On the general theoretical foundations of methods for optimum hybrids

487

Journal of Molecular Structure (Theo&em), 169 (1988) 487-508 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

ON THE GENERAL THEORETICAL FOUNDATIONS METHODS FOR OPTIMUM HYBRIDS* GIUSEPPE

OF

DEL RE**

Cattedra di Chimica Teorica, Universiti di Napoli, Via Mezzocannone 4,80134 Napoli (Italy) (Received 4 December 1987)

ABSTRACT This paper is intended as an attempt to establish a badly needed common theoretical background for different criteria for optimum hybrids. First, it is shown that hybridisation can be defined within the CI scheme and is not a specific feature of either the VB or the MO schemes. Next, using the MO scheme as a reference model, a description of the way in which hybrids follow from the search for bond orbitals is given. After analysing various physical and mathematical conditions, the hybrids thus defined are shown to be maximum-overlap hybrids, so that optimum hybrids determined by the maximum-local&&ion condition and hybrids determined by the maximum total overlap criterion coincide up to a choice of weights which is free in either case. Secondary hybridisation associated with lone pairs and redundant A0 bases is shown to fit within the same general scheme. Its relation with orbital occupation is also discussed. Finally, the physical meaning of the various conditions and parameters (maxim&ration of overlap, orthogonality of hybrids, limiting form of lone pair hybrids, etc.) is discussed.

FOUNDATIONS

OF THE I-M3RIDISATION

CONCEPT

Hybridisation was introduced into the theory of chemistry because it provided a link between the hydrogenic model of atoms and stereochemistry. Questions answered with the help of the hybridisation concept were, for example: how can the properties of isolated atoms be used to explain why water is a bent molecule; why are the bonds around the carbon atom in alkanes always arranged tetrahedrally; and why are certain complexes of nickel planar, while others are tetrahedral? Some quantum chemists claim that these are pointless questions, since the stereochemistry of any specific molecule can be predicted, in principle, by computation. This view (discussed in particular by Maksic [ 1] ) has been quite effective in discouraging work on those general concepts and rules, introduced and made into corner stones of the modern theory of chemistry by pioneers such as Linus Pauling. As regards more specifically hybridisation [ 2-81, there * Dedicated to Professor Linus Pauling. ** Present address: Lehrstuhl fUr Theoretische Chemie der Universitiit Erlangen-Nthnberg. 0166-1280/88/$03.50

0 1988 Elsevier Science Publishers B.V.

are a few recent contributions in the literature [g-28]; but they are scattered here and there, and often appear not to be aware of most of the earlier and present work by other authors this seriously hinders progress in the general theory of chemistry. Therefore, an attempt to specify the unifying general theory on which all studies of hybridisation are implicitly based, by rewording the ideas of Pauling, Slater and Mulliken, in the language and mathematics of current record is in order. Such an attempt is the object of the present paper. Hybridisation is one of those general concepts required for classifying and explaining the common features of large groups of molecules. Such concepts cannot be found by inspection of a few computations, however accurate and reliable they may be, but must be introduced by reference to an approximate model. An illustration of this statement is provided by the concept of a Rydberg state, which is quite popular in computational chemistry [29,301, and was introduced by Mulliken with reference to the united-atom model of a molecule [31,32]. The word “model” can be applied to mean analogy, simplification, correspondence, etc. (a list is given in refs. 1 and 33); or can be used to indicate the whole sequence of techniques for solving the Schriidinger equation of a molecule that has led from the Heitler-London scheme to contemporary computational chemistry [ 341. However, as explained elsewhere [ 351, a model used to construct a theory must be a model in a very limited sense, which corresponds to what is currently called a “physical” model. It is by specifying the conceptual and mathematical features of the physical model on which hybridisation is based that the aim of this paper will be reached. THE MVAO-MODEL AND HYBRIDISATION

Following Mulliken [ 31,321, the model underlying the hybridisation concept can be called the Modified Valence Atomic Orbital (MVAO) model which is the penultimate step in a hierarchical scale of models. The first step is the wellknown fixed-atom model, which can be justified in terms of the Born-Oppenheimer approximation and the consideration that classical amplitudes of the vibrations of atomic nuclei around their equilibrium positions are of the prder of 0.8208 m s-1’2 [u/ (Mv) ] ‘I2 ( u = vibrational quantum number, v frequency in cm-‘, A4reduced mass in atomic mass units), which gives, in the worst case, a few tenths of an angstrom. Within the fixed-atom model, the original goal of quantum chemistry consisted in establishing some sort of mapping between the chemical description of a molecule as an edifice of well specified atoms belonging to a small number of different species, and its quantum mechanical picture as a set of semi-free electrons moving in the field of close-lying atomic nuclei. Such a mapping was expected to provide answers to the textbook questions: why are two close-lying H atoms a more stable system than two separated H atoms; why are the double bonds of benzene comparatively unreactive, while butadiene-1,3 is a strongly

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bonds between atoms usually and why the carbon in all hydrocarbons coordinated cording to rigid tetrahedral whereas there no distinct corresponding to around a CC bond? concept of was the to questions the last Pauling’s idea based on A0 concept. a certain in recent the statement among quantum that atomic were merely objects. This probably a to emphasise the A0 makes sense within a specified reference a language AOs are real is only if model is formulated, is consistent with mechanics, serves purpose for it was and leads unique (well-defined) Now, the model of theory of valency does the required A definition into account progress made almost fifty involves a of points. are (a) electronic states a molecule be described one or Slater determinants constructed single-particle states p) . (b) The simplest physically significant description of type (a) is one where the one-particle states lj) are chosen so as to represent some basic general feature of molecules. (c) The hydrogenoid model of atoms is adopted, and every one-particle reference state lj) is assumed to be an atomic one-electron state ImA), namely the m-th Valence Atomic Orbital (VAO) ImpA) of atom A. The VAOs, or better, the MVAOs defined below, form the so-called minimal valence-orbital (Slater or hydrogenlike) basis, which is an essential feature of the model we are studying. (d) A state / mA) is actually to be seen as a Modified (or in situ) Valence Atomic Orbital (MVAO). The modifications (with respect to a free-atom VAO) are of two types: radial distortion and hybridisation. The spirit of the model requires that they should be determined by some general laws or rules before construction of the molecular wavefunction. (e) Radial distortion is a change in the radial part of the wavefunction u,A representing 1mA ) in configuration space. (f) Hybridisation is a mixture of pure MVAOs giving directional orbitals. It can be described as a block-diagonal linear transformation U of the VA0 basis IPA) = (IlpA)

I~PA) . . . I~APA) 1

(1)

where nA is the number of VAOs of atom A. Each block of U has the size of the basis subset belonging to a single atom, namely nAX nA. Hybridisation must satisfy two main requirements: (i) U must be an orthogonal transformation, i.e. the nA VAOs of an atom are mixed together so as to give nA new orthonormal orbitals and (ii ) the new AOs must provide, as far as possible, pairs which can be associated with specific bonds. HYBRIDISATION AND CONFIGURATION INTERACTION

Statement (a) above means that the definition of hybridisation is not a specific feature of either the Valence Bond or the Molecular Orbital scheme. To

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show this, let us consider a general CI expansion. Consider any atom A and the n AOs which are associated with it. In a general CI basis over atomic orbitals of A, B, C, . . .. we shall have a set s of Slater determinants consisting of subsets containing determinants which are identical to one another except for one spin orbital, which goes through the nA possibilities, say, for spin alpha (+), 12sA+), 12p,A+), 12p,A+), 12p,A+). Call SA one such subset, and CB&A(Pu’l, -a.,nA) the coefficients with which the Slater determinants which are the elements of SAappear in the CI expansion. By a well known property of determinants, we can then group those determinants into a single determinant IS), where one column contains a linear combination of the AOs of A (with coefficients C,,/N (N-- normalisation factor). That linear combination of pure AOs of A is a unique hybrid spin orbital (in our case with spin + ), IhA,+

>-

By applying the same procedure to the other subsets (as well as to their counterparts with spin beta), we shall obtain a number of hybrids of A. We can then divide those hybrids into nA groups, and determine four orthogonal “average” hybrids, which are as close as possible to the various hybrids of each group. It will then be possible to decompose the determinants IS) forming each group into a common leading term containing the “average” hybrid, and a residue which is hopefully (at least in a closed shell ground state) just a minor correction. An example of this was given in ref. 36. By repeating the operation on the other atoms, we shall find that the CI expansion contains a leading part formed by determinants containing occupied “average” hybrids of the various atoms. The energy will depend largely upon the self-energies and interactions of those hybrids, since the major contributions to the energy of a molecule always come from one- and two-centre terms. (This analysis does not prove that the hybrids thus found are transferable, but that is not required at this stage.) Since, as is well known, the usual VB and MO schemes can be obtained from general CI over AOs by treating the ionic configurations in a special way, the above argument is also valid for both those schemes. In order to clarify the above, rather complicated, argument we can proceed in the opposite direction, starting from hybrids, and show that, (a), their use amounts to a special choice of the coefficients of configurations over pure AOs, and, (b), that that choice is compatible with either the VB or the MO scheme. To see this, consider the example of two atoms A and B, each having two orbitals (ISA), IzA), and IsB), IzB), respectively). Let (hA) =a1 ISA) +azlzA)

(2)

JhB) =bl IsB) +b, IzB) Now write a CI state vector JAB) limited to the ground state covalent and ionic configurations over IhA) and IhB ) . Using the notation where a Slater

491

determinant is represented by its diagonal, without explicit indication of the electrons, the result can be written \I)=hA+*hB-, /III> =hA+.hA-,

III)=hB+.hAIIV) =hB+*hB-

(3)

Substitution of (2 ) into (3 ), gives

II) =alb,(sA+-sB-)+a&,(sA+-zB-) +a&,(zA+.sB-)+a,ba(zA+.zB-)

III) = II) with spins interchanged ~rII)=u,2(sA+~sA-)+u,u2[(sA+~zA-)-(zA+~sA-)]+a22(zA+~zA-~ 1IV) = 1111)with a replaced by b and A replaced by B. Thus, a linear combination

is equivalent to a full CI over the four orbit& appearing in the two hybrids, with four free coefficients o.Ja,, b2/bl, c3, c4; a, and b, being determined by the nonnalisation conditions. . Since there are at most nine free coefficients (those of the nine singlet configurations over the pure orbitals), we see that using hybrids is a restriction on the CI different from that of the scheme adopted: the VB scheme would have c3=c4- 0, the MO scheme would assign c3 and c4 special values, but the hybrid orbital choice would be unaltered. There may be one difference: the best hybrids would depend on the kind of reduced CI chosen. Indeed, the hybrids could be optimised within a general CI scheme by a MC-SCF procedure but one ought to resist the temptation to introduce one more computational method into quantum chemistry: what is needed with the present state-of-the-art are criteria for determining optimum hybrids from a priori physical considerations, so as to contribute to the understanding of molecular reality. RADIAL DISTORTION, HYBRIDISATION,

LOCALISATION

A study of radial distortion can be made from double-zeta ab initio computations (see (e) above). We do not know of any systematic study, but there are examples of the role it can play, for instance, from studies of hydrogen motion in a hydrogen bridge where it appears that the A0 of the bridge hydrogen changes its shape according to the location of that atom so as to keep its electron population constant [ 281. Radial distortion could be used to make hybrids satisfy multiple conditions, such as cancellation of atomic moments [ 37,381. However, at the STO-3G level of approximation, free atom radial parts are a comparatively good general choice for the ground states of molecules in

492

their equilibrium geometries. For this reason, a general study of hybridisation does not require a preliminary analysis of radial distortion. This is why the abbreviation VA0 is often used here instead of MVAO to denote “pure” basis orbitals. Point (f) (ii) contains the condition for the choice of hybrids, since point ( f) (i) allows for an infinity of different possibilities: we can say that the Hybrid Atomic Orbitals (HAOs) must give as “localised” a picture of the molecule as possible. In other words, the hybrid basis lb) = (IhA)

IbB) ... W))

(which we represent as a row matrix formed by sub-rows associated to the individual atoms A, B, .... Z of the molecule), must be formed by orbitals which participate essentially in only one bond. A bond will normally be a two-centre sigma bond, but a sufficiently general treatment should allow that word to designate also a two-centre pi bond, as well as many-centre sigma and pi bonds (an example of the latter being, of course, the pi system of benzene). LOCALISATION AND THE MAXIMUM OVERLAP CRITERION

The question is now: how can that localisation be ensured? The intuitive answer provided by the genius of Pauling was based, as is well known, on the so-called Maximum Overlap Criterion (MOC by several in particular 211 and Liu 191. Further questions arise at Is the on the of the or can it in terms of a Molecular Oris overlap to energy? to the is well It stands to reason be shared by two A and B, 1vB) are simultaneously non-zero and the higher the values those wavefunctions take in a compact measure of of those in general, if the so chosen is positive a very large internuclear distance, then, in a molecule with several bonds, maximisation of of the At least or ambiguous be found in this on closer is It it will be cancellations of positive be a good measure of the

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actual overlapping of the two wavefunctions ufi and Us. Why not use some other measure of overlap, like the second-degree overlap integral j-1upAl 1hB 12 dV? This is a quantity where no cancellation is possible. (ii) We maximise only over “bonds”. Therefore, we have to decide beforehand where the bonds are, and then adjust the hybrids so as to fulfill the MOC. But would it not be more reasonable to let the maximum overlap hybrids determine where the bonds are? (iii) Finally, is it justified to maximise the sum of overlaps sic et simplititer, instead of weighting each overlap by a suitable factor? We shall try to answer these questions as far as possible, basing our study on our 1963 paper [ 391, where the general theory was formulated, and on later work, where special questions like lone pairs were treated [ 12,40,41]. HYBRIDS, LOCALISATION, AND BOND ORBITALS

In order to proceed, a standard approach must be chosen. We adopt here the MO scheme for two reasons: (i) it is a reasonable approximation for the ground states of molecules in their equilibritmr geometries and (ii) it is now the scheme currently used. Since the “real” hybrids should be those which fit best the general CI picture, as discussed above, we thus imply that the hybrids associated with the bestMO picture of the ground state of a molecule at equilibrium are also the best hybrids for a CI treatment. This seems to be acceptable since the SCF wavefunction is an excellent approximation for atoms and ordinary molecules. We shall pause later on a generalisation of the treatment given below. Granted the legitimacy of referring to the MO scheme, we return to the point that hybrids should provide the most localised picture of a molecule, and ask: in what way can such a condition be imposed on the MO-LCAO scheme? The answer is comparatively simple. In the spirit of the orbital scheme of the elementary theory of valence, we describe a localised two-centre-two-electron bond by a doubly occupied orbital formed by just two AOs IA(B) ) and IB (A) > which do not mix with other AOs. The condition for the existence of such a bond orbital (BO) consists in the requirement that the block of the Hamiltonian one-electron matrix obtained from the AOs (A(B) ) and (B(A) ) should be connected to the other AOs of the VA0 basis only by zeroes, and that the same should hold for the overlap matrix. To see this in practice, consider atoms A, B, and C and the block of the associated Hamiltonian matrix H (or of the overlap matrix S ). In general, that block will be full but, with the ideal hybrids, it should have the form illustrated below in the case of two AOs per atom (say, a linear system) and two bonds AB and BC.

494

A AXBx c

B x

-X_ -

-X-

-

-

-

C xx 1c -

-

-

-

-

x-

X

Scheme 1: X is a diagonal element; x:is an unspecified non-zero element. The two off-diagonal blocks associated with AI3 and BC should contain only one non-zero element, while the AC block should be entirely empty, since there is no AC bond. As mentioned, the picture given in Scheme 1 should hold for both the Hamiltonian matrix H and the overlap matrix S. The ideal hybridisation transformation U is a block diagonal transformation of the VA0 basis whose blocks UA, Un, UC are transformations of the AOs of A, B, C that give the S and H matrices the form suggested by Scheme 1. We must now answer two questions: (i) do such transformations exist and (ii) how can the mathematical conditions for finding them be formulated? The answers (to be discussed below) are: (i) no, if the condition is to be met rigorously; yes, if the condition is to be valid only to first order, and then only if localisation is actually possible (the answer is again “no” if the bond to be described is, say, a three-centre bond). (ii) It is necessary to find local maximum overlap hybrids first, and then to relax a little the “local maximum overlap” condition to allow for orthogonality: the end result is essentially the same as that obtained by application of the MOC in most cases. In addition, it allows a clear-cut correlation with energy considerations, a greater flexibility in the mathematical form of the MOC, a treatment of lone-pair hybrids on the same footing as binding hybrids, and a matrix formulation excellent for computer programming even in the case of very high azimuthal quantum numbers (d, f, g, .. orbitals). MAXIMUM-LOCALISATION

MAXIMUM-OVERLAP

HYBRIDS

The first part of the answer to (i) is easy to prove by comparing the number of conditions with the number of free parameters. If the transformation U were a general linear transformation, there would be no problem in obtaining as few non-zero off-diagonal elements of H and S as required, since one could diagonalise the two matrices and then apply a suitable transformation to generate off-diagonal elements at specific locations, However, that kind of localisation procedure would produce a basis whose elements are not genuine atomic orbitals, but general molecular orbitals, although they can be made very close to

495

atomic orbitals by taking advantage of the many remaining degrees of freedom. If, as in our problem, U is to be a block-diagonal matrix so that no mixing of AOs of different atoms is allowed at the hybridisation level, reduction of H and S to the form of Scheme 1 is not allowed. The second part of the answer to question (i) requires a much longer analysis, which will also be a more detailed proof of the first part. Let S,.,ndenote the block of the overlap matrix (S ) which connects the basis subset of A to the basis subset of B. That block will be in general non symmetric, and will have nAX nB elements. Let VA and UB be as before the two blocks of U associated with atoms A and B, respectively. The AB block of the transformed S matrix will be S'AB

Ma)

=(U+ASABUB)

In order to obtain just one non-zero element, let us derive expressions for the products of SAB from eqn. (4a) by its Hermitian conjugate on the left and on the right. We find T AA=S'AEIS'BA =U+ASABUBU+BSBAUA

(4b)

=U+ASA&BAUA

TBB= S’BAS’m =U+BSBAUAU+ASABUB =U+

(4c)

BSBASABUB

where we have taken into account two facts: (i) the blocks of U are unitary because we want the hybrids of each atom to be orthogonal to one another and (ii)S’BA must be the Hermitian conjugate of S’- and SnA must be the Hermitian conjugate of SAs, if S’ and S are Hermitian matrices. The two matrices TAAand TBBare square, and they will be diagonal (with the same diagonal elements) for a suitable choice of VA and Ug. Then, as was shown long ago [39], S IABand S’nA will also have just one non-zero element per row and per column (not necessarily on the diagonal), the non-zero elements being the same for both, and equal to the square roots of those of TAA and TBB.The ambiguity in the signs of those square roots can be removed, and the elements in question made positive, if the signs of the columns of UA and UB are conveniently chosen. The locations of the non-zero elements of S’ An and S’nA depend on the order of the columns of UA and Ug. No degree of freedom is then left, so the other blocks of S’ are uniquely determined; for example, S ’ is uniquely determined Ao

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by UA and Uc, which are obtained by diagonalising (according to the procedure outlined above) the blocks of S that correspond to bonds formed by A, B, and C. If this were the end of the story, after determination of UA, Un, and Uc from the condition that the T-matrices should be diagonal, Scheme 1 would take the form: A

B

A ;xy ;y

x;-?

;x

B

ix

-

: -

?

c ;ww;-

:wwi?

C -

;ww :ww

;Xy ;-? iy xix

x

-

-

;xy iY

x

Scheme 2.

In Scheme 2 X is a diagonal element representing the one-electron energy for the corresponding HAO; 3cis an unspecified, possibly large, non-zero coupling element; w is an element which is normally very small or zero; ? corresponds to an “eigenvalue” of S’ that is hopefully small (see text); y is zero in the S’ matrix (if we were dealing with the H’ matrix, it would be not equal to 0 if the two diagonal values of the corresponding block of H are not equal). Of course, x, x, ... are not the same everywhere. OVERLAP, LOCALISATION AND THE ORTHOGONALITY

PROBLEM

We now proceed to discuss Scheme 2 with specific reference to overlap, under the implicit assumption (to be discussed later) that overlap is a measure of physical interactions. We shall also ignore, for the time being, the lone-pair problem, and assume that each atom forms as many bonds as it has VAOs. If the elements other than those denoted by X and x are actually small (which depends on the geometry of the molecular system under study) Scheme 2 represents at first sight a choice of hybrids which generate two-centre-two-A0 bond orbitals up to small corrections, because neglect of all the “small” elements would give Scheme 1. Unfortunately, there is a further difficulty: sup-

497

pose that we determine UA and Us so as to give the AB block the form shown in Scheme 2, then we shall have to satisfy eqn. (4~) U + a SsA SABUs = diagonal matrix

(5a)

Since we also want to give the BC block the form shown (which is diagonal up to an interchange of the rows), two new transformations must be determined according to the procedure given above, which will correspond to the atoms B and C. This means that a new matrix Uc will be obtained, which is fine, but we shall also get a new matrix Us, since the condition U + n Snc Scs Un = diagonal matrix,

(5b)

must be satisfied and gives rise to a dilemma, because in general the Un defined by eqn. (5a) will not be the same as the Us defined by eqn. (5b). A simple interpretation of this situation can be given as follows. Let 1hX( Y) ) be the hybrid of X involved in the bond of X with Y. Equation (5a) predicts two hybrids for atom B, one of which, 1hA(B) ), will be involved in the AB bond; the other is not directly connected with bonds, and serves to ensure that the element of the S’ matrix connecting 1hA (B) ) with 1hB (C) ) vanishes. This is not equivalent to choosing the second hybrid of B so as to ensure the required form for S’nc, because eqn (5a) does not contain information about the BC bond. From eqn (5b), we shall obtain, for the latter, two new hybrids, one being the )hB (C ) ) we actually want, the other another spurious hybrid resulting from the diagonalisation procedure. In other words, in trying to satisfy the conditions for Scheme 2, we find a different Us matrix for each bond formed by B, and this gives us not one, but several sets of hybrids (several different Us matrices) for that same atom. This new difficulty can be overcome as follows. What really interests us in the matrix Us, obtained for the bond AB, is the first column; namely the column which is associated with the largest element of S’ac, and therefore contains the coefficients of expansion of the bonding hybrid 1hB (A) ) in terms of the pure orbit& of B. Similarly, what interests us in the Un matrix, obtained for the bond BC, is the second column, which represents the bonding hybrid 1hB (C ) ) ; and so on, up to as many bonds of B as it has pure orbitals. Supposing for the moment that B has four bonds (to atoms A, C, D, E), why not define our final UB as that matrix which consists of the four “important” columns in question? A WAY OUT OF THE ORTHOGONALITY

PROBLEM

The objection to this suggestion is that the four hybrids )hB (A) ) , 1hB (C ), IhB(D) >, W(E) >, do not form an orthogonal set. However, an orthogonalisation procedure which would change them as little as possible, and which would finally yield the “maximum localisation” hybrids we are looking for can

498

be devised. As we shall see, this procedure contains the principle of maximum overlap. First of all, let us illustrate the orthogonalisation procedure with the example of two non-orthogonal unit vectors v and w in a plane. Let V and W be the orthogonal unit vectors which are expected to be as close as possible to v and w. Evidently the angles between v and V and between w and W must be as small as possible. This is the same as requiring that the cosines of those angles, i.e., the scalar products v-V and w* W, be as large as possible. The easiest way to formulate such a condition mathematically is to require that the sum of the two scalar products be as large as possible, subject to the condition that V*V=l, V*W=W*V=O, W*W=l. If this is done, two new unit vectors will be obtained which form an angle of 45” with the bisector of the angle formed by v and w. This means that the angles between each initial vector and the final vector corresponding to it will be the same. This is not completely satisfactory; in fact, suppose that v and w actually represent two hybrids, v forming a strong bond, w a weak one (this might mean, in our approach, a large and a small overlap, respectively). Then, it would be more reasonable to orthogonalise the two vectors in such a way that V was much closer to v than W to w. Using a physical language, we could say that between hybrids of the same atom there is a special kind of repulsion, which tends to displace them from the directions of the bonds to which they belong, until they become orthogonal, and hybrids more strongly involved in bonds will resist that repulsion more effectively than weakly binding ones. This physical interpretation obviously connects overlap with energy, and is particularly useful in the discussion of “secondary” hybridisation, i.e., the hybridisation of lone pairs. The required modification of the orthogonalisation procedure can be realised very easily by multiplying each scalar product by a weight factor related to the corresponding overlap, so that the “weaker hybrid” will contribute less in the sum to be maxim&d. In conclusion, we shall construct the final hybrids IHX (Y) ) by the following procedure. First, we choose a suitable weight factor for each hybrid. Next, we write down the weighted sum of the scalar products of the starting hybrids with their unknown orthogonal counterparts (each scalar product being weighted by the appropriate factor). Finally, we maxim& the expression obtained, subject to the condition that the latter in fact form an orthogonal set. The mathematical formulation allowing programing of the method on a computer has been given in ref. 39. Applications have been given, e.g., in refs. 9,12 and 41 and it is contained in a slightly modified version in the program package of the PCILO method [ 42,43 1. MAXIMUM OVERLAP AND MAXIMUM LOCALISATION

The general procedure outlined above is now readily summarised. First, for every XY bond formed by the atom X the matrix SxySvx is diagonalised; its

eigenvector corresponding to the highest eigenvalue will represent the ideal hybrid of X associated with the XY bond, )hX (Y) ) , in the basis of the pure VAOs of X. Repetition of the same procedure for the other bonds formed by X will yield a set of non-orthogonal hybrids, each having as high an overlap as possible with a suitable partner of the other atom. Finally, we construct the final hybrids II-IX(Y ) ) by a suitable orthogonalisation procedure. We have thus completed the analysis of the maximum localisation condition, and can now show that the latter is essentially equivalent to the MOC. In fact, the hybridisation problem is solved in two steps: (i) find the ideal maximum overlap hybrids IhX (Y) ) of each atom X by maximising the overlap separately for each bond XY and (ii) find the orthogonal best hybrids IHX (Y) ) by applying an orthogonalisation procedure which changes them as little as possible to the ideal hybrids obtained in step (i) for each atom X. The latter condition implies that the total overlap (sum of (HX (Y) IHY (X) ) over all bonds) will stay as close as possible to the sum of the individual maximum overlaps ( hY (X) 1hX (Y) ) : this is essentially the proof of equivalence we were looking for. With the above notation, the MOC can be expressed as the requirement that the optimum hybrids /23X(Y) ) should satisfy the equation (6)

C(HY(X)IHX(Y)>=max

Now, if the procedure outlined above is followed, we also have IHXW

> = IhXW

>cx(y, + IYXW)

>

(7)

for an optimum hybrid (regardless of the procedure used for obtaining it) where CX(Y) =

<~(Y)lhXW)

>

(8)

is as close to unity as possible because, were it not for the orthogonality problem, the hybrids IhX (Y) ) already satisfy both the MOC and the maximum localisation condition, and lyX (Y) ) is a vector orthogonal to (hX (Y) ) whose norm is as small as possible for the same reason. From eqns. (6) and (7) we obtain (again for the real case)

Since we know that the limit superior of the right hand side is obtained when the c-coefficients are unity, maximising the left-hand side is conceptually equivalent to maximising the c coefficients with appropriate weights and under some suitable condition. The weights will clearly be the square roots of the appropriate overlap integrals. It stands to reason that, if one comes sufficiently close to the ideal hybrids JhX (Y ) > , the choice of the weight will matter little; nevertheless, this seems to be the major difference between direct application

500

of the MOC, on one side, and weighted procedures like our own and that of Randic and Maksic [ 45,461. Thus, the MOC amounts to a simplified recipe for finding the coefficients cxty, without going through the explicit treatment of maximum localisation. It is an interesting illustration of the power of intuition helped by simple pictorial models that the pioneers did not go through such an elaborate procedure and saw at a glance the final result. However, explicit rules and criteria always have some advantage, and we now proceed to show how the general mathematical scheme and physical interpretation described above can be utilised to that end. There are five points which have either not been touched on or require further discussion: (i) Choice of the weight factors. (ii) Lone pairs and redundant atomic orbitals. (iii) Determination of bonded atom pairs. (iv) Meaning of the orthogonality condition. (v) Relationship between overlap and physical interaction. WEIGHT FACTORS AND NON-BONDING HYBRIDS

Choice of the weight As has been mentioned, it might be useful to replace the sum of overlaps for the various bonds by a quantity which would normally follow the same trend, but would change some quantitative details in particular cases [44,45,46]. Use of higher order overlap integrals is compatible with the scheme being discussed, but, as long as hybrids are determined solely from the overlap matrix, that would amount to adding an extra feature. Limiting the choice to quantities that come directly into the computations, either the first or the second power of the overlap integrals can be used as weights. It is difficult to justify one particular choice, unless it can be related to energy (and even then, as will be seen, only educated guesses are possible). In our previous work we used the overlap square which was obtained directly from the determination of ideal hybrids; and this does provide good approximations to the localised bond orbitals used [ 17,441. Of course, it is very likely that hybrids determined according to the MOC, i.e., using first-power weights, will be practically the same in most cases, but further work in this direction might be illuminating on several conceptual aspects of the bond-orbital description of the chemical bond. Non-bonding hybrids and molecular geometry The interpretation of the MOC in terms of weighted orthogonalisation of best-localisation hybrids allows treatment of non-bonding hybrids [ 40,471. Let

501

us consider here the practical side of the question, postponing to the next section a discussion of the physical significance of the procedure. Consider a molecule where the number of bonding hybrids formed by a given atom is less than the number of VAOs available (e.g., ammonia and its nitrogen atom, which forms only three bonds, and has four valence orbitals). In order to apply the orthogonalisation procedure described above so as to obtain a unique complete set of hybrids, we have to include non-bonding hybrids, i.e. to take into account the existence of hybrids that cannot be determined on the basis of overlap with the AOs of neighbouring atoms. In order to include those non-bonding hybrids, denoted by IHX (. ) ) , in our procedure, we need, as for the other hybrids, an initial form jh (X (. ) ) , and a weight. Empirical considerations reported in the literature suggest that the ideal (i.e. pre-orthogonalisation) non-bonding hybrid )hX (. ) ) should be a pure-s orbital [ 471. This will have no relevance on the final results if the weight assigned to )hX (. ) ) in the orthogonalisation procedure is zero, and will have a significant effect if the weight is large. Now, as is well known, the stereochemistry of simple molecules, ions and radicals depends on the occupation of their non-bonding orbitals. Therefore, it is reasonable to assume that the weight to be assigned to non-bonding hybrids in the orthogonalisation procedure should be chosen according to their expected occupations. The observed trends can be reproduced, and satisfactory predictions concerning the geometries of various compounds containing heteroatoms, beginning with water, can be obtained by giving standard weights 0.25 to doubly occupied non-bonding hybrids. Taking this as 0.125 per electron, weights 0.125 and 0 can then be assigned to singly occupied and empty non-bonding hybrids, respectively. The results are encouraging [9,10,12]. Thus, the interpretation of the MOC in terms of localisation and weighted orthogonalisation provides a solution to the problem of so-called secondary hybridisation. TRANSITION ATOMS AND REDUNDANT ATOMIC ORBITAL BASES

Redundant A0 bases are found when d and higher orbitals are included in the computations, as is the case for transition metal atoms. The solution of the redundancy problem, i.e., the question of taking into account the existence of many non-bonding orbitals not necessarily associated with lone pairs, is contained in the solution given to the problem of non-bonding hybrids. We can have redundancy in the A0 basis in two cases: (i) when, as happens in a transition metal atom, we have d and f orbitals in addition to s andp ones, so that the number of VAOs exceeds the coordination number. Even in the case of coordination eight such an atom will have one A0 not engaged in binding. As is well known (but often forgotten) even the coordination complexes of those atoms can be treated by hybridisation, thus supplementing standard pic-

502

tures of binding in those complexes, such as the crystal-field picture. However, it is also necessary to have a procedure for obtaining hybrids in cases where symmetry conditions do not determine them from purely geometrical considerations. (ii) When a double-zeta basis set is introduced to allow for radial distortion. An extensive discussion of both cases has been given in previous papers [9,10,12]. A particularly interesting example is a rectangular cluster of five iron atoms [ 121, one in the centre and the others on the corners. That system may be expected to simulate either a group of iron atoms in the crystal, or an actual metal cluster. In ref. 12 the weights assigned to the hybrids of the corner iron atoms not involved in the bonds with the central iron atom were taken to be 0.125. This is a way of simulating to some extent a situation where those hybrids are occupied by just one electron because they are engaged in bonds with other atoms of the solid. By taking different weights those hybrids could be made to represent bonds to the neighbouring iron atoms (along the shorter sides of the rectangle): that would change the picture, as can be expected and is confirmed by computations under way. WHERE ARE THE BONDS?

The most interesting aspect of the above rationalisation of the MOC is that, in principle, it does not require knowledge of the location and nature of the bonds. For instance, in butadiene-1,3, it should provide the standard chemical description in terms of nine sigma and two pi “bonds”. An example which is chemically less interesting but contains several theoretically important features is the system H-O*** H***O=CH2 (with H in the middle for the case of a short H-bond or a transition state), where there are, at least in principle, a three-centre sigma bond and a double bond. For this system the ideal S’ matrix should correspond to the pattern H x:x

.:.:. -.:.:. .:.:. ,:.I. .:.:. --

.

.

.!.I.

.

.

.:.

*

.

. x . .

. . x *

. * . .

.:. .:. .I. .:.

. . * .

. . . .

x

.

.:T:. .:x:. .i.:. x:.:. .:x:x -

* . * *

H

i-:x .I. .:. .:. __ .:.

.

.

.:.

.

.

.:.:. --

0

I;:

. . .

. I): . . .:. . ,I. . __------

. . .

. . . * .

.:x:x .:.:. .:.:.

* I .

. x * I .

. . x

. . x . .

.:. .:x .:.

XI. ,:x .I. .I. .I.

. . x

. . x * .

. . .

. . . x .

,:.:. .:.:. I .:.

. I--. I . .:.i. ,I.:. .1.:x x:x:. --

H

.:* ----

.

.

.:.:.

.

.

*:.

.

.

x:x:. --

H

.I.

.

.

.\.I.

.

*

.:.

.

x

,:.:x

0

.L.

.I.

c

,:.:. . :,I. *:,I. *:.:. .:.:.

.

Scheme 3

In Scheme 3, in the off-diagonal block associated with CO, we expect not one, but two large elements: they should correspond to the sigma and to the pi

503

bond between carbon and oxygen. This is a case where one of the questionmark elements of Scheme 2 is definitely not small. We can generalise this by saying that we shall have a multiple bond whenever an off-diagonal block of S’ has more than one large element. Another feature of Scheme 3 is that one orbital of the formaldehyde oxygen is coupled to one orbital of the water molecule because both are coupled to the orbital of the bridge hydrogen. Here again, we can expect that this is an intrinsic feature of the overlap matrix: upon attempting to find the optimum hybrids, we shall find that it is impossible to get a single large overlap for the bridge hydrogen orbital because the system is close to local symmetry. Therefore, the three-centre bond is respected by the attempt to localisation. With reference to Scheme 2, we can state that there are cases when one or more of the w elements are not small: then, after application of the above procedure, certain atomic orbitals will remain strongly coupled to several partners, as is characteristic of many-centre bonds. In practice, the procedure which consists in “diagonalising” all the off-diagonal blocks of S, and examining the highest eigenvalues will provide a picture of localisation in a given system as far as overlap can tell: there will be no arbitrariness left in the determination of the number and nature of bonds, except for the threshold we want to impose in order to discriminate between “large” and “small” elements. That threshold is no drawback: it only means that, as is already understood in the general theory of chemistry, the localisedbond picture, even when it works, is an approximate one. THE PHYSICAL MEANING OF OVERLAP-BASED ANALYSES

We have explicitly or implicitly attributed a physical significance to concepts and approximations on at least three points: (i) the validity of operations carried out on the overlap matrix for the Hamiltonian matrix; (ii) the orthogonality requirement and (iii) the idea that, besides the geometrical interpretation of the weighting procedure, there is a physical justification for the way in which non-bonding hybrids have been treated, which will now be briefly discussed. Relation between energy.and overlap

The early notion that overlap measures the coupling of atomic orbitals has continued to attract the interest of a number of quantum chemists until very recently, the more so as the Extended Htickel Method is based on that assumption. The argument can be summarised as follows. Consider the bond integral

504

where we have assumed real values for the sake of simplicity. Now expand (T+ V) )ux) as a linear combination of the elements of a complete basis set consisting of )ux) and of arbitrary state vectors orthogonal to ]ux) , and neglect the contribution of the latter. Repeat the same procedure for 1uy ) . The resulting approximate expressions can be written (T+V)Iu,)=(a+bSxy+ (T+V)IUy)

=(a’+b’Sxy+

...)Iux)

@a)

...)Iuy)

(9b)

multiplication of eqns. (9a) and (9b) by ( uy I and ( ux I, respectively and substitution of their average into eqn. (8) gives BI=SXY(A+BSXY

+ ...)

(10)

which is the general form of the various approximation formulae proposed in the past. The whole problem, which is still only empirically solved, consists of justifying the truncation, eqn. (9), and determining the forms and dependence on other parameters of the coefficients a, b, .. . References, tests, and discussions of various concrete forms of eqn (lo), e.g. the well known Wolsberg-Helholtz formula, can be found in several studies, even recent ones [ 48,49,50]. The authors of ref. 50 use the expression of the kinetic energy in terms of the contributions of the atomic orbitals to the natural orbitals to investigate the assumption that the bond energy predicted by a bond-orbital scheme is proportional to the overlap of the two atomic orbitals involved. They confirm the validity of that assumption, at least in the limit for large interatomic distances, and for Slater or Gaussian orbitals having approximately the same orbital exponent. The proportionality constant contains the average orbitals exponent and the interatomic distance. Reference 50 thus suggests very strongly that .the empirical expressions under consideration are justified at least asymptotically, and hence are physically significant. This conclusion is very important, because doubts may arise from the fact that those expressions are not invariant under a zero-point shift and under a hybridisation transformation [ 481, and therefore the specific form adopted in concrete cases will depend on the type of basis and on the energy zero point. Even though this remark must be kept in mind when adopting specific approximation formulae, the arguments of ref. 50 and the comparative success of the approximate expressions under consideration strongly supports the validity of the general approximation eqn. (lo), which stipulates that the off-diagonal element of H associated with atomic orbitals of different atoms can be expected to be large or small depending on whether the corresponding overlap integral is large or small. This means that overlap is in fact a measure of orbital coupling, and a basis transformation which reduces certain overlap values and

505

increases other overlap values will have roughly the same effect on the Hamiltonian matrix, as was to be proven. Orthogonulity

The whole ability of hybridisation to explain directed valency depends on the orthogonality condition imposed on the hybrids. As has been mentioned, at first sight it is not clear why orthogonality should be so important. Nevertheless, it is possible to justify qualitatively the idea that it is the mathematical counterpart of a physical effect when imposed on a basis which is itself physically meaningful. Mathematically, a non-orthogonal basis is as good as an orthogonal one, provided it is linearly independent. However, if the hybrids were not orthogonal, there would be not only a non-zero overlap integral between them, but a non-zero coupling element of the Hamiltonian matrix. This would also mean that in the Hamiltonian matrix the elements y of Scheme 2 would be non-zero: this is tantamount to claiming that there is some effective interdependence of those orbitals. Now, we expect this as long as bonds are concerned (see Scheme 1 ), because we are looking for a description where ideally the AOs are coupled by the molecular field to form bonds; but other interactions are extraneous to the description. It is important to reformulate the last point in terms of occupations. What we are looking for is atomic orbitals which (i) would be singly occupied if they were not engaged in bonds, and (ii) will only “share” their electron. with a partner on another atom. This excludes intra-atomic couplings, because that would involve fractional occupations of the hybrid AOs not as a result of bonding, but as a result of intra-atomic redistribution. Using an objective language, we could say that the hybrid AOs will change as a result of that redistribution, until they become orthogonal, by a process similar to the mutual readjustment described numerically by the SCF procedure. This interpretation of hybridisation has the advantage of providing a foundation for the promotion process which made Mulliken at first quite sceptical of the explanation of directed valency in terms of hybrids [51]. Originally, promotion was the name given to a process in which electrons would be first shifted from one pure atomic orbital to another until the electron assignment required for the “valence state” was reached. Only afterwards would the coupling with other atoms take place. This view is not realistic, because this is not the way in which a bond would form even in the simplest case, and would anyway involve an enormous “activation energy”. On the other hand, if we look at the process of molecule formation as the physical counterpart of an SCF iteration procedure, we see that what really happens is that the molecular field couples the atomic orbitals to one another, so that, say, an s orbital of carbon is coupled indirectly to the p orbitals, because both are coupled to the same orbital of, say, oxygen. Therefore, even within the atom, it will not be

506

legitimate to assign integral occupations to pure AOs. Therefore, the very notion of an electron jumping from an s VA0 to a p VA0 loses its meaning. If we think of in situ atomic orbitals as those orbitals which as far as their atom is concerned carry an integral occupation, then we must consider atomic orbitals which are (as far as possible) neither directly nor indirectly coupled to one another. Those orbit& will be hybrids arising from the gradual coupling of pure VAOs: as one atom comes closer to the other, its orbitals defined in terms of integral occupation will be more and more distorted, and finally become hybrids, without ever increasing in energy, if the bond is a stable one. It is proper to mention here a very interesting way of imposing orthogonality which has not been given enough importance probably only because of its revolutionary character. It is not necessary to apply a compromise procedure as described above if the hybrids are allowed to have complex coefficients [ 521. This opens a completely new perspective in orbital studies, although it has the drawback that final overlap values are distinctly smaller than with the hybrids obtained as above [ 531. In refs 37,38 and 54 other alternative or supplementary aspects of orthogonalisation procedures are discussed. Non-bonding hybrids and occupation

Consider first of all the coupling scheme for pure VAOs, say of oxygen. In general, the formation of a bond will couple the s and the p orbitals to one another. Uncoupling will be realised as far as possible by localising and orthogonalising. However, since oxygen only forms two strong bonds, we shall be left with two hybrids which are not imposed by localisation. Shall we leave them to the play of those forces which manifest themselves through the mathematical orthogonality condition? The answer is negative, because here we have a different electron population with respect to the binding hybrids. The orthogonality condition does not involve populations directly, and therefore the presence of a double population must be somehow introduced. Now, firstly, if there were no other constraints, the electrons would occupy preferentially a lower energy orbital, namely the s VAO. Secondly, the double occupation will impart to the ideal hybrid, in this case the s orbital, a resistance to mixing with p orbitals which is essentially a problem of promotion energies. Therefore, a population-dependent weight must be introduced. Strictly speaking, that weight should be related to some relative activation energy. As has been mentioned, it should be possible to define that activation energy in a more physical way than promotion prior to bond formation. Pending further studies in this direction, we may assume that it is approximately the same for all atoms (as a relative quantity per electron): this is the essence of our assumption of standard weights. F’urther clarification of the whole question could come from the deduction of hybrids by ab initio treatments based directly on the specific form of the

507

first-order density matrix [55 1. Also the a posteriori analysis in terms of hybrids of canonical and localised orbitals obtained from ab initio computations may be very instructive [ 13,281. CONCLUSION

We do not claim that the analysis presented here cannot be improved, indeed it is hoped that some researchers will find time to do so. However, it seems to us that it can already serve as a reference study for those researchers who are interested in connecting their contributions with previous ones, so as to work towards a unitary theory of directed valency. To that end, methods must be not alternative, but “compatible”. Now, the considerations given seem to make that compatibility evident as concerns the major procedures for determining and discussing hybrids. The most important conclusions are: (i) the almost trivial one that hybrids are not specific to either the VB of the MO approach and (ii) the maximum overlap criterion for hybrids can be considered essentially equivalent to the maximum localisation condition, and interpreted as meaning rigorously the same thing. These conclusions open new wide perspectives to Pauling’s hybrids: for one thing, the above analysis indicates the lines along which the MOC can be applied starting from a separate study of individual bonds and atoms so as to determine a priori the number, the nature, and the locations of bonds, and to select a physically significant treatment of lone pairs and redundant AOs. The references given prove that in this way many valuable contributions of the past, instead or remaining just episodic remarks, become building blocks in the construction of the still incomplete edifice of the theory of valence, whose foundations we owe to pioneers of the stature of Linus Pauling. ACKNOWLEDGEMENT

This paper was written in the context of research projects Sponsored by MPI (Italy), CNR (Italy) and DFG (West Germany) whose support is gratefully acknowledged. REFERENCES 1 2 3 4 5 6 7 8

Z.B. Maksic, Croat. Chem. Acta, 57 (1984) 5. L. Pauling, J. Am. Chem. Sot., 54 (1932) 3570. L. PauIing, The Nature of the Chemical Bond, Cornell University Press, Ithaca, NY, 1939. L. Pauling, Proc. Natl. Acad. Sci. U.S.A., 35 ( 1949 ) 495. L. PauIing and V. McClure, J. Chem. Educ., 47 (1970) 15. J.C. SIater, Phys. Rev., 37 (1931) 481. J.C. Hater, Phys. Rev., 38 (1931) 1109. J.C. Slab, Phys. Rev., 41 (1932) 255.

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