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Quantum theory of chemical valenceconceptsKeith R. Roby a ba Research School of Chemistry, The Australian National University,Canberra, A.C.T., 2600, Australiab School of Mathematical and Physical Sciences, Murdoch University,Murdoch, Western Australia, 6153, AustraliaPublished online: 22 Aug 2006.

To cite this article: Keith R. Roby (1974) Quantum theory of chemical valence concepts, MolecularPhysics: An International Journal at the Interface Between Chemistry and Physics, 28:6, 1441-1456, DOI:10.1080/00268977400102721

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MOLECULAR PHYSICS, 1974, VOL. 28, No. 6, 1441-1456

Quantum theory of chemical valence concepts

IL Hybrid atomic orbitals and localized molecular orbitals studied by projection operator techniques

by K E I T H R. ROBY

Research School of Chemistry, Th e Australian National University, Canberra, A.C.T. 2600, Australia t

(Received 7 May 1974)

The processes of the hybridization of atomic orbitals and the localization of molecular orbitals are treated in a unified manner through the introduction of a criterion of maximum or minimum projection. This is an extension of the maximum overlap approach to hybridization. It leads, first, to a proposal for a general index of the extent of hybridization or localization and, secondly, to a new method of finding hybrid atomic orbitals and localized molecular orbitals using projection techniques. Results are presented for the diatomic molecules Be~, HF and CO, and a comparison with previous methods is made.

1. INTRODUCTION

Hybr id atomic orbitals (HAO) and localized molecular orbitals ( L M O ) are of interest because of their actual or potential use in the qualitative description of molecular electronic structure, in the chemical interpretat ion of calculated molecular wave functions, in finding quantities transferable f rom one molecule to another, and in estimating approximate wave functions. Further , hybr id atomic orbitals may have some advantages as basis sets for molecular calculations, while localized molecular orbitals may be the most suitable starting point for calculations including electron correlation.

Since the initial proposals by Pauling [1] and Slater [2], the ' c r i t e r ion of maximum over lap ' has been central to studies of hybridization. Mulliken [3] and Maccoll [4] were first to propose the atomic orbital overlap integral itself as a measure of the extent of hybridization. Later work has been concerned with finding maximum overlap hybrid orbitals in various special cases [5-13]. These methods depend only on knowing the molecular geomet ry ; other methods have been devised, depending on a knowledge of the molecular electron density [14-17], for which a prior calculation of the molecular electronic s t ructure is necessary.

Localization procedures have been developed somewhat independent ly of the maximum overlap approach. The re now exist several criteria for localization given a set of delocalized molecular orbitals, notably the maximization of the distances apart of the centroids of orbital electronic charge (method of Boys [18, 19]), maximization of orbital self-repulsion energy (method of Edmis ton and Ruedenberg [20]) and maximization of various parts of a Mulliken populat ion

]- Present address : School of Mathematical and Physical Sciences, Murdoch University, Murdoch, Western Australia 6153, Australia.

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1442 K . R . Roby

analysis (method of Magnasco and Perico [21]). Following the work of Adams [22] and Gilbert [23] there has also been interest in the direct calculation of localized molecular orbitals because of their potential use in solid-state calcula- tions. Gilbert [23] and Ruedenberg [24] have developed unified treatments of localization methods, and there have been recent reviews by Weinstein et al. [25] and by England et al. [26].

In this paper we use projection operator techniques to develop a unified approach to both hybridization and localization. Some previous suggestions in this regard [27] are taken to the point where calculations become possible. The work is an extension of the maximum overlap approach, and starts from the idea that an atom in a molecule can be represented mathematically by a projec- tion operator on its atomic orbitals, and projection operators for pairs of atoms, atoms in threes, etc., can also be defined [27, 28]. In w 2 we show how this leads to a useful measure of the extent of hybridization or localization whatever the method involved, and in the following sections we set out a new method of generating hybrid atomic orbitals and localized molecular orbitals by projection techniques.

2. T H E PROJECTION NORM AS AN INDEX OF HYBRIDIZATION OR LOCALIZATION

Any single orbital or set of linearly independent orbitals may be represented by a projection operator P having by definition the properties

p 2 = p , p * = p . (1)

We have defined previously [27, 28] the projection operator P~ for any atomic orbital I/z> :

(2)

and the projection operator PA for any atom A in a molecule :

PA = Z ttEA

= Ix ><xAI, (3)

where ]X~,> is a row matrix whose elements are the atomic orbitals Itx> on A, and <X-~l a column matrix of the corresponding bras </x]. The atomic orbitals on any two atoms A and B may be combined in the set ]XAB>, for which the projector is

PAB = ]XAB>SAB-a<XAB I (4)

and the appropriate overlap matrix SAB has elements.

S ~ = <~lv>, It,>EA or B, ]v>zA or B. (5)

In a similar way, projectors may be defined for other pairs of atoms PAc, PBC, etc., for atoms in threes PABC, and so on. We may add to these the projector PMo on the molecular orbitals, which may be those found in the Hartree-Fock approximation or in some more accurate calculation :

PMo = ~. Ii></l = ]0><0l , (6)

I~> being the row matrix of molecular orbitals li>.

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Quantum theory of chemical valence 1443

Now each projector that we define represents uniquely some subspace of the molecular Hilbert space, and so we may speak of the subspace for atom A, the AB subspace, and so on. In Hilbert space mathematics, if P is the projector for some subspace and If) is any vector in the total space, the component of If) in the subspace is P[f), and its magnitude is measured by the projection norm ][Pf[I given by [29]

Upfl [ = ( (f [p tp l f ) )l/2 =(<flPl[>) 1/2 (7)

using the properties, equation (1). If If) is normalized, the following limits apply :

0~< ]]P]]] ~< 1, (8)

where ]lPf]] =0 if [f) has no component in the subspace.

[]P~H = 1 if [f) belongs wholly to the subspace.

In view of these limits and their meaning, the projection norm would seem to be an appropriate measure of hybridization or localization. The optimum directed hybrid orbital on atom B relative to atom A is then that orbital Iv) on B having maximum component PA] v) about A and therefore maximum I[PAv[I. Similarly the degree of localization of any molecular orbital [i) about atom A is measured by [IPAiII, approaching unity for complete localization. A value of unity has the additional meaning that molecular orbital [i) can be given an exact expansion in terms of the atomic orbitals on A alone. Similar remarks apply to other projection norms such as IIPABiH, although if HPABiH approaches unity, [i) could be localized in the AB bond or on either of the individual atoms. Hence ItPABilI, IIPAill and IIPnill have to be considered together in order to determine precisely what type of localized molecular orbital [i) is.

In effect the physical idea of maximum overlap is here being translated into the mathematical idea of maximum projection. The relation to the overlap integral itself is seen in the following examples. Let S~ be the overlap integral between atomic orbitals I/z) and Iv), and let B ~ = <~[i> be the overlap integral between atomic orbital [/~) and molecular orbital [i). Then the following results obtain :

(a) comparison of single atomic orbitals :

liP,,vll = llP= }l -- I sz, t, (9)

(b) measure of hybridization about A :

IIPA"II=( E s,,2) (10)

(c) measure of localization about A :

IIP /ll=( E (11) /xCA

(d) measure of localization in the AB bond :

IIe Bitl--( E E * ~ B, yeA or B. Br (SaB ),,B~r ,/~e:A or /z v

(12)

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1444 K . R . Roby

Thus in (a) above the present t reatment is identical to the overlap integral measure, while the other cases are generalizations of this.

Some of these points are illustrated in tables 1 and 2. Results are listed for the diatomic molecules B%, H F and CO, the appropriate projection norms being given for the initial delocalized molecular orbitals of Ransil [30] (table 1), for localized molecular orbitals obtained f rom these by Magnasco and Perico [21] according to their population criteria (table 1), and for the localized molecular orbitals found by Edmiston and Ruedenberg [20] using their self-repulsion energy criterion (table 2). T h e same molecules will be considered from the maximum projection viewpoint in w 4.

Delocalized orbitals Magnasco-Perico localization Molecule Projection norm relative to Projection norm relative to

AB MO PA PB MO PA PB

Be2

HF

CO

log 0"7132 0"7132 /Be 0"9995 0-0285 lou 0-7116 0"7116 /Be' 0'0285 0"9995 2o 9 0"9483 0.9483 loBe 0'9431 0-7294 2ou 0'7119 0'7119 loBe" 0"7294 0"9431 lo 0.0585 1'0000 iF 0'0055 1"0000 2o 0"6140 0-9911 loF 0"1190 0"9796 3o 0'5637 0"8791 boHF 0"8270 0"8919 1 ~v 0 1"0000

lo 0"1027 1"0000 iO 0"0221 0"9998 20 1"0000 0.1009 iC 0"9996 0"0350 3or 0'8586 0'9800 loO 0"2038 0"9948 4o 0'4627 0"9053 loC 0"9947 0"0681 50 0'9144 0"3730 boCO 0"8758 0'9663 1 zr 0"6666 0"8915

Table 1. Projection norm analysis of delocalized and localized molecular orbitals for Be~, HF and CO (Ransil STO calculations).

Because the calculations involve the LCA O approximation and we are using the same atomic orbitals for the analysis, all ]IPABi][ are unity. Th e effects of localization are apparent in table 1. For these and other diatomic molecules studied but not listed here, inner shell and lone pair localized molecular orbitals commonly have projection norms greater than 0.99 about the atoms concerned. The y are, thus, to all intents and purposes completely localized in the projection sense.

T h e only exceptions to this rule are the orbitals classified in references [20, 21] as lone pairs on Be in B% (/aBe and /aBe'). The i r projection norms are considerably below 0.99 (0.94) about one atom and indicate a large component (0.73) about the other. Such values are similar to those of bond orbitals in other molecules (for example, b~ H F and ba CO), suggesting that a reclassifica- tion may be appropriate.

In the case of bond orbitals, the projection norms show the relative distribu- t ion of the orbital about each atom. For instance, the ba orbital in CO has a larger component (0.97) about the oxygen atom than about the carbon atom (0.ss).

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Quantum theory of chemical valence 1445

Molecule Localized Projection normt relative to AB MO PAB PA PB PAB(i) PA(i) PB(i) PAB(V) PA(V) PB(V)

Be2 iBe 1 0"9994 0"0397 0"9946 0-9946 0-0002 0"1188 0'0977 0'0392 iBe' 1 0"0397 0"9994 0-9946 0-0002 0.9946 0"1188 0'0392 0"0977 loBe 1 0-9428 0"7294 0-1033 0-1033 0'0024 0"9958 0"9394 0-7279 loBe' 1 0"7294 0"9428 0-1033 0-0024 0"1033 0"9958 0-7379 0"9394

H F i F 1 0'0023 1-0000 - - - - 0"9907 0-1287 0'0023 0'1096 ltlF 1 0.1160 0.9960 - - - - 0.0768 0.9984 0.1160 0.9947 lt2F 1 0.1160 0-9960 - - - - 0-0767 0-9984 0.1160 0.9947 lt3F 1 0.1160 0.9960 - - - - 0.0768 0.9984 0.1160 0.9947 boHF 1 0.8110 0.8826 - - - - 0.0289 0.9995 0.8110 0.8803

CO iC 1 0.9997 0.0381 0.9934 0-9934 0.0006 0-1353 0.1160 0.0377 iO 1 0.0174 0-9998 0.9919 0-0005 0.9919 0.1556 0.0172 0-1133 IoC 1 0"9936 0"0468 0"t031 0-1026 0"0107 0"9965 0"9906 0-0436 hrO 1 0"0230 0"9876 0"0947 0"0188 0"0929 0"9972 0"0171 0'9860 b t lCO 1 0'7524 0"9201 0"0565 0"0273 0'0495 0'9986 0'7511 0'9185 bt2CO 1 0"7526 0"9202 0"0565 0"0274 0'0494 0'9986 0"7513 0"9185 bt3CO 1 0"7526 0"9202 0"0565 0"0273 0'0494 0"9986 0"7513 0"9185

J" i denotes projector on inner shell atomic orbitals only. v denotes projector on valence atomic orbitals only.

Table 2. Projection norm analysis of Edmiston-Ruedenberg localized molecular orbitals for Be2, H F and CO.

By and large, t he M a g n a s c o - P e r i c o L M O ' s ( tab le 1) and the E d m i s t o n - R u e d e n b e r g L M O ' s ( t ab le 2) are ve ry s imi la r ( c o m p a r e t he resu l t s for Be2). D i f f e r ences arise, s ince the f o r m e r m e t h o d m a i n t a i n s the s i g m a - p i s y m m e t r y s e p a r a t i o n whe rea s t he l a t t e r a l lows m i x i n g ; th is l eads to b e t t e r lone pa i rs on f luo r ine in H F in t ab le 2, and to d i f f e r en t b o n d o rb i t a l s ( ' b a n a n a b o n d s ' ) in CO.

I n t ab le 2 t he ana lys i s is ca r r i ed f u r t h e r to show c o m p o n e n t s in t he i n n e r shel l and va lence a t o m i c o rb i t a l subspaces . As expec t ed , t he i n n e r - s h e l l L M O ' s have smal l ( b u t no t i n s ign i f i can t ) m a g n i t u d e s in the va lence o rb i t a l subspaces , whi l e lone pa i r a n d b o n d L M O ' s have smal l m a g n i t u d e s in t he i n n e r - s h e l l subspaces . T h e t ab le shows tha t the d iv i s ion of a tomic p ro j e c to r s in to i n n e r shel l and va lence pa r t s is a usefu l ex t r a s tep in the c h a r a c t e r i z a t i o n of m o l e c u l a r o rb i ta l s .

W e shal l n o w c o n s i d e r t he pos s ib i l i t y of f i n d i n g h y b r i d a t o m i c o rb i t a l s a n d loca l ized m o l e c u l a r o rb i t a l s in such a w a y t ha t the r e l evan t p r o j e c t i o n n o r m s are m a x i m i z e d .

3. DETERMINATION OF HYBRID ATOMIC ORBITALS AND LOCALIZED MOLECULAR ORBITALS BY PROJECTION TECHNIQUES

In w 2, hybridization and localization were seen as involving relations between subspaces. Suppose P and Q are projectors representing such subspaces, the subspaces being distinct and non-orthogonal :

P Q 4 Q , P Q # P , PQ#O. (13)

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1446 K . R . Roby

Then the following statement is derived in a straightforward way [27] from the ' outer projection theorem ' of L6wdin [31 ] and the pairing theorem of Amos and Hall [32] and L6wdin [33] :

" The operators PQP and QPQ have non-vanishing eigenvalues in common. The associated eigenvectors are " paired " such that if Ifi) and Igi) are the eigenvectors of PQP and QPQ respectively belonging to the common eigenvalue a~, that is,

PQPIh) = a~ [fi), (14)

QPQIg~) =~lg~) , (15)

then Q If,> = ail/2lg,> (16)

and PI g , ) = o~i 1/2 Iti)" (17)

Furthermore O~i 1/2 is equal to the projection norms :

~1/2 = II Pg, II = II Q[, II (18)

and these norms are extreme for the two subspaces concerned." That eigenvector ]fi) of PQP having'maximum eigenvalue ~i is therefore

the vector in the subspace of P having maximum projection norm relative to Q and hence maximum component in the subspace of Q. It therefore satisfies our projection criterion for optimum hybridization or localization. Direct application of this result gives the following hybridization and localization schemes :

(a) Bond hybrid atomic orbitals. The projectors concerned are the atomic projectors PA and PB. After finding the eigenvectors of PAPnPA, one chooses that orbital (or orbitals) I/x) on A having maximum eigenvalue and thus maximum IIPBt~II about B. This orbital is the optimum directed hybrid from A to B in the projection sense. A corresponding orbital on B is found from PBPAPB .

(b) Inner shell and lone pair hybrid atomic orbitals. In this case a criterion of minimum projection is appropriate. For inner shell and lone pair orbitals on atom A one takes the projector PA and a projector/DR on the remaining atoms in the molecule. PR involves the total set of atomic orbitals on atoms other than A and requires the inverse of their overlap matrix. The eigenvectors of PAPItPA having minimum projection norms and thus minimum components in the re- mainder subspace will be optimum inner shell and lone pair orbitals on A relative to this subspace.

(c) Molecular-density-determined hybrid atomic orbitals. In procedures (a) and (b) above only the molecular geometry need be known. Given as well a prior solution for the molecule, the operator PAPMoPA may be used to give maximum projection hybrid orbitals on A relative to the molecular subspace.

Another interesting case is PaBP~IoPAB, which leads to optimum bonding combinations of the atomic orbitals on A and B.

(d) Localized molecular orbitals. Corresponding to (c) above, the eigen- vectors of PMoPAPMo having maximum eigenvalues will be the optimum

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Quantum theory of chemical valence 1447

localized molecular orbitals on A. Best inner shell or lone pair LMO's are obtained if Pa is replaced respectively by the projector P~(i) on the inner shell atomic orbitals of A, or PA(v) on the valence atomic orbitals. Best two-centre, three-centre . . . . bond LMO's are found by substituting in turn PAB, PABC, etc., for irA.

(e) Direct calculation of localized molecular orbitals. Following the work of Adams [22] and Gilbert [23], one may derive Hartree-Foek-like equations for the direct calculation of LMO's. For example, the set of LMO's I~A) on atom A results from the solution of the equation :

[ F - Pz~o(F- P,)e~m] l ~,A> = I~A)= 1/2, (19)

where ~t 11~ is a diagonal matrix of the projection norms ((@,IPMo{@~))112, and F is the Hartree-Fock hamihonian operator. Again, one may find like equations for two-centre, three-centre, etc., bond LMO's by using PAB, PABC, etc., instead of PA.

The computations necessary in any of the procedures (a)-(d) are quite simple. The only integrals requiring computation are overlap integrals. There are however some aspects of the method that require further elaboration.

Returning to the general projectors P and Q, we suppose that P has an orbital expansion

P=Ix>S- I<x l , $=<XIX> (20)

while Q has the expansion

Q= I~>T-I(~], T = (qb]~). (21)

To find the new set ]X") in the subspace of P having extreme projection norms with respect to Q, one first orthogonalizes IX> by some suitable method such as symmetric orthogonalization :

IX'>= 1X5$-~/2, (X' IX'5=I (22)

and then finds the unitary matrix U that diagonalizes <x'lQIx'>, where,

<x'lP Ix'> = $-112<x 1~)'1"-1(~ 115 $-112" (23)

Then

IX"> = IX> $-1/~ U. (24)

Similarly one obtains a set ]qb") in the subspace of Q having extreme projec- tion norms with respect to P :

tqb") = Ic~>T-zl~V, (25)

where V is the unitary matrix that diagonalizes the matrix product

T-z/2(~ IX) S-z (7. I c~)T-Z/2.

At this stage the connection with the maximum overlap hybridization methods of King et al. [34] and of Murrell [5], Golebiewski [6] and Gilbert and Lykos [7] is clear. If the orbitals in the sets IX") and I~") are placed in order of decreasing eigenvalue, then L6wdin's ' outer projection theorem ' means that

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1448 K.R. Roby

the trace or any ordered subtrace of the corresponding projection norms is a maximum. But

T r (<x" IQ Ix">) ~/~ = T r ((X" I ~ " ) ( ~ " IX")) 1/2

= E Y ( < x , " l r t 3

E E I<x/'lr (26)

However, according to the pairing theorem, there is a one-to-one correspondence between the sets IX") and Iqb") such that each IX") has non-zero overlap with only one ]r Therefore,

Tr ( (x" lQlx")) l /2= ~. I<x/'l~/'>l = T r <X"Ir (27)

Thus the trace or any ordered subtrace of the overlap (X"[~") is a maximum, which is the result obtained by King et al. Their discussion of the relation with the work of Murrell, Golebiewski and Gilbert and Lykos, who dealt with hy- bridization in the special case of a central atom surrounded by a number of ligands each contributing one orbital, is thus also applicable here.

There are also some problems of non-orthogonality that arise. Thus the final set of hybrid atomic orbitals on atom A found using procedures (a) and (b), or the final set of LMO's found from procedure (d), is non-orthogonal because it results from repeated applications of the procedures using different projectors (for example, PAPRPA, PAPBP,, P~PcPA, etc.). As pointed out by Adams [22], the important point is that the final sets be linearly independent and of the same dimension as the starting sets. The optimum LMO's or HAO's are in fact likely to be non-orthogonal. Orthogonality becomes a constraint which may be useful if further calculations are to be done, but which moves the HAO's or LMO's away from the optimum.

Therefore two procedures have been derived, one of which produces linearly independent but non-orthogonal hybrid or localized orbitals while the other maintains orthogonality. We shall consider the case of localization and then indicate how the methods may be adapted to hybridization.

Procedure A. Non-orthogonal localized molecular orbitals. (i) Find optimum inner shell and lone pair LMO's for each atom in turn,

using the projector P~lo and projectors such as PA(i) and P,(v) on the inner shell and valence atomic orbitals on atom A.

(ii) Form the projector on these LMO's and project from the total set of molecular orbitals. If I~b) is the starting set of canonical molecular orbitals and 10) the set of inner shell and lone pair LMO's from step (i), we form a new set 1~ ' ) :

10'>=(1-Po)1r Po=lO>(<OlO>)-1;2<Ol. (28)

(iii) The set I~ ' ) is linearly dependent and non-orthogonal. Convert I~ ' ) to a reduced orthogonal set I~") by canonical orthonormalization [35] :

1~"5 = f~ ' ) vd-1;2, (29)

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Quantum theory o[ chemical valence 1449

where V is the rectangular matrix of eigenvectors of ( ~ ' 1 ~ ' ) having non-zero eigenvalues and d is the diagonal matrix of those eigenvalues.

(iv) The projector on Idp") is formed and used with atomic pair projectors PAB, PBo etc., in turn to find optimum bond LMO's. These are added to the set [0) of LMO's.

(v) If the dimension of 10) is not yet equal to the dimension of the starting MO's I~) , then steps (ii), (iii) and (iv) may be repeated, projecting the bond LMO's from Idp"), reorthogonalizing and finding three-centre LMO's from the projectors P ABO PBCD, etc.

(vi) If higher-order LMO's are necessary, steps (ii)-(iv) may be repeated using projectors for greater numbers of atoms. Alternatively the procedure may be terminated at any time by adding delocalized MO's left after step (iii) to the LMO's previously found.

Using this procedure, the most localized possible set of MO's is built up in the sense that individual atoms are dealt with first, then pair bonds, then three- centre bonds and so on, until the required number of LMO's is obtained. A threshold value of the projection norm above which an MO is regarded as localized and is included in the set of LMO's, below which it is excluded, is a necessary input to the procedure. Orthogonality properties of the final set are as follows : inner shell and lone pair LMO's on the same atom are orthogonal (or nearly so, depending on the value of the threshold chosen) ; LMO's on one atom are not necessarily orthogonal to LMO's on another ; all individual atom LMO's are orthogonal to all bond LMO's ; bond LMO's of different order are orthogonal (e.g. all pair LMO's are orthogonal to all three-centre LMO's) ; bond LMO's of the same order are not necessarily orthogonal.

Procedure B. Orthogonal localized molecular orbitals Symmetric orthogonalization [35] is applied to the atomic LMO's between

steps (i) and (ii) in procedure A, and is included in step (iv) for bond LMO's. The final set of LMO's is then orthonormal and, by the resemblance theorem for symmetric orthogonalization, resembles the non-orthogonal set as closely as possible.

By similar procedures one may form a set of non-orthogonal or orthogonal HAO's for an atom A, using the projector PA in turn with the projector PR on the remaining atoms to select inner shell and lone pair HAO's, then the pro- jectors for individual atoms PB, Pc, etc., beginning with nearest neighbours and continuing until the required number of HAO's is found.

4. HYBRID ATOMIC ORBITALS AND LOCALIZED MOLECULAR ORBITALS FOR DIATOMIC

MOLECULES: Bee, HF, CO

The projection methods of w 3 are here applied to the sample diatomic molecules Be 2, H F and CO, whose LMO's as found by other methods were discussed in w 2. In all tables referred to in this section, the 2s' orbital is that found by Schmidt orthogonalization against the Is atomic orbital.

Hybrid atomic orbitals determined from atomic projections (w 3, proposals (a), (b)) are given in table 3. For diatomic molecules, of course, proposals (a) and (b) coincide so that those orbitals on an atom having maximum projection norms relative to the other atom will be optimum bond hybrids, while those

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1450 K . R . Roby

Projection norm relative Hybrid atomic orbitals

Atom to other atom ls 2s't 2pa 2wr 2p~

Be in Be2 0.8869 0.12044 0.77517 0.62017 0 0 0.3376 0 0 0 1 0 0.3376 0 0 0 0 1 0"0952 0"43529 0.52022 -0.73477 0 0 0-0066 0"89220 -0.35845 0.27476 0 0

F in HF 0"5615 -0.09766 -0.84089 0.53231 0 0 0 0"99130 -0.03475 0.12697 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 - 0"08827 0'54008 0.83697 0 0

C in CO 0.7940 0.10593 0.67191 0"73302 0 0 0"2567 0 0 0 1 0 0.2567 0 0 0 0 1 0.0609 0.64708 0.51315 -0"56388 0 0 0.0053 - 0.75502 0"53405 - 0"38043 0 0

O in CO 0-7940 -0.11842 -0.82217 0.55678 0 0 0"2567 0 0 0 1 0 0"2567 0 0 0 0 I 0-0609 0.34591 0.49145 0.79926 0 0 0.0053 0'93076 -0.28725 -0.22620 0 0

t 2s' = Schmidt orthogonalized 2s orbital.

Table 3. Hybrid atomic orbitals from atomic projections for Be~, HF and CO (STO's).

having m i n i m u m projections will be o p t i m u m inner shells and lone pairs. T h e total set of such orbitals on any one a tom is or thonormal .

Fur thermore , the hybr id orbitals on the two atoms correspond in the sense of the pairing theorem. Each orbital on one a tom has non-zero overlap only with the orbital on the other a tom that has the same projection norm. Th i s no rm is equal to the absolute value of the overlap integral be tween the two orbitals. T h e equality of the projection norms is evident for CO in the second column of table 3.

Maximizat ion of the overlap is seen by compar ing the m a x i m u m value of the overlap integrals (projection norms) in the table with the m a x i m u m values for the original atomic orbitals before t ransformation. Thus the value of 0.89 in Be is to be compared with the prior m a x i m u m value of 0.51 (2SBe-2SBe') ; in HF , 0"56 compared with 0.47 ( l s n - 2 s v ) ; and in CO, 0-79 compared with 0-49 (2p~c-2So).

T h e orbital having m i n i m u m projection norm would be classified in each case as an inner-shell hybr id orbital. T h e orbital(s) having next to m i n i m u m projection norms are the lone pair orbitals. (For F in HF , the inner shell and lone pair H A O ' s are degenerate.) Note that there are significant contr ibut ions of the ls atomic orbital to the valence HAO's , and of the 2s and 2p~ orbitals to the inner shell HAO's .

We have also computed valence H A O ' s for CO allowing only the valence atomic orbitals to mix and maintaining the respective ls atomic orbitals as the

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Quantum theory o[ chemical valence 1451

inner shells. T h e o p t i m u m bond hybr id orbitals now have projection norms of 0-7839, only slightly below these of table 3, and are given by

b e = 0-67767 (2Sc') + 0-73537 (2p~c),

b o = 0.83083 (2So') - 0.55652 (2p~o).

T h e coefficients are also similar to those in the table. Sigma lone pair orbitals also have a reduced projection no rm (0.0422) and are given by

1~ c = 0.73537 (2sc') - 0.67767 (2p~c),

I % = 0-55652 (2%') + 0.83083 (2p~o).

In table 4, hybr id atomic orbitals for the same molecules but found by molecular projection (proposal (c) of w 3) are given. T h e r e is now the possi- bility of projection no rms reaching unity, whence the relevant H A O has an exact expansion in te rms of the molecular orbitals. Th is is the case for the Be inner shell HAO, F inner shell and sigma and pi lone pairs, and C and O inner shell and sigma lone pair HAO's . Where degeneracies occur in the projection norms (as in F, C and O) any linear combinat ion of the degenerate orbitals has the same projection no rm and is an equally valid solution. T h e r e is no obvious connect ion between the H A O ' s of table 4 and those of table 3. T h e r e is less

Projection norm relative to molecular Hybrid atomic orbitals

Atom orbitals I s 2s't 2pa 2pTr 2p~

Be in Bez 1.0000 0"99878 0"04751 0.01315 0 0 0.9897 0"04927 -0"97042 -0.23636 0 0 0-6645 -0.00153 -0-23672 0-97158 0 0 0 0 0 0 1 0 0 0 0 0 0 1

F in HF 1"0000 0.20236 0.97486 0.09326 0 0 1"0000 0 0 0 1 0 1"0000 0 0 0 0 1 1"0000 0"97927 -0.20055 -0.02849 0 0 0.8689 0"00907 - 0"09709 0"99523 0 0

C in CO 1 0"53873 -0-77708 0"32545 0 0 1 0-84238 0-50278 -0-19394 0 0 0-8932 -0.01292 0 " 3 7 8 6 3 0.92546 0 0 0.6666 0 0 0 1 0 0"6666 0 0 0 0 1

O in CO 1 0.05498 -0"96111 -0-27062 0 0 1 0"99818 0"05966 - 0.00911 0 0 0.9639 0.02490 -0"26963 0.96264 0 0 0"8915 0 0 0 1 0 0.8915 0 0 0 0 1

Table 4.

~" 2s" = Schmidt orthogonalized 2 orbital.

Hybrid atomic orbitals from molecular projection in Be2, HF, CO (Ransil STO calculations).

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1452 K.R. Roby

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Quantum theory of chemical valence 1453

contribution of inner-shell atomic orbitals to valence HAO's, and of valence atomic orbitals to inner shell HAO's in the former.

Localized molecular orbitals calculated according to proposal d of the previous section are given in tables 5-7. Procedures A and B developed in w 3 have been followed and where applicable both non-orthogonal and orthogonal LMO's are listed. In the case of the HF the same orthogonal set results from either procedure because there are no orbitals localized on hydrogen alone and iF is automatically orthogonal to laF. The threshold value selected to signify localization was 0.985. Note, however, that all localized orbitals listed excepting /aBe and leBe' have projection norms greater than 0.997 and are thus almost completely localized in the projection sense.

Projection norms LMO PH Pv Hk Fk Fs' Fpo

iF 0.0524 1.0000 0.00471 -0.99965 -0.00251 0.00362

loF 0"4409 1"0000 -0"00006 -0"24357 1'02467 0"09708

boHF 0"7079 0"8689 -0"59809 -0"01567 0"20388 0"68607

Table 6. Localized molecular orbitals for hydrogen fluoride by the projection method. (Based on Ransil STO calculations.)

As expected, the non-orthogonal LMO's are more localized than the ortho- gonal LMO's. It is the latter which are closest to the LMO's of Magnasco and Perico and Edmiston and Ruedenberg, as is evident from a comparison with tables 1 and 2 of w 2.

In fact the non-orthogonal LMO's iBe, /Be', iC, iO, leC and leO together with iF and laF have projection norms of unity about their respective atom and are therefore completely localized. Notice the extremely small, almost negligible, values of coefficients of atomic orbitals on the other atom in these LMO's. The projection norm value shows that it would be possible to eliminate these completely.

Upon orthogonalization projection norms for orbitals localized on single atoms are lowered. It is interesting in the light of the discussion in w 2 that the most marked case is the lone pair LMO on Be in B% (table 5). Here the non-orthogonal leBe LMO has a projection norm about Be of 0-9898, which qualifies it as a ' lone pair LMO ' according to the threshold value used. Ortho- gonalization however reduces this to 0.9435, which classifies it rather as a sigma bond LMO.

One difference from the Edmiston-Ruedenberg LMO's is that the present method maintains the sigma-pi symmetry separation. For example, there are no ' banana bonds ' in CO ; rather we find the beCO LMO as listed in table 7 while the 7r molecular orbitals as obtained by Ransil remain unchanged. These orbitals are, however, degenerate with respect to the projector Pco. Hence any linear combination of them is also an acceptable solution. It would there- fore be possible to obtain ' banana bonds ' but it is not essential to the method.

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Quantum theory of chemical valence 1455

5. CONCLUSIONS

The criterion of maximum or minimum projection is a natural extension of the maximum overlap approach to hybridization. It places the latter in the context of Hilbert space mathematics and permits a unified treatment of hybridi- zation given only molecular geometry, hybridization given the molecular electron density, localization by transformation of canonical molecular orbitals, and direct calculation of localized molecular orbitals.

Because of its mathematical significance, ease of calculation, and performance with respect to the example molecules studied, the projection norm index seems a very suitable general measure of hybridization or localization.

In the determination of maximum or minimum projection hybrid atomic orbitals or localized molecular orbitals, only atomic orbital and atomic orbital- molecular orbital overlap integrals are required. Although certain complications are introduced into the procedures by the need to maintain linear independence, the method has some other advantages over previously existing methods. For instance it offers the choice of optimal but non-orthogonal HAO's or LMO's, or slightly less than optimal but orthogonal HAO's or LMO's. When applied to localization, the method is not restricted to single determinant Hartree-Fock wave functions but is equally applicable to more approximate or more accurate wave functions. Further, any suitable set of atomic orbitals may be used to effect localization (for example, it does not have to be the same set as may have been used in an LCAO calculation of the molecular orbitals).

The performance of the method has been demonstrated by its application to simple diatomic molecules. Some interesting results are obtained particularly in the calculation of LMO's. The superior localization in the non-orthogonal case (as measured by the projection criterion) is evident, and it has been shown that the orthogonal LMO's are closest to those found by other methods. Values of the projection norms obtained indicate when contributions from other atoms or bonds can be eliminated entirely (projection norm unity) and when equivalent localizations exist (projection norms degenerate) leading to equivalent descrip- tions of the bonding (for example, ' banana bond ' or cr-Tr descriptions).

REFERENCES [1] PAULINe, L., 1931, ̀ 7. Am. chem. Soc., 53, 1367. [2] SEATER, J. C., 1931, Phys. Rev., 37, 481. [3] Mt~gLmrN, R. S., 1950, .7. Am. chem. Soc., 72, 4493. [4] MACCOLL, A., 1950, Trans. Faraday Soc., 46, 369. [5] Mgam~LL, J. N., 1960, .7. chem. Phys., 32, 767. [6] GOLEmrWSKI, A., 1961, Trans. Faraday Soc., 57, 1869 ; 1963, Acta phys. pol., 23,

243 ; 1970, Ibid., A, 37, 319. [7] GILBERT, T. L., and LYKOS, P. G., 1961, y. chem. Phys., 34, 2199. [8] LYKos, P. G., and SCHMEISING, H. N., 1961, .7- chem. Pkvs., 35, 288. [9] COULSON, C. A., and GOODWlN, H., 1962, .7. chem. Soc., 2851.

[10] DEL RE, G., 1963, Theor. chim. acta, 1, 188. [11] RANDI~, M., and MAKSI6, Z. B., 1972, Chem. Rev., 72, 43. [12] VALrr L., and PELXKAN, P., 1969, Theor. chim. Acta, 14, 55. [13] HUBA6, I., LAUaINC, V., and KVASm6Ka, V., 1972, Chem. Phys. Lett., 13, 357. [14] RUEDENBF.RC, K., 1962, Rev. rood. Phys., 34, 326. [15] PF.TERS, D., 1963, ft. chem. Soc., p. 2003. [16] McWEENY, R., and DEL RE, G., 1968, Theor. chim. Acta, 10, 13. [17] POLAK, R., 1970, Int. `7. quant. Chem., 5, 317.

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1456 K . R . Roby

[18] BOYS, S. F., 1960, Rev. mod. Phys., 32, 296 ; 1966, Quantum Theory of Atoms, Molecules and the Solid State, edited by P. O. L6wdin (Academic Press), p. 253.

[19] FOSTER, J. M., and BoYs, S. F., 1960, Rev. rood. Phys., 32, 300. [20] EDMISTON, C., and RUED•NBERC, K., 1963, Rev. mad. Phys., 35, 457 ; 1965, J. chem.

Phys., 43, $97 ; 1966, Quantum Theory of Atoms, Molecules and the Solid State, edited by P. O. L6wdin (Academic Press), p. 263.

[21] MAGNASCO, V., and PF.RICO, A., 1967, J. chem. Phys., 47, 971. [22] ADAMS, W. H., 1961, J. chem. Phys., 34, 89 ; 1962, Ibid., 37, 2009 ; 1965, Ibid., 42,

4030, 1971, Chem. Phys. Lett., 12, 295. [23] GILBERT, T. L., 1964, Molecular Orbitals in Chemistry, Physics and Biology, edited by

P. O. L6wdin and B. Pullman (Academic Press), p. 405. [24] RUEDENBERG, K., 1965, Modern Quantum Chemistry, Vol. 1, edited by O. Sinano~Iu

(Academic Press), p. 85. [25] WEINSTEIN, H., PAUNCZ, R., and COHEN, M., 1971, Adv. atom. molec. Phys., 7, 97. [26] ENGLAND, W., SALMON, L. S., and RUEDENBERG, K., 1971, Topics Current Chem.,

23, 31. [27] ROBV, K. R., 1973, Wave Mechanics--The First Fifty Years, volume of honour of

L. de Broglie, edited by W. C. Price, S. S. Chissick, and T. Ravensdale (Butter- worths), p. 38.

[28] ROBY, K. R., 1974, Molec. Phys., 27, 81. [29] VON NEVMANN, J., 1955, Mathematical Foundations of Quantum Mechanics, English

edition (Princeton University Press). [30] RANSIL, B. J., 1960, Rev. mod. Phys., 32, 239. [31] L6WDIN, P. O., 1965, Phys. Rev. A, 139, 357. [32] AMOS, A. T., and HALL, G. G., 1961, Proc. R. Soc. A, 263, 483. [33] LCSWDXN, P. O., 1962, ft. appl. Phys., 33 (suppl.), 251 ; 1964, Preprint No. 125,

Quantum Chemistry Group, Uppsala. [34] KING, H. F., STANTON, R. E., KIM, H., WYATT, R. E., and PARR, R. G., 1967, ft.

chem. Phys., 47, 1936. [35] LbWDIN, P. O., 1970, Adv. quant. Chem., 5, 185.

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