On the use of localised molecular orbitals in some triatomic radicals

Chemical Physics 52 (1980) 23-31 © North-Holland Publishing Company

ON THE USE OF LOCALISED MOLECULAR ORBITALS IN SOME TRIATOMIC RADICALS T.A. CLAXTON Department of Chemistry, The University, Leicester LE1 7RH, UK Received 8 April 1980

The population localisation method of Verwoerd is extended to an ab initio framework and is applied to molecular systems of low symmetry for which there is evidence of delocalisation and which contain a large number of non-bonded electrons . Three isoelectronic radicals are chosen, HCN -, HCO and HBO - , which, for example, have large experimental proton isotropic hyperfine coupling constants of 13 .75 mT, 12 .6 mT and 9 .42 mT respectively, suggesting that the unpaired electron is extensively delocalised in each radical . The purpose of this paper is to investigate whether this delocalisation can be usefully discussed within the concept of localised orbitals . The rephrasing of the population localisation method into one of solving an eigenvalue problem eliminates numerical difficulties encountered elsewhere, but produces a procedure which is non-uniform in nature . Some of the difficulties inherent in the procedure are illustrated, but careful application of the method produces highly localised, and similar, orbitals for all three radicals even though the delocalisation of the spin density in each, inferred from ESR, is very different . Some advantages of the method are discussed .

1. Introduction Ever since Fock [1] pointed out that the properties of single determinantal wavefunctions, constructed from doubly occupied orbitals, are invariant to unitary transformations of the orbitals, there have been many attempts to define criteria to find those transformations which will produce localised orbitals that reflect the conceptual nature of chemical bonds . The limitations of symmetry criteria [2] were overcome using self-energy localisation methods [3,4] as were other methods more appropriate to ab initio type calculations [5, 6] . The term localised molecular orbital (LMO) was first proposed by Edmiston and Ruedenberg [4]. Another approach to localisation was notably developed by Magnasco and Perico [7] which seeks to maximise orbital populations in a uniform manner. A general feature of population localisation methods requires some guess as to the nature of the LMO's, but instances of convergency difficulties have been reported [8]. Still

within the spirit of population localised orbitals, Verwoerd [9] has proposed a non-uniform mathematical procedure which involves the solution of eigenvalue equations . A major objective [9] was to provide highly localised orbitals which can be transferred from one molecule to another. The necessity for orthogonal LMO's is abandoned [5] as this in itself may limit transferability . However, Verwoerd indicates how his method can be used to obtain a set of orthogonal LMO's by "progressive orthogonalisation", but does not investigate it further . In this paper the Verwoerd eigenvalue method is extended for use with non-orthogonal basis sets and the progressive orthogonalisation procedure is investigated . This is part of a study into the use of localised orbitals in doublet state radicals and has been applied to isoelectronic radicals for which there is experimental evidence of extensive delocalisation of the unpaired electron .



4 2

T.A. Claxton / Localised molecular orbitals in triatomic radicals

2 . Determination of Verwoerd localised orbitals Let E be an cigenvector matrix of which the columns are the LCAO coefficients of the m occupied molecular orbitals. The metric tensor of the space defined by the N basis orbitals is generally known as the overlap matrix, S, and must be included explicitly in all expressions . In the case of MINDO/3 this tensor is the unit matrix (9] . Atomic (basis) orbitals which belong to a "chemical bond" are called local orbitals and all others non-local, To study a given bond all reference to the local orbitals in E are removed by setting the corresponding rows to zero : the resultant matrix is called A . It is convenient to define matrices E and R in an orthogonal basis, 8 and a?, say, where d'=S' 12E and l# a S'r2R. There Is now no necessity to repeat Vcrwoerd's development since it is identical as S1/2R in place of his E long as we use $' t E and and R, so that eq . (2 .6) of ref. [9] becomes

Non ~ AX,

(2.1)

where A is a diagonal matrix with elements A, . A r . . . A„„ where It is assumed that A 1 s A2 :: : #: A,„. The lowest cigcnvaluc, A,, corresponding in the first column vector, X,, of X, is used in give the best localised orbital, L,, L, ~ EX,,

(2 .2)

where A, Is interpreted as a measure of the residual non-local content in L 1 . If orthogonal • fly between the LMO's is not important, the most Incallsed set of LMO's which can be obtained Is to choose R ' by defining j succ°a'sivc new sets of local orbitals using the same E in each case . From (2 .1) a A ;'' and X'I are found . Combining all the X', (' into a matrix U, AV' forms the Jib column of U, the "best" lncali tl nrl)itnls tire L fR XU,

(2 .3)

whttirn cnrrcspnnding to eq . (2 .2) we have IUtV' .

(2.4)

ftrr,crdttre wilt not, in general, make U an rafrn nnl matrix . "ore this procedure will be

labelled as the non-orthogonal localisation method. We require all the LMO's to be orthogonal and so the progressive orthogonalisation scheme is now described . The following is repeated m -1 times successively, with the understanding that E(°) = E and j starts at 1. From the matrix E lf- " a residue matrix R't' can be obtained once the jth set of local orbitals is chosen. Solving the eigenvalue equation R(i)tSR(')X(n = A (u)XU ', (2.5) from which A ;D can be extracted . Since L(l) = E (1-n X (n,

(2 .6)

L ;' is the required jth LMO and E (t' is now L'" with the first column L oll) removed, that is, E" is an n x (m -j) matrix . The procedure is now repeated for the (j +1)th set of local orbitals . It is still possible to use an equation similar to eq . (2.3) but the matrix U is now understood to be orthogonal . U can be constructed from the X (t) matrices but is more easily evaluated as described below [eq . (3 .1)] . The A ' retain the same meaning as before but it is to be expected that, apart from A ;", generally A j'1 will be larger in the progressive orthogonalisation scheme. It does not matter if the lowest eigenvalue in the progressive orthogonalisation scheme is degenerate, since the "lost" eigenvector can be picked up from the (j + 1)th set of local orbitals (if chosen to be the same as the jth) . It seems logical [9] to expect that the first LMO in the progressive orthogonalisation scheme will be more localised than all future LMO's (that is, A ~" < A V' ; j > 1 or more generally, A (10 < A ;'' ; i < j) on the grounds that, because of orthogonality constraints, E", from which the future LMO's can be selected, corresponds to a reducing space . This has been found to be not generally true and that this fact has a significant bearing on the order of selection of the sets of local orbitals. For m LMO's found by progressive orthogonalisation, there are m 1 ways of choosing the order in which they are selected . The claim by Verwoerd to have



TA

stn )

low

o"" #4w a s am

overcame . in h non r as found [fit in the umfon i method frA Magnum Perish [71 to be vkvvd fans comparing Iiks with bke no way of finding the best order of mf or&n seem to he available . try that OxAd he mood tart in the Progtessive is to minimise the sum of the ei A'00 1. but we offer no procedure how to do this. T major difference in V d's mettiod compared with many others is its r r-moll in the determination of the LMO°s. S

3. AppBadlm to

.

° rAO

UHF type serf 'tt a at, 114 a

by

Three radicals have been used to study the application of Verwoerd's method in the ab initio framework. They are HBO - . HCW and HCO. This choice has been made for a number of reasons. Although they are isoeles nic and are thought to have similar geometries . the interpretation of electron spin resonance experiments suggests that there is a significant difference in the distributions of the unpaired electron. For example, the proton isotropic hyperfine coupling constants are 9 .42 mT. 12.6 mT and 13.75 mT far HBO - [ lOJ, HCO [11] and HCN - [121, respectively . In addition. minimal basis set ab initio UHF calculations already exist for these radicals [131 which reproduce the experimental electron-nuclear isotropic hyperfine coupling constants quite well. For example, the proton coupling constants were calculated as 9 .16 mT. 11 -25 mT and 14.03 mT for HBO HCO and HCN , respectively . These minimal basis set calculations are used here . since they are much easier to interpret in chemical terms than extended basis set calculations and also retain some similarity with MINDO/3 calculations [9L There are no mathematical reasons why Verwoerd's method should not be applied to extended basis set calculations. Instead of using UHF orbitals it was decided to use natural orbitals, which were obtained from each UHF calculation by diagonalising the first-order density matrix . It is simpler to study

r

d, (of .not terra hi a

of l

-

*k radkoh

25

v

of the natural bxy Arm cn -=

ion

UHHFerateftnnthrn -

h the f th the oM u

t I I and . a In the with tlon ntm*ff equal to 1. i oa h q vely r to the spin ed open dwg SC? nvefunetion for a . Tht rsortnal practice is to e the y ly. that is, the set of natural ofttah of fl-spin, kaving the singly occupied orbital t feed. This embles the single sal wavdunttlan in canonical [4j, or oabitah, to continue to be czprcncd as a single determinantal wavcf - in 1)4O's. However, if there is reason to believe that the unpaired electron is involved in the bonding orbitals, it may be argued that the singly orbital should be included in , that is, the set of the total' natural orbitals by electrons of a-spin should be locafse+d- To clarify the terminology ii is emphasised that the set of natural orbitals containing electrons of a-spin is identical to that set made up of the natural orbital containing the unpaired electron plus those natural orbitals occupied by electrons of $-spin . For example, if H 2CO is described in terms of IMO's and an electron is removed to form H.,CO . it is of interest to know if this electron is removed from a localised orbital . This should be obvious once the natural orbitals occupied by a-spin electrons in H TCOO arc localised. Unfortunately . if all the orbitals occupied by aspin electrons are locabsed . the original single determinanual wavefunction must now be expressed as a linear combination of single determinantal wavefunctions in I)O's to reproduce the original total electron density . Similarly for a radical such as HCW, it is relevant to ask if the electron added to HCN is localised in HCW . particularly if it is in on orbital ro account for the bent structure . Again this type of question is readily answered if the b

r



26

T.A . Claxton / Localised molecular orbitals in triatomic radicals

orbital: containing the unpaired electron is included in the localisation procedure. If the unpaired electron is in an orbital of different symmetry to the doubly occupied orbitals, the two possible sets of LMO's, that is, using either the a-spin orbitals or the fl-spin orbitals, will lead to identical results . If the unpaired electron occupies an anti-bonding orbital and all doubly occupied orbitals are either bonding or non-bonding, the difference between the two sets of LMO's will probably be very small. The same will apply if it so happens that the unpaired electron is in a very localised orbital. Generally localisation of the a-spin orbitals will be expected to produce much better LMO's than those from just the 13-spin orbitals . Verwoerd's method should immediately distinguish between these two possibilities because of the localisation measure . A . Verwoerd's method also enables an assessment of the best way to choose the sets of local orbitals since just one LMO at a time is studied . It will be useful to know, if the a-spin orbitals are localised, the amount that each LMO overlaps with the orbital of the unpaired electron (the natural orbital with occupation number 1) . The square of this overlap will be the probability that the unpaired electron will be found in each localised orbital [16]. From eq . (2.3) E;SLi = Ui ,

(3 .1)

where E; is the ith column of E, that is, equivalent to the ith canonical MO, ¢ i, and Li is the jth column of L, that is, equivalent to the jth LMO, Xi. Therefore eq . (3 .1) can be written equivalently as

f

X 0 i xi dr=U;,

may have some value as a measure of, the suitability of LMO's in the non-orthogonal localisation method. The important quantities in this paper are Uki; where k is the orbital containing the unpaired electron and j is one of the LMO's, since Uki is then the probability of finding the unpaired electron in the jth LMO .

4. Discussion of localisation procedure HXY shall be the general term to refer to one of the radicals HCN -, HCO or HBO -. The geometry of the radical and the labelling of the LMO's are described in fig . 1 . Since the HCO results are intermediate between those of HCN and HBO-, as expected, only the latter two will be described in detail . The non-orthogonal localisation method was used initially to find the best way of selecting the suitable combinations of basis orbitals to form the LMO's . These conform to chemical expectation and are given in table 1 . 4.1 . Localisation of the f3-spin orbitals

Unitary transformation of the orbitals occupied by electrons of f3-spin leaves the single determinantal wavefunction unchanged with respect to all physical observables and would be the simplest way of interpreting these radicals in terms of LMO's . Including the two is orbitals on atoms X and Y, and one orbital of it-symmetry, there are eight possible LMO's but only seven orbitals occupied by electrons of f3-spin . So one of the LMO's must be empty of 6-spin

(3 .2)

so that U; is simply the overlap of the ith canonical MO and the jth LMO . Similarly eq . (2 .3) can be written equivalently as Xi = E

bond

Uiclb=,

Ynbh

(3.3)

t

from which it follows that 2: ; U' =1 . If U is orthogonal, as in the progressive orthogonalisation procedure, then L U24 =1 . This latter sum

Fig . 1 . Geometry and labelling for LMO's for the radical HXY (nbh = non-bonding hybrid orbital, p=p-type atomic orbital) .



T.A . Claxton / Localised molecular orbitals in triatomic radicals

Table 1 Comparison of residues (A) from the non-orthogonal localised method using either the $-spin orbitals or the a-spin orbitals for the radical HXY Spin

orbitals :

9

a

HBO - HCN - HBO - HCN -

Radical :

Local orbitals 2s(X), 2p r (X), 2p,,(X) 2s(X), 2p .(X), 2p„ (X), ls(H) 2s(Y), 2p.(Y) 2p,(Y)

Xnbh

0 .2700 0 .1558 0-0000 0 .0000

XH

0 .0000 0.0000 0 .0000 0 .0000

Ynbh YP

20), 2p1 (X), 2s(Y), 2pr (Y)

XY

0 .0002 0 .0005 0 .0000 0 .0000 0 .1270 0 .5316 0 .0086 0 .0156 0 .0000 0 .0000 0 .0000 0 .0000

electrons . In order to find which one, each LMO in turn was determined from the complete set of doubly occupied orbitals (orbitals containing Q-spin electrons) and the residues compared (table 1) . It is clear that neither Xnbh nor the Yp LMO's are entirely suitable for either HBO - or HCN - . It is significant that if any orbital is to be neglected in HBO - it is Xnbh, but in HCN - it is Yp , reflecting a big difference in the closed shell electron distributions in these radicals. It is concluded that localisation of the doubly occupied orbitals is unsatisfactory and will not be considered further except to point out that the existence of the chemical bonds XH and XY is well established . 4.2. Localisation of the

a-spin orbitals

As table 1 shows, all LMO's, except Y p (2p y (Y)), can be obtained with perfect localisation (A = 0) from the set of the eight orbitals occupied by electrons of a-spin, using the nonorthogonal localisation method . There are 8! ways of selecting the order by which the LMO's are determined from the progressive orthogonalisation procedure, each way potentially giving a different set of LMO's . As mentioned above it is not necessarily true that the residue of an LMO will be smaller than the residues of all LMO's determined later in the

27

procedure . For example, if the HBO - local orbitals are chosen in the order O p , Onbh, BO, Bnbh, BH, the residues are 0 .0086, 0.0, 0.0, 0.0498, 0.0447, respectively . This observation in fact proved useful, after obtaining experience with many other approaches, in deciding how to find the "best" of the 8! ways . The following method seems to have sufficient generality to be applicable to other radicals, as well as giving that set of LMO's for which the sum of the residues was the least of those ways studied . The 8! ways are reduced to 7! if account is taken of the fact that one of the a-spin orbitals is of ;,-symmetry and therefore it is already localised . It was found that the remaining 7 local sets could be divided into groups which drastically -reduce the number of ways still further. One group is obviously the inner shell (1s) orbitals on atoms X and Y which are already very localised and are left until last in the progressive orthogonalisation . The other groups also arise naturally . For example Xnbh C XH, that is, the local orbitals 2s(X), 2p,(X), 2p y (X) which are used to obtain the LMO Xnbh are included in the XH LMO as well . If the local orbitals for XH are used first, doubly degenerate solutions (A =0) are obtained which are linear combinations of the X nbh and XH LMO's . Similarly Ynbh c XY, so this forms another group . To obviate the difficulty of degenerate solutions the following procedure was used. Suppose that A, B and C are sets of local orbitals such that A c B c C and C gives a triply degenerate solution in the non-orthogonal method, set A must be used before set B which must be used before set C in the progressive orthogonalisation method. So the local orbitals corresponding to the LMO Xnbh are used before those corresponding to XH and similarly for Ynbh and XY . Using the criteria that the sum of the residues, A, should be least, it was found that it was better to localise the orbitals about the atom X before the atom Y . Presumably it is better to localise first about those atoms with the least number of non-bonded electrons . The order is now completely determined but for the Y p LMO . Using just the 2p,,(Y) local orbital in the non-orthogonal localisation



TA . Clarion / Localised molecular orbitals in afatomic radicals

28

method gives an LMO which includes a significant amount of the local orbitals from XH and X„ b h . In fact the local set 2s(X), 2p,(X), 2p y (X), ls(H), 2p y (Y) gibes a triply degenerate solution (A=0) in the non-orthogonal localisation method . The chemical significance of the involvement of the 2p,(Y) orbital with both the XH and the X„ bh orbitals is that it is reminiscent of the model for hyperconjugation . In table 2 the residues are listed for a Y p LMO from the local set 2s(X), 2p x (X), 2py(X), Is(hL 2p y (Y) case being and afro the local set 2p y (Y), in a determined after the XH bond . It was hoped to obtain LMO's on the Y atom corresponding to non-bonded sp Z hybrid orbitals using the local set 2s(Y), 2p„(Y) and 2p y (Y) . This was partly frustrated by the involvement of 2p,(Y) with the X atom, but also illustrated the potential asymmetry that the Verwoerd method can produce In the residual density, since only one of the spZ hybrids can be determined at a time . Of course the sp Z hybrids are easily constructed by taking linear combinations of the

Y,w. and Y P LMO's . The choice of local sets and the order in which they are used must be governed to some extent by a need to minimise the possibility of unwanted asymmetries appearing in the residual density .

5. Discussion of loealiaadon results The orbital coefficients for the valence LMO's, obtained by progressive orthogonal nation of the orbitals occupied by a-spin, are given in table 2 for HCN- and HBO - . Since it is clear that just the doubly occupied orbitals cannot be well localised, there is no advantage in trying to retain a single determinantal description of the radical . To add to this view is the comparison of the orbital coefficients of the LMO's for the two radicals. This is further emphasised by comparing the isometric projections of the density associated with each LMO, in figs. 2 and 3, clearly showing the degree of localisation that can be achieved .

Table 2 Orbital coefficients of local orbitals using the progressive orthogonalisation scheme of Verwoerd for the a-spin natural orbitals for radicals of the type HXY Radical :

I

Eland : Local orbitals :

HBO B n b,, 2,3,4

HCNBH 2,3,4,9

op

Onbh

2,3,4,8,96,7

BO 2,3,6,7

Cnbh 2,3.4

Np N„ bh CH 2,3,4,9 2,3,4,8,96,7

CN 2.3,6,7

Local orbital no . 1 2 3

4 5 h 7 H 9

type ISIX) 2s(X) 2p, (X) 2p,(X) Is1Y) 2s(Y) 2p, (Y) 2p,(Y) 15U1)

Residue (A)"

Percent occupancy of unpaired electron

-0 .0007 -0 .0074 -0 .0001 -0 .0045 0 .1518 0 .2340 -0 .0492 -0.1026 0 .7582 0 .1211 0 .1916 -0 .0359 -0 .0279 0 .7587 0 .0014 -0 .5063 -0 .3831 -0 .0842 -0 .0920 -0 .0019 -0 .3621 -0 .3990 -0 .0694 -0 .1056 0 .5104 -0.0851 0 .0096 -0 .0622 -0.5150 0 .5611 -0 .1154 -0 .0110 -0 .0723 -0 .5272 0 .0016 -0 .0002 0 .0023 -0 .0013 0 .9158 -0 .1010 0 .9288 -0 .1868 0 .4273 0 .6796 0 .3945 0 .7996 0 .0370 1,0046 -0 .0028 0 .0227 1 .0135 0 .0020 .5770 U .2501 -0 .0038 0,0494 0 .5297 0 .3252 -0 .0090 0 .1090 (1 0 .0000

55 .44

0 .0000

12 .44

0 .0031 0,0000 0 .0015 (0,0720) (0 .0015) (0 .0031)

28 .64

0 .00

3 .29

0.0000

16 .07

0 .0000

0 .0032 0 .0100 0 .0000 (0 .1191) (0 .0032) (0.01%)

6 .40

75 .05

0 .04

2.21

" 'the figures In parentheses are the residues obtained if orbital 8 is chosen as the Y(p) bond instead of orbitals 2, 3 .4. 8. 9 .



9 .+R 41frP ara, ? d . .m is

a "4ENWn n4miSb *w

L

Yt #,"?40o

Fig 2 . Isa=rr= pw .m a of d= cktct>= u sc rhr P1atst cd drr r 33CC Szy d lvo!rsed ortssr.t t o CH b=d . '.j+ tea ~ lfy r gr4 n cxt'!err, s " sacsta tp-gpc t arbrz rya 9 sir a twtrt nrtra n !cu g am rah Chi hurl? +ea= asps p-npt t~ e rv Chi Es S i3 rrrss s . tas t a~ 3 r ~.t +r~ da, tetras c t r4 r t ; s x9 z 9 r + : tsn'"7 " ~ • a 3ar & TrS A ~ ~xrds rsrtta e

Since cherntsts prefer to talk about tndr5sdw i bonds in molecukrs. the marked smnbrrt-# rn the °,&g ) an 1..MO's for these two r..~ttrl$ is > diJ . invaluable result_ In fact the

between the XH and XY bonds can be o f .' rationalised in terms of the a H, X and Y . For example . the

teeter e -

tronegativity of carbon compared with boron can be correbted with a smaller Sib

of

in th-- XH bond for H i swrnniparc.d math HW) . Again the nngcb 9argrr c€sctrnn zlrH11y th4 rrrnrc brtnc_+t b attd C) than 14"CIffi_-Arnt

in s rr . mp scan C and is r in the 2p, Coeffirlam in the XY h for HCC are band, The u=pwtcd r u ine rrn HUN similar and are ittt rthmcr with etrerrune and HBO' w oy"Altrrat t

.

30

TA . Claxton / Localised molecular orbitals in triatomic radicals

Fig. 3 . Isometric projection of the electron density in the molecular plane of the radical HBO - for the following completely localised orbitals : (i) BH bond, (ii) non-bonded sp-type hybrid orbital on boron, (iii) non-bonded sp-type hybrid orbital on nitrogen (co-linear with BO bond) . (iv) non-bonded p-type atomic orbital on nitrogen (perpendicular to BO bond) . The orientation of the radical in each diagram can be found by rotating diagram (v) until the *'s are in corresponding positions .

From table 2 it may be concluded that the most significant difference between these radicals is in the percentage occupancy of the LMO's, rather than differences in their detailed structures. For HCO the percentage occupancies of the Cnbh, Op and CH LMO's are 46 .0, 32 .2 and 17.3, respectively . Although acidity and

basicity are macroscopic properties, it is tempting to try to correlate these distributions with the facts that HCN - behaves as a base [17] and H2CO+ is extraordinarily acidic [18]. The unpaired electron in HCN - is largely concentrated in N P, whereas in HCO the unpaired electron predominates in C nbh . Therefore HCN-

7:A. Claxton / Localised molecular orbitals in triatomic radicals

and H2CN will be expected to have similar electron distributions, but H2 COT will be totally different from HCO . Although hyperfine coupling constants are calculated from the total spin density, the major component in cr-radicals is normally the contribution from the electron density of the unpaired electron . The contribution to the hyperfine coupling constant of the proton from each LMO will be approximately proportional to the square of the ls(H) orbital coefficient times the percentage occupancy of that LMO by the unpaired electron . The high proton coupling constant in HCN - is clearly associated with the 75% occupancy of the N P LMO, since it is also the dominating contribution . For both HBO and HCO the contribution from the XH LMO is dominant. Although other localisation procedures can produce similar interpretations [16], Verwoerd's procedure definitely highlights certain features which are obscured in uniform localisation methods. For example, the non-orthogonal localisation method clearly excludes the suitability of localising the doubly occupied orbitals by themselves for these radicals . Also the non-orthogonal localisation method enables the sets of local orbitals to be selected by trial and error rather than chemical intuition . This should prove invaluable for extended basis set calculations .

6 . Conclusions Verwoerd's eigenvalue solution for obtaining population localised orbitals is very easy to implement in the ab initio framework . Although there may be some doubt as to the suitability of a particular LMO, the determination of each LMO singly and independent of all others (the non-orthogonal localisation method), can give invaluable information . The inability to obtain an approximately isolated 2p,, non-bonded orbital on atom Y as an LMO is clearly an example of a hyperconjugative interaction . This

31

result is not forced on the LMO due to orthogonality constraints . On the other hand, although non-orthogonal LMO's, carefully chosen, must be used if the objective is to transfer orbitals to other molecules, it is very dangerous to use these orbitals to discuss properties within a molecule . The progressive orthogonalisation scheme cannot be used without some care due to the dependence on the order of selection of the groups of local orbitals and the inherent non-uniformity of the method.

References [1] V. Fock, Z . Physik . 61 (1930) 126 . [2] G .G. Hall, Proc . Roy . Soc . (London) A202 (1950) 336 ; G.G. Hall and J .E . Lennard-Jones, Proc. Roy. Soc. (London) A202 (1950) 155 ; A205 (1951) 357 . [3] J.E . Lennard-Jones and J .A . Pople, Proc. Roy. Soc . (London) A202 (1950) 446 ; A210 (1951) 190 . [4] C. Edmiston and K . Ruedenberg, Rev. Mod . Phys. 35 (1963) 457 ; 43 (1965) 397 . [5] W.H . Adams, J . Chem . Phys . 34 (1961) 89 ; 37 (1962) 2009 ; 42 (1965) 403 . [6] S.F. Boys, in : Molecular orbitals in chemistry, physics, and biology, eds . P.O . Lowdin and B . Pullman (Academic Press, New York, 1964) p . 253 . [7] V. Magnasco and A . Perico, J . Chem. Phys . 47 (1967) 971 . [8] C . Trindle and 0 . Sinanoglu, J . Chem. Phys . 49 (1968) 65. [9] W.S. Verwoerd, Chem . Phys . 44 (1979) 151 . [10] M .C.R. Symons and H .W. Wardale, Chem . Commun. (1968) 1483 . [11] J.A. Brivati, N . Keen, M.C.R . Symons and P .A. Trevalion, Proc. Chem . Soc. (1961) 66 . [12] J.A. Brivati, K .D.J . Root, M.C.R. Symons and D .J.A. Tinling, J. Chem. Soc. A (1969) 1942[13] T.A . Claxton, Trans . Faraday Soc . 67 (1971) 897 . [14] A .T. Amos, D .R. Beck and LL Cooper, Theoret . Chim . Acta 52 (1979) 329 . [15] Y. Ellinger, R. Subra, B . Levy, P . Millie and G . Berthier, J. Chem. Phys . 62 (1975) 10. [16] T.A. Claxton, to be published . [17] 1 .S. Ginns and M .C.R. Symons, J . Chem. Soc . Dalton (1972) 185 . [18] S.P. Mishra and M .C.R . Symons, J . Chem. Soc. Chem . Commun . (1975) 909 .