO and thus no maximum in the latter case is found. The main purpose of fig. 5 is to demonstrate the corrections as compared with second-order perturbation theory. They are twofold: first, due to the factors FO(tiP) and F,(Ix,) and second, due to the coupling between intra- and inter-bond correlation as well as to the coupling between oneand two-particle excitations. The polarity LYEis rather small for the systems XH; with X = Li, Be, B, C and thus second-order perturbation result for the intrabond correlation (4.10) holds in this case. The corrections of f T” due to one-particle excitations are rather small, as can already be seen in fig. 1. They are equal to 2.5 and 6.1 percent, for OH:+ and FH:+, respectively. In the case of interbond correlation one obtains a substantial reduction from EP for each molecule. The corrections of E* are TP due to one-p article excitations seen in fig. 5 only for the (hypothetical) OHi+ and FH:+ systems, where they amount to 2.8 and 6.4 percent, respectively. More realistic systems with CT~i- 0 are the III-V and II-VI semiconductors of the wunite structure [30] to which the above theory may also be applied due to the use of the BOA. The reduction in double occupancy of a hybrid

Table 1 Values of the bond polarity [I~ and electronic interaction parameters to, V,, V, obtained in the SCF INDO calculation for the XHZ molecules. The factors &(a,) and F,(cu,) describe the changes of intrabond and interbond correlation energies from the result given by perturbation theory [eqs. (4.10)]. Electron correlation in the ground state (LA) is described by the reduction of double occupancy ( n5 f ns 1 ) and charge fluctuation (An”,) from their SCF values, as well as by the correlation strength A Parameters

obtained f. (W

z----Be B c N 0 F

0.067 - 0.088 - 0.085 - 0.022 0.141 0.392 0.613

8.09 9.34 10.53 11.95 13.57 13.98 14.49

within INDO model V, (eV)

3.29 3.55 4.00 4.68 5.35 6.42 7.20

V, (ev)

- 0.21 - 0.27 - 0.07 0.42 0.81 1.57 2.30

Results within BOA &,(a,,)

0.998 0.997 0.997 l.ooo 0.992 0.939 0.871

F,(+

0.870 0.885 0.851 0.790 0.763 0.714 0.628

A

(nStn51) SCF

LA

0.218 0.296 0.294 0,261 0.185 0.093 0.037

0.168 0.250 0.248 0.213 0.139 0.059 0.022

0.230 0.222 0.222 0.203 0.247 0.367 0.423

(Add SCF

LA

1.99 1.98 1.99 2.00 1.96 1.69 1.25

1.72 1.76 1.65 1.44 1.33 1.09 0.93

et al. / Electron correlationsin moleedes. I

W. Borr~nn

21

where

0.6 A

(An;)

= 2(1 -a;).

(4.12)

Again Q, and q1 are given by eqs. (4.6) and actual numbers resulting from eq. f4.11) can be found in table 1.

5. Intraatomic correlations

0.12 -

/

i --w 3

I

(b)

/’

I

1

0.08 s 3 5

0.04

1

-I

0 Li

Be

B

C

N

0

F

X Fig. 5. Intrabond correlation energy e,, (a) and interbond correlation energy c, (b) of XH3 molecules in BOA found in second-order perturbation (c,‘, cf), without one-particle excitations (fz’, fTP) and with one-particle excitations (co, (,). Bond polarities aP and electronic parameters from the SCF INDO cafculations (see table I).

due to correlations can be again calculated by using eq. (3.20a) and (3.20b) but now with 9 replaced by Q as given by eq. (4.6). The values of double’occupancy at H atom in the SCF state and correlated state for XHZ are given in table 1. It is maximal in BeHiand BH;, where one has a charge transfer from X to H atoms (see table l), while the double occupancy is strongly decreased and FH:+). The correlain polar systems (OHS’ tion strength A is larger than that of unpolar systems (see table 1). This agrees with the result found in the one-bond case (fig. 3). The charge fluctuation at the X atom is determined from ($0 IAn:, I J/d =(A~:)[l-t(1-(~~)(1)0+61),)],

(4.11)

The correlated ground state I&) was calculated within the minimal (valence) basis set which is used in the INDO calculations. Therefore it cannot contain intraatomic correlations because their treatment requires much larger basis sets. Thus we must find a different way of taking them into account. It has been demonstrated before on small molecules [7,13] that to a good approximation intraatomic correlations are independent of each other for different atoms. Therefore we may calculate them separately for each atom of the molecule. The same conclusions were reached independently by Lievin and co-workers 1151, who formufated an “atom-in-molecule” approach to the calculation of intraatomic correlation energies. Here we will use a modified version of that approach which Gas appiied before to the calculation of intraato~c correlations in hydrocarbon molecules 191. In an atom the number of valence electrons n is fixed. They are in the Hund’s rule ground state term of a given configuration i. For example, in a C atom: n = 4, the configuration is s2p2 and the ground state term is 3P. When the atom A is part of a molecule n is no longer fixed. Instead one has a distribution P,(n) of finding rt valence electrons. Furthermore, for a given value of n there is also a distribution of different configurations i present which we denote by w,fn, A). Both, P,(n) and w,(n, A) can be obtained from I+,> by a population analysis. We shall assume, though, that the relative weight wi(n, A) of the different configurations i for fixed value of n is the same in 1qo) as it is in ] +s,,o). This assumes implicitly that the fluctuations of s electrons are suppressed equally strongly as those of p electrons due to correlations. This will generally not be the case as for example 71

22

W. Borrmrtnn et ul. / Electron correlations in molecule.~. I

and u electrons are differently strongly correlated. However, the changes are not very important for the final results and we avoid in this way an elaborate determination of the weights of the different configurations contained in I#“). Furthermore, when a given configuration is considered we must know the relative contributions of the different terms to it. An analysis of I#,,) with respect to different terms is complicated and has not been performed by us in the present work. Here we assume that the terms appear with relative weights w,(n, A) according to their degeneracies. This leads to an overestimation of the correlation energy because the Hund’s rule term within each configuration, which has relatively small correlation energy due to the parallel alignment of the spins, has a larger relative weight in I$,), than the one following from its degeneracy. This error was shown to be essentially independent of the molecular environment, being = 0.15 eV per carbon atom [31]. It is neglected in the following. We calculate the intraatomic correlation energy from

P,““‘(n)

-p>“-“.

P,=f(l+(Y,)2-,(l-~;)TJ,. P, =+(1

-a;)

+a(1

PO= a(1 - a,)‘-

YE)))@

+(1 - “$jO,

respectively. The probability then given by

(5.3) PA(n) is

distribution

4! ‘A(‘>=

c

m,!m,!(4

m1.mz ??I,+2m,=n

- m, - m,)!

~p,4-“I-“2p,~‘pz”‘~ 05 V&y

0

aa ,’

0.2

:a

0.1

5

0 LiiiI 0.7

A I

0.5 0.4

fa

0.2

’

I

V&=1.0

0.6

1

’

y&o.5

0.3

’ -is ‘h a -’

(5.4) I

0.4

nr

The energies e’,::(A) are obtained from atomic calculations with fixed valence electron number and configuration. Calculations are done for different terms and the results are averaged with weights according to the term degeneracies. The <:‘;I:” have been listed for a number of atoms [15,16]. When we apply eq. (5.1) we assume implicitly that renormalization effects are small. The electronic wavefunction in a free atom has a larger spatial extent than when the atom is part of a molecule. Therefore one could have modifications in the intraatomic correlations when one goes over to a molecule. These modifications are largest for highly ionic configurations [32]. Their weights, in ]lclO), however, are very small. Let us turn to the determination of P,(n). We consider an atom of the first or second full row and treat first the SCF ground state 14S,-F). When there are Z electrons on the average on the atom, the probability of finding one of the N = 8 spinorbitals occupied is p = Z/8. In the BOA the probability distribution for I +,,,) is therefore

(5.2)

It is modified when I$,,) is used instead of I c#J~~~).Let us make the simplifying assumption that the different bonds are independent of each other. In that case the probabilities P2, P, and PO of finding two, one and zero electrons in a hybrid of the X atom are

1 A

= (f)p”(l

.r

V&=1.5

A l-_ .

0.3

0.1 0 0

2

4 ” __-

6

80

2

4 n--

6

0

Fig. 6. Probability distribution in BOA PA(n) (points) and obtained from the normal distribution QA( n) (lines) for different values of electron-electron interaction &/to. Parameters: ii = 4 and ap = 0.

23

U? Bomnann et al. / Electron correlations in molecules. I

In order to obtain an impression how strongly P,(n) is changed due to correlations we assume that (Y*= 0 and i? = 4 (e.g., as in CH,). Furthermore we approximate ne by &/to. In fig. 6 we show the results for PA(n) when 1&-,> is used (V&‘r, = 0) and when Fe/t, + 0 and correlations are taken into account. One notices a considerable reduction of the charge fluctuations with increasing value of Vo/to. The population analysis can be elaborate when LY*# 0 and one is dealing with non-covalent bonds. For that reason we would like to approximate P,(n) by a simple function. It turns out that the normal (gaussian) distribution

comes narrower than that of the normal distribution QA( n). There is one further point which requires attention Excitations of the form s2pnV2 -+ p” at an atomic site are already taken into account within a minimal basis set and therefore have been included when treating the interatomic correlations. They must be therefore subtracted from the calculated atomic values [2] when e:?(A) is inserted into eq. (5.1). The so-called “pseudo-correlation energies” given in ref. (163 take this into account. For an application of that equation to a number of molecules we refer to III.

Q,(n)

6. Limitations of the calculational scheme

= c, exp[ -(n

-n)*/2rl

(5.5)

is well suited for that purpose. The three parameters C,, n and F are determined by the requirements Ce*W

= 1,

C&(n)

= n,,

&Q&) n

=z.

(5.6)

The choice of the width F of the normal distribu‘tion guarantees that Ani is reproduced correctly. It is seen from fig. 6 that the normal distribution approximates the binomial distribution quite accurately. The agreement between the two probability distributions is rather good in the weakly correlated case (V,/t, = 0.5). It becomes somewhat poorer in the case of more strongly correlated bonds, where the maximum of P,(n) be-

The limitations of the above procedure to determine correlation energies are rather obvious. Consider first the interatomic correlations. Here the crucial approximation has been the expansion S = 1 + S when (2.14b) is evaluated. This excludes a tr~t~ent of bonds with strongly correlated electrons as they appear, e.g., in chemical reactions when the bond lengths become very large. Such a limitation is also present when a linearized perturbation expansion within a coupled-cluster ansatz is made (this has been called CEPA-0 in ref. 131). For an estimate of the critical interaction strength beyond which the above expansion breaks down consider a single bond as in section 3 with (Ye= 0. The ground state energy E, in units of 2t, is given within the LA by ELA = 0

-++$x-$2,

(6.1)

Table 2 Ground state energy I& and double occupancy (nl t n, , > in a model of H, molecule as a function of x = Vo/rO x

LA 0.5

1.0 1.5 2.0 2.5 3.0 4.0 5.0 -

(nz,ntr)

Eo - 0.3906 - 0.3125 - 0.2656 -0.2500 - 0.2656 -0.3125

CPE

exact

LA

-0.3169 -0.2188 -0.1782 -0.1532 -0.1182 - 0.0963

- 0.3904 - 0.3090 - 0.2500 - 0.2071 -0.1754 -0.1514 -0.1180 - 0.0963

0.1875 0.1250 0.0625 0.0

CPE

exact

0.1016 0.0600 0.0432 0.0265 0.0179

0.1894 0.1382 0.1000 0.0732 0.0548 0.0420 0.0264 0.0179

24

W. Borrmann

ef ul. / Electron correlatrons m molecules. I

where x = &/to. The same notation is used as in section 3. A comparison of this energy with the exact one corresponding to the model hamiltonian is shown in table 2. It is seen that up to x = 1 the LA gives good results. For x much larger than one the Heitler-London state is a more appropriate starting point. By a canonical perturbation expansion (CPE), configurations with doubly occupied hybrids are generated in virtual transitions [33]. The ground state energy becomes then E?PE = _ ix-’

+

fx-3

_

y5. (6.2)

As it may be seen from a comparison with the exact ground state energy which is also contained in table 2, eq. (6.2) provides a possibility to describe the correlation energy in strongly correlated bonds. Similar findings hold true for the expectation value of double occupancy for each part of the bond. One finds the following expressions (6.3)

(n ,~n,,>,,=a-& (n

ltnll

>CPE=i

(n It~,~L=x,/2(x,

‘x-2(1

- 3x-2 + 9x-4), + 1)3

(6.4) (6.5)

where x, = a i (x” -t 4)“* - 11’.

(6.6)

The subscripts LA, CPE and ex refer to the two different approximation schemes and to the exact result, respectively. In the presence of interbond correlations the.n parameters may diverge for particular values of the interactions. This is seen by considering eqs. (4.2) with E = 0 which apply to the XH: molecules. When one solves for Q one finds no = C/Z with a denominator z=l

-;(l

unless one considers dissociation processes. From section 2 it follows that Z = 0 also implies that the matrix ( 01, HO,,), becomes singular. Therefore, negative eigenvalues of that matrix always indicate a breakdown of the linear expansion of exp S. The same difficulty has also been found in the linearized coupled-pair many-electron theory (LCPMET) [34]. When the intraatomic correlation energies are calculated similar limitations exist. There it is in particular the assumption that the relative weights of different terms within a configuration are given by their respective degeneracies which breaks down when the correlations are strong. The charge fluctuations are reduced in that case and the Hund’s rule ground state within each configuration should have a larger weight than it corresponds to its degeneracy. Therefore, the procedure applied here overestimates intraatomic correlations.

-~~)‘(V,/t,)*F,(a,)F,(a,).

(6.7)

This denominator vanishes for a particular value of V,/to, where the numerator C does not vanish. For (or + 0 one finds V,/to = (8 + 4 Vo/to + $ V,/tJ’*.

(6.8)

Usually V, < V, and therefore the requirement that Z > 0 is generally not a very severe one

7. Conclusions and final remarks We have discussed a simple method for calculating correlation energies of molecules with special emphasis on the role of bond polarities and one-particle excitations. One of the main results is that the interatomic correlation energies are proportional to (1 - (ri)5/2. The BOA has been used in this paper in order to derive analytical results. But the equations derived in section 2 can be applied as well without making that approximation. When the correlation calculations are coupled to semi-empirical SCF calculations the computations become very simple and can be applied to large classes of molecules. Furthermore, it has been pointed out that a useful measure of the strength of interatomic correlations is given by the parameter A introduced by eq. (2.23). It characterizes the suppression of charge fluctuation. The latter can also be studied by calculating the probability distribution PA(n) for finding a given number of valence electrons at a given site A. Interatomic correlations narrow this distribution function. It was shown that PA(n) can be well approximated by a normal (gaussian) distribution.

W. Borr~ann

25

et al. / Efeciron cortelations in molecules. I

The intraatomic correlations were discussed in terms of an ‘*atoms-in-molecule” approach. The limitations of such a description as well as the one of the interatomic correlations were outlined. Altogether, we have presented an efficient method to calculate the correlation energy of molecules. This method is transparent, does not involve extensive numerical procedures and provides considerable physical insight into the phenomenon of electron correlation. The practical usefulness of the present approach has been demonstrated before [7,9] and will be used for further applications in part II and III. In particular, we shall present an analysis of the different correlation energy contributions for a number of molecules involving C-C, C-N and N-N bonds.

tegrals for the two bonds. va,, V,, and W,, will stand for the respective combinations of the intrabond (Vol. V,,) and interbond (I+‘,,) interaction elements K,,, as defined by eqs. (3.8) and (4.3). The correlation energy which results from the interbond coupling (A.l) is then

E corr,l2

(1 - “;J3’2(1

_2 t

1 - &)“*t,

Interbond

correlation

between

polar

As a further extension of the qualitative analysis of electron correlations in the frame of BOA, we consider here the dependence of interbond correlation between two polar bonds on the bond polarities (or, and CQ. This correlation is described by the operator O,, = rz,n2 in the LA (2.13), where n, and n2 are the respective electron density operators for the local regions within the two bonds. The contribution of the interbond correlation to the correlation energy has the form Ecorr.l2

=

-

I’

where (1 - a;t,)i’2(1 - a;t)1’2

F;‘=1+2

+ (1 - a;,)“2t2

x[a(l-1Y~,)V0,+t(l-Q~2)V02

The financial support by the Stiftung Volkswagenwerk is kindly acknowledged by MC. Biihm and A.M. OleS.

Appendix: bonds

+ (1 - a;J1’21*

(A-2)

(1 - ff;J’*t, Acknowledgement

- (Y;2)3’2w:2 F

=

~~,2~~~/~0~2~U,~~~.

+ 5pl~p2v2]

1

(A-3)

Its dependence on the bond polarities a(,,i and aP2 reduces to the previously introduced factor (1 (Y;)~‘~ in the case of equivalent polar bonds. Fl describes the reduction of the interband correlation energy from the one found within secondorder perturbation theory. As an illustrative example, let us consider the isoelectronic molecules X2H:(“-@, having the same geometry as C,H,, where X stands for an atom of the first full row and n is the nuclear charge. They are analysed in more detail in IL The interato~c correlation energy may then be written as follows

(A.1)

We have neglected here the coupling terms (O,HO,,), and (O,,HO,,), to the intra-orbital correlations within two bonds (Oi = nit n, &, i = 1, 2). This simplification is made in order to obtain analytic results for the interbond correlation energy. It leads only to small quantitative changes in the final result. Let us denote by t, and t2 the hopping in-

(A-4)

where the indices 0, (I and n refer to XH, XX(o) and XX(T) bond, respectively, LYEis the polarity of the XH bond and

X[rg+(l-a$)l’2t,]-‘j-i, (A-5) F,,(o,)=(lif(l-a~)[(l-a~)cb+

V-1

x [to + (1 - u~)“‘t~]

-‘}-‘,

(~.6)

F,,=[l~%(VO+~V,)/(r,+t,)]-‘,

(A.7)

F,,*=(l+fVJI,)-‘.

(A@

The interbond correIation energy cont~butions which refer to the interbond correlation between a polar XH bond and non-polar XX(a) and XX(v) bond, respectively, are effectively proportional to (1 - CX~)~,where 1 6 y G 3/2. The result of BOA for Z$!zF (A.4) makes it easier to understand qualitatively the changes of different parts of Ei!$ along the isoelectronic series X,H,, as described in II.

E.J. Baerends and P. Ros, Intern. J. Quantum Chem. Symp. 12 (1978) 169; B.I. Dunlap, J.W.D. Connolly and J.R. Sabin, J. Chem. Phys. 71 (1979) 3396; 71 (1979) 4993. 171 G. Stollhoff and P. Fulde, J. Chem. Phys. 73 (1980) 4548. isI G. Stollhoff and P. VasiIopouIos. J. Chem, Phys. 84 (1986) 2744, 191 F. Pfirsch. M.C. Biihm and P. Fulde, Z. Physik 860 (1985) 171. P. Fulde and MC. 1101 A.M. OleS, F. Pfirsch, W. Borrmann, Bohm, Chem. Phys. 106 (1986) 27. 1111 A.M. Oles, F. Pfirsch, P. Fulde and M.C. Bishm, to be published. J.A. PopIe and D.L. Beveridge, Approximate molecular m orbital theory (McGraw HiII, New York, 1970). [I31 B. Kiel, G. Stollhoff, C. Weigel, P. Fulde and H. Stall, Z. Physik B46 (1982) 1. and K. Rupnik, Croat. I141 Z.B. Maksid, M. Eckert-Maksit Chem. Acta 57 (1984) 1295. [15] J. Lievin, J. Breulet and G. Verhaegen, Theoret. Chim. Acta 60 (1981)_339;

[16]

[17] f18] [19] [20]

References [21]

Ill J. Bardeen,

L. Cooper

and J. Schrieffer,

Phys. Rev. 108

(1957) 1175.

PI D. Pines and P. Nozibres, The theory of quantum

liquids

(Benjamin, New York, 1966). [31 W. Kutzelnigg, in: Methods of electronic structure theory. ed. H.F. Schaeffer III (Plenum Press, New York, 1977) p. 129. 141 P. Hohenberg and W. Kohn, Phys. Rev. B136 (1964) 864; W. Kohn and L.J. Sham, Phys. Rev. A140 (1965) 1133. 151H. Stall, C.M.E. Paulidau and H. Preuss, Theoret. Chim. Acta 49 (1978) 143; H. StoII, E. Golka and H. Preuss, Theoret. Chim. Acta 55 (1980) 29; B. Delley, A.J. Freeman and DE. Ellis, Phys. Rev. Letters 50 (1983) 488. 1610. Gunnarsson, J. Harris and R.O. Jones, Phys. Rev. B15 (1977) 3027; J. Chem. Phys. 61 (1977) 3970; J. Harris and R.O. Jones, J. Chem. Phys. 68 (1978) 1190; Phys. Rev. Al8 (1978) 2159; R.O. Jones, J. Chem. Phys. 71 (1979) 1300; 72 (1980) 3197; 0. Gunnarsson and R.O. Jones, J. Chem. Phys. 72 (1980) 5357;

[22] [23] [24] [25] 1261 [27]

1281 [29] [30] [31] 1301 1331 [34]

J. Lievin, J. Breulet, P. Clercq and J.Y. Metz. Theoret. Chim. Acta 61 (1982) 513. G. Verhaegen and C.M. Moser, J. Phys. B3 (1970) 478; J.P. Desclaux, C.M. Moser and G. Verhaegen, J. Phys. B4 (1971) 296. i. Gksuz and 0. Sinanoglu, Phys. Rev. I81 (1969) 42. W.A. Harrison, Electronic structure and the properties of solids (Freeman, San Francisco, 1980). G. StoIIhoff and P. Fulde, Z. Physik B26 (1977) 257; 29 (1978) 231. M. Scholz and H.J. Klihler, Quantenchemie, Vol. 3 (VEB Deutscher VerIag der Wissenschaften, Berlin, 1981). M.C. BGhm and R. Gleiter, Theoret. Chim. Acta 59 (1981) 127, 153. J.M. Foster and S.F. Boys, Rev. Mod. Phys. 32 (1960) 300. P. Horsch and P. Fulde, 2. Physik B36 (1979) 23. S. Horsch, P. Horsch and P. Fulde, Phys. Rev. B28 (1983) 5977; 29 (1984) 1870. P. Horsch, Phys. Rev. B24 (1981) 7351. A.M. OleS, J. Phys. Cl5 (1982) L1065; A.M. OIeS, Phys. Rev. B23 (1981) 271; 28 (1983) 327. G. StoIIhoff and P. Thalmeier, Z. Physik 843 (1981) 13; A.M. OIeS and G. Stollhoff, Phys. Rev. B29 (1984) 314; A.M. OIeS and P. Fulde, Phys. Rev. B30 (1984) 4259. S. Diner, J.P. Malrieu and P. Claverie, Theoret. Chim. Acta 13 (1969) 1. W. Meyer, Intern. J. Quantum Chem. 5 (1971) 341: J. Chem. Phys. 58 (1973) 1017. W.A. Harrison and S. Ciraci, Phys. Rev. BlO (1974) 1516. F. Pfirsch, Thesis, Darmstadt, FRG (1986). K.F. Freed, Accounts Chem. Res. 16 (1973) 137, and references therein. K.A. Chao, J. SpaIek and A.M. OIes, Phys. Letters A64 (1977) 163: Phys. Rev. 318 (1978) 3453. K. Jankowski and J. PaIdus, Intern. J. Quantum Chem. 18 (1980) 1243.