Molecular orbital theory of transition metal complexes

7 MOLECULAR ORBITAL T H E O R Y OF T R A N S I T I O N METAL COMPLEXES

David A. Brown, W.J. Chambers, and N.J. Fitzpatrick

Department ol Chemistry University College Bellield Stillorgan Road, Dublin 4, Ireland CONTENTS I. II. III.

IV. V.

VI. VII.

VIII. IX.

Introduction . . . . . . . . . . . . . . . The Hartr~'-Fuck Equations . . . . . . . . . . . The Ab-lnitio LCAO-SCF Method . . . . . . . . . . A. The LCAO Approximation . . . . . . . . . . . B. Roothaan s Equations . . . . . . . . . . . . C. Analytical Expressions for Atomic Orbitals . . . . . . D. Population Analysis Schemes . . . . . . . . E. Correlation Effects . . . . . . . . . . . . F. Koopmans' Theorem . . . . . . . . . . . . G. The Calculation of Electronic Transition Energies . The Scattered-Wave Model . . . . . . . . . . . . Semi-C~antitative Methods . . . . . . . . . . . . A. The Need for Approximate Molecular Orbital Methods B. The Mulliken/Rucdenher8 Method . . . . . . . . . C. Zero-Diferential Overlap Methods . . . . . . . . . Semi-Empirical Methods (The SCCC-MO Technique) . . . . . Results and Discussion . . . . . . . . . . . . . A. Some General Comments . . . . . . . . . . . B. The Transition Metal Fluorides . . . . . . . . . . C. Ab-lnitio Calculations . . . . . . . . . . . . D. Calculations using the MulIiken/Ruedenherg Method . E. Calculations using the ZDO Approximations . . . . . . F. Semi-Empirical Calculations . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . References

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I. INTRODUCTION The aim of this review is to present in a comparative and critical manner the various theoretical approaches to the molecular orbital t h ~ r y of transition metal complexes proceeding from the rigorous ab-initio type to the widely used semi-empirical type, such as the Self Consistent Charge and Configuration Method (SCCC-MO). The results o~ the application of these various methods to typical co-ordination complexes such as metal fluorides and to typical organometallic complexes such as metal carbonyls and ~-complexes are discussed. It is hoped that the review will be of general interest to the practising chemist and will enable him to evaluate more critically the ability, of molecular orbital theory to 'explain' or 'rationalize' observed experimental data such as bond energies, photoelectron orbital energies, force constants and even relative reactivities in terms of theoretical quantities such as orbital energies, overlap populations, charge densities, etc. The rapid development in computer software and hardware in recent years has resulted in significant advances in the field of simulating basic quantum mechanical laws in many-electron atomic and molecular systems. The day draws nearer when computation of electron-structure dependent properties of atoms and molecules will be measured more accurately, more easily and more economically

Reviews 1972

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7 7 9 9 10 10 11 12 13 13 14 15 15 15 17 19 21 2t 21 22 24 26 27 28 28

by the theoretically rigorous simulatory programs than the direct experimental observation. A number of general reviews have appeared 1'2'3'4 and one 3 is closely related to the subject of the present paper although special place has been given in our review to the stepwise development of molecular orbital schemes and their application to the study of organometallic molecules. With regard to the latter no review has appeared since those of Richardson I and Brown s although the reader is referred to the excellent lecture of Fenske s for a recent short critique of semi-quantitative and semi-empirical methods.

II

THE HARTREE-FOCK EQUATIONS

Attention may be focussed on the behaviour of the N-electrons associated with a particular N-electron molecule if, using the Born-Oppenheimer approximation, # the nuclear configuration of the molecule is assumed to be fixed. Neglecting spin-orbit and spinspin interaction as well as relativistic effects the wave functions ~ for the stationary states of an N-electron system are solutions of the equation (1) and have energies E,. H~=E,~,

(I)

8

DAVID A BROWN, W.J. CHAMBERS, a n d N.J. FITZPATRICK

The wave functions q)i are of the form (2) where • , = O,(xi(i),yi(t),zl(t),o',(l)

....

(2)

xN(N),yN(N).z.(N).o'.(N))

xi(j), y,(j), z,(j) and (r,(j) are the x,y,z and spin coordinates of electron j respectively. The spin coerdinate cri(j) can be either ~. or {3 depending on whether the spin is up or down. The total hamiltonian operator in equation (I) is defined (in atomic units) by

-=-

(v,v.'+E z'----)+ z

1

were A and B label the M nuclei in the molecule, Z, is the charge on nucleus X, ~ and ,~ label the electrons, rs,A is the distance between electron i~ and nucleus A and r~, is the distance between electron Ix and electron v. The first term on the right hand side of (3) is the kinetic operator for the Ixth electron, the second term is the potential between the Ixth electron and the nuclear charges on the M nuclei, the third term is the electronelectron potential between the Ixth and vth electron and the last terms is the nuclear potential with RAs being the distance between nuclei A and B. This potential is constant for a given geometrical configuration. Equation (3) may be rewritten as H=Z H~'+ Z I - L + ~E ZAZ. ~-1

~>* r~.

A>s

R~

Thus where ~b in the MSO, q) the MO and cr the spin function. The total N-electron wavefunction for a given steady state is now built up as a normalised antisymmetrical product of MSO's:

O=(ND -~

~.(1)~N(2)...

(7)

Thus where dx is the one-electron volume element including the spin and 8, is the Kr6necker Delta. For a closed-shell structure, consisting of 2n electrons distributed over n MO's, the MSO's are given by

~,,_,=cp,ot,

¢~,=q,,~

(8)

Now, since the MSO's are orthogonal, it follows that ~O~jdhjdx=0, (i#j)

(9)

But in a closed shell structure if,, and ~ have identical spins so that integration over the spin factors cannot be zero. Hence ~q),q)ldx---0, (i#j)

(5)

~, (l)q, (2) . . . . . ¢,i (N) (1)~ (2) . . . . .~: (N)

~~I/,I]/jd'%"= ~,,

(4)

where H ~ is the hamiltonian operator for electron IX moving in the field of the nuclei alone. In order to obtain a solution O to equation (1) which would obey the Pauli principle in being antisymmetrical in every pair of electrons it was suggested by Fock7 and SlateP that a wave function be assigned to each electron which would contain the space and spin coordinates of that electron only. Such a wave function is called a molecular spin orbital (MSO). Since magnetic effects are being neglected each MSO factors into a spin function and a wave function which is dependent upon the space coordinates of the electron only. This latter function is known as a molecular orbital (MO). = ~(x,y~,~)= ~x,y,z)~ =

This form of the wave function clearly statisfies the Pauli principle because if the labels of any two electrons are interchanged the two columns of the determinant are interchanged and so the determinant changes sign and so also does ~. Furthermore, all the MSO's must be linearly independent since otherwise the determinant vanishes. In particular no two MSO's can be the same so that the MO's can only be the same when the corresponding MSO's have opposite spins. Thus no two electrons can have the same set of quantum numbers--an alternative formulation of the exclusion principle. The MSO's may be assumed to form an orthonormal set since if they do not they can be subjected to a transformation which will make them orthonormal3

where dx is now the one-electron volume element without the spin. Thus the MO's which form a closed shell also form an orthonormal set. The total energy of an electronic state represented by (I) is given by E= ~@*H~d'~

E= 2ZH,,+ ZZ(2Itr--K,j)+E,a I

I j

(12)

where the Hii terms are the nuclear field orbital energies (13) H. = ~q)l*(l)Htq~l(1)d'~t

(13)

and the coulomb l,j and exchange KI, integrals are defined by

(6)

This determinantal form of the wavefunction is usually referred to as a Slater determinant.

(11)

Substituting the antisymmetrical product for a closed shell system for • in (I1) leads to expression (12) for E

l,J = ~ f

~(N)

(10)

K,j= ~~

qh"(1)qh(1)qh*(2)q~'(2)dxtd't2 ru qh'(1)cpi(2)qh*(2) qh(l)dx,dx: ra

(14)

(15)

where one electron is labelled by the numeral 1 and the second by the numeral 2. E,.A is the nuclear lnorganica Chimtca Acta

Molecular Orbital Theory o! Transition Metal Complexes

repulsion energy E~=

Y-

Z.~Z~

A>B

Rkll

(16)

To obtain the best value for * the energy is minimized by varying the MO's within the constraints that they form an orthonormal set. This can be done by the method of lagrangian multipliers I° and leads to the conclusion 9 that the 'best' MO's satisfy the Hartree-Fock equation. F~p,(l) = e,q~,(l)

(17)

where the Hartree-Fock operator F is defined by F = H + G with H being the nuclear field operator defined by H=H t

(19)

where H i = -1/2 W2_y. ZA &-I

(20)

l'A

and the total election interaction operator G is defined by

9

sarily true for an open shell system in which case it may be necessary to represent • b ~ g ~ ' ~ i n a tion of Slater determinants m or0g_,k-~.-~l~t~,yce~6~i't'~on

. i . T . E A.-I.mO ,e A.

• ,

ll'a

~- (21,--K,)

.oD

/

)f

:-i

THE LCAO A

:4' For molecular systems direct s o ' o n of t h e ~ a r tree-Fock equations is impraticai and approximations to the best MO's are employed. The most commonly used is that of representing each MO as a linear combination of atomic orbitals (LCAO): cp,=Zc,,X,

(25)

where X. is a normalised atomic 3rbital (AO) Therc are two common types of analytical expressions for the AO ×a. Both will be discussed in more detad and are mentioned but briefly here The most common representation of an AO is due to Slate? 2 who described atomic orbitals with analytical functions of the type: X (s) = N'r"e ~t, Y,m(0,q))

G=

,I

(26)

(21)

t-I

The coulorhb and exchange operators 1, and K, are given by I,(l)l)j(l)=(f.

~,*(2)e,(2) dx2)~)(1) rn

K,( 1)~p,(1) = ( I


dx0q~,(1)

(22)

(23)

and e, is the orbital energy associated with q~,. Thus, by equation (17), the MO's which give the best antisymmetric product are all eigenfunctions of the operater F which in turn is defined in terms of those MO's. It is noteworthy that, when j = i in equations (22) and (23), the term (2L - K,) q~, (1) reduces to l,cp,(1) and so the electron ,in question interacts with the other electron in q~, but not with itself. In order to solve equation (17) a set of MO's is assumed, the G and F operators calculated and equation (17) solved for the n-lowest eigenvalues. The resulting MO's are compared with the initial assumed ones and a new set of MO's chosen. The procedure is repeated until the assumed and calculated MO's agree. This method for solving equation (17) is called the Hartree-Fock Self-Consistent Field (HF-SCF) method. In addition to being an eigenfunction for the operator H, the wavefunction • must also be an eigenfunction of the operator S~, satisfying equation (24), where S is the total spin and h is Planck's costant. sz~=s(s+ l)h2(2.g)-l~

(24)

While closed shells can always be represented by a single determinantal wavefunction which is an eigenfunction of S2 with S = O n, the same is not neces-

Remews 1972

where the origin of i is the atomic nucleus, Y~m(0,q)) are normalised spherical harmonics appropriate to the hydrogen atom, N' is a normalisation constant, n can be integral or non-integral and ~ is known as the orbital exponent. ~ dictates the distance of the maximum radial electronic density from the origin and is related to the screening effect of the other electrons on the one being considered, a relationship quantified by Slater's RulesJ z.13 Functions of the type (26) are known as Slater-type orbitals (STO's) A second type of analytical expression involves an expansion of the radial part of the AO m terms of Gaussian functions e -~'2 with retention of spherical harmonic angular functions or the form (27) first introduced by Boys.14 xcc)= x)y,.z,e_yr2

(27)

These functions differ from the Slater type m having r 2 instead of r in the exponential argument. Such functions are referred to as Gaussian-type orbitals (GTO's). The set of AO's needed to construct the MO's of a given state in a given molecule is called the "basis set" tor that state and molecule and the individual component AO's are referred to as "basis functions" When the basis set has been decided upon the MO q~, in equation (25) can be obtained by optimizing the coefficients c,a and the orbital exponents ~, by varying them until the total energy is minimized. Often the exponent variation is omitted and values appropriate to the free-atom or ion are used. Such procedures lead to the equations of Roothaan 9 which will be discussed shortly. The accuracy of the MO's obtained by Roothaan's method depends, inter alia, on lhe size of the basis set used in the expansion (25). The best results

10

DAWD A. BROWN,W.l. CNAMnERS, and N.J. FITZPATRICK

would be obtained using an infinite basis set but, since this is clearly impossible, one of three types of basis set is usually employed.~s These are: (1) Minimal Basis Set: this consists of AO's up to and including the valence orbitals of each atom of the system; (2) Extended Basis-Set: this consists of a minimal basis set as well as any number of AO's lying outside the valence shell of each atom; (3) Valence Basis Set: this consists of AO's in the valence shell of each atom in the system. The inner shell electrons are considered to occupy AO's which, together with the nuclei, make up an unpolarisable core. The valence shell orbitals must be made orthogonal to the core orbitals. '6

B. ROOTHAAN'S EOUAT1ONS The electronic structure of a molecule with a closedshell ground state can be represented by a singledeterminantal wavefunction. If each MO in the determinant is expressed in LCAO form then energy minimization arguments9,H lead to the following set of simultaneous equations: I:cl.(F.,.---elS..) = 0

(28)

n

where m varies from 1 to x, the total number of basis set functions, and e, is one solution of the secular equation (29) which also yields the AO coefficients Cm

IF_--e,S..I =0

(29)

Finn in equation (24) is defined by F~=H.,.+Gm

(30)

where (31) Gm= ~EP,.[ (mnluvb-V2(mvinu)'i

(32)

(mn[uv)= J Jx.'(l)X.(I) r-~--X.*(2)X,(2)d'r,dx2

(33)

and P.v is an element of the bond order matrix defined by oae

P,,~= 2 | T. cl.*civ, ~!

(34)

the summation being over the occupied MO's only. Stun is an element of the overlap matrix defined by S_= JX.*(l)g.(l)d'r,

potential energy of the charge cloud ×m~ in the field of the core. If a minimal or extended basis set is used then Z^ is the actual nuclear charge of atom A. If a valence basis set is used then Z^ is the effective nuclear charge of atom A. Gmn is the matrix element of the potential due to (i) the other valence electrons if a valence basis set is in use or (ii) all the other electrons if a minimal or extended basis set is in use. The Hmn and Sin. elements are one-electron integrals in contrast to the two-electron Gmn integrals. For the rest of this paper the coulomb operator over AO's will be written as ( Imn) and the corresponding exchange operator as ( m I n). Thus, for example, the G operator of equation (32) can be written as G= ~P,,(

luv)-S/2( vl u[)

(36)

If a molecular system belongs to a symmetry point group and R is one of the operations of the symmetry group and (I) an eigenfunction of the total hamiltonian H with eigenvalue E then Rq) is an eigenfunction of H with eigenvuale E. Thus, in additions to equation (24), • also satisfies the equation (37) HRq~= ER~

(37)

Furthermore, if, instead of using isolated AO's to build up a MO, combinations of AO's are used which are symmetry-adapted to act as bases for the irreducible representations of the point group to which the molecular system belongs, then many of the integrals in equation (29) may be set equal to zero by symmetry with consequent saving of computing time in solving (29). Assuming that all the terms in equation (29) can be evaluated for an initial guessed charge distribution the eigenvectors for each MO can be obtained and atomic charge densities calculated by one of the population analysis schemes to be discussed shortly. Guided by these new charges and configurations a new choice can be made and the method repeated until eventually self-consistency of the eigenvectors is obtained. This method is usually referred to as the Roothaan LCAO - SCF method. When the total energy has reached a minimum with respect to increasing the size of the basis set the corresponding wavefunction is called a Hartree-Fock wavefunction. On extending the above procedure to open shell systems one of the difficulties encountered is that a single determinantal function is not generally an eigenfunction of S2 in such systems. A procedure applicable to certain types of open shell states has been given by Roothaann and this has been extended to all classes of open shell states by Huzinaga.18 Birss and Fraga 19 and Roothaan and Bagus?° Reviews of open shell methods have been given by Berthier,21 Amos and Snyder~ and Sando and Harriman.z3

(35) C.

e, is identified with the orbital energy of the MOq~,. H~.. is the matrix element of the one electron hamiltonian which includes the kinetic energy and the

ANALYTICAL EXPRESSIONS FOR ATOMIC ORBITALS

The one-electron function of an electron in an AO lnorganica Chimica Acta

Molecular Orbital Theory o] Transition Metal Complexes

can be obtained numerically with the HF method24 or analytically using Roothaan's method. 17 Such functions can be obtained as linear combinations of large numbers of STO's with angular factors identical to those for the hydrogen atom. Watson,~ for example, has computed SCF-AO's for first-row transition metals and has expressed Is, 2s, 3s and 4s AO's as combinations of up to 10 STO's. While such elaborate functions are necessary to describe atomic properties fairly accurately they are very unwieldy for use in MO calculations due to the large number of integrals involved. Simple representations have been provided by Zener~ and Slater Iz who proposed the use of a single exponental function to simulate an AO. Rules for calculating the orbital exponent were given by Slater) ''13 More recently Richardson et alz7 have constructed approximate radial functions for first-row transition metal atoms and ions using a small set of normalised STO's. The parameters necessary for these functions were obtained by maximizing overlap with the more elaborate Watson functions and by energy minimization methods. These results indicate, for example, that the representation of a 4s function can be reduced to a combination of four STO's and so Richardson's functions are extremely useful in molecular calculations. Similar minimal STO representations are now available for the AO's of the second and third row transition elements and for nofi-transition metal atoms containing from 2 to 86 electrons,w Such truncated functions have been shown3° to reproduce quite accurately the main features of the more elaborate functions in regions not too close to the nucleus. While STO's give an efficient description of electronic distributions their use in molecular calculations can be very time-consuming because of the difficulty in evaluating two electron integrals. Even though methods for evaluating such integrals are available3~ many workers now employ GTO basis sets. The use of Gaussian-type wave functions was first suggested by Boys~ who sho~/ed that these functions allowed all the integrals required in molecular calculations to be evaluated in closed form. Unlike the Slater type orbitals, Gaussi~ns have the disadvantage of possessing incorrect cusp and long range behavmur but the use of very large GTO basis sets can overcome this difficulty. Moreover, the disadvantages of such elaborate functions are not as pronuneed as those mentioned earlier for STO's because of the increased ease of computation. To ameliorate the situation further some workers32'~ have employed a large GTO basis set to evaluate important interactions, for example, one-electron and one- and twoelectron integrals, and a smaller set for the less important three- and four-centre two-electron integrals. An alternative approach~ is to use a mixed basis set, evaluating one-electron and one-centre two-electron integrals in a STO basis while evaluating the manycentred two-electron integrals by a small GTO expansion of the STO basis. However, the usual procedure in using GTO's is to expand each MO in terms of a basis set built either from a linear combination of GTO's, sometimes R~iews 1972

11

known as uncontracted Gaussians,~ or from a special linear combination of GTO's, sometimes known as contracted Gaussians.3s The various schemes for cvaluating the expansion coefficients and exponents in a Gaussian representation of X~ have included: (l) minimization of total atomic energy36 and (2) obtaining a least squares fit to corresponding STO functions37'-~'39 or more elaborate HF AO's?° GTO's are now available for many of the first and second row atoms3~''z'4t and also for the third row atoms, including the first-row transition metals?" Small Gaussian expansions for the STO's Is through 5g have been reported by Stewart. 3~ Methods for evaluating all molecular integrals using Gaussians have been given by a variety of workers./4'43'" It is also possible to repre sent an AO by a linear combination of purely radial Gaussian 'lobes'.45 The angular dependence is simulated by positioning the functions at different points in space determined in part by the symmetry of the orbital being expanded Such expansions do not appear to have been used to date in MO calculations on transition metal complexes. It is noteworthy that any MO calculation on a transiUon metal complex will almost certainly yield non-integral charges and configurations on the metal and hgand even though the radial functions for atoms and ions are always derived for an integral configuration and charge. The usual procedure is to use radial functions appropriate to a configuration as near as possible to the calculated one.

D.

POPULATION ANALYSIS SCHEMES

Analysis of the AO coefficients in LCAO wavefunctions can yield information on the distribution of electronic charge among the atoms in a molecule and among the different orbitals on each atom. To illustrate this let the MO tp, be composed of two AO's, X~ and Xba, centred on atoms A and B respectively i.e. ~, =

ClaXaA-I" ClbXbV

(38)

From equation (38) it follows, taking the occupancy of ¢p, as two electrons, that 2 J'qh (l)~,(I)d'r, = 2ca,. JX.,(l)X~(l)dxt+2C',b j Xt,e(l )Xba( I )dx, + 4C,.C,b J X~,( 1)XbB(l )dx,

(39)

and, since the various functions are assumed to be normalised, equation (39) reduces to 2

= 2C=,a "{- 2C2,b -{'- 4C,,C,bZ,,b

(40)

where S,s is the overlap integral. According to the Mulliken population analysis method the charge fraction 2c,fl is associated with the atom A and the fraction 2c,bz with atom B. The 4c,,~ c,h S,b term is identified with the 'overlap population', the electron density between A and B. For the purposes of calculating total atomic charge Mulliken'~ proposed that the overlap charge be equally divided

12

DAVID A BROWN, W.I. CHAMBERS, and N I- FITZPATRICK

between A and B whose charges thus become 2c,~z + 2c,~ 2C,b Sab and 2C,b 2 -l-- 2C,a C,b Sab respectively. Mulliken's method seems very reasonable if the overlap charge distribution ga~(1)XbB(1) has a centre of gravity approximately half-way between A and B. However, if X~A and Xb~ are quite different in their radial and angular extensions then equal partition of the overlap charge seems inappropriate. This deficiency of the Mulliken method has led to reported cases of negative orbital populations"'~ and orbital populations greater than two electrons." A number of workers have proposed alternative ways of distributing the overlap population. It has been shown ~ that the distribution X.A(I)Xb~(1) can be written as a weighed sum of individual AO population densities, X.A(I)XbB(I)=)~.X~(1)X.A(I)+~,bX, m(I)Xb~(1)

(41)

with ~.a"l-~.bmSab. Mulliken's partitioning method is equivalent to taking ~.~=)~b= t/2S,b. However, Lowdin~ has proposed a more general method by evaluating )~ and ~.b as in equations (42) and ( 4 3 )

)~, = S.b( I/2-- ~-)

(42)

~=S.~-k,

(43)

L= S.b-~JX(I)X.A(l)Xbu(l.)d'~

(44)

where

R is the internuclear distance and the x-coordinate in (44) is measured from the mid-point of the internuclear axis and points from A to B. Clearly if the centroid of the distribution )C~(1)XbB(1) lies, for example, closer to B than to A then L is positive and .~ S,b ~.a< and kl,> 2 ' L6wdin's method has thus the advantage of partitioning the overlap charge so as to preserve the dipole moment. When XaA and XbB are identical apart from being on different centres then L = O and L6wdin's approximation reduces to that of Mulliken. However, despite the shortcomings of the Mulliken technique, it has found far more widerspread use than the more correct Lowdin method. Pollak and Reins° have proposed an approximation similar to Lowdin's in which the overlap charge between A and B is divided by a plane mid-way between them and perpendicular to the internuclear axis. This is now referred to as the midplane method. The overlap charge density on either side of the plane can be evaluated by integration between suitable limits and ~.~ and ~.b evaluated. Another method~L52 of partitioning the overlap charge to reflect the relative importance of the various AO's in an MO is shown in equation (45) X.~(I )X.(I ) =

S.b

c2,,+c2~b

[ c~,.X,.,(1) X.~(I)+ c ~,bxb~(I)X~( l)]

(45)

This method results in atoms A and B having charges AA and BB reslaectively, where

_

2

C2

ta

A A - 2c ,~& _---:----:--_ r4c..c,~S.b]

(46)

C2jb

(4~)

c,,,+c,,b ~

-

BB = 2c2,~+'-27--"~-7-, _2 [ 4C..c,hS.b]

Cusachs53 and Davidson5~ eliminated the partitioning problem of working with a basis of symmetry orthogonalised AO's. However, a~ pointed out by Doggett," these functions are no longer mono-centric and so the allocation of atomic populations becomes unclear. It has demostrated by a number of workers that quite different MO pictures for a given compound can be obtained deponding upon the population analysis method used.53'ss's~'57 However, it is generally agreed that trends in results for a related molecular series using a given population analysis method are chemically meaningful.

E.

CORRELATION EFFECTS

In the "one-electron approximation 'n,~ a MSO is assigned to each electron and is assumed to depend on the space and spin co-ordi,,.ate of that electron only. This is an approximation since electrons repel each other because of the coulomb potential E2/ru. The HF wavefunctions differ from the exact wavefunctions in that they fail to take account of correlation effects. Each electron is assumed to act on another only through its averaged charge density. While electrons with parallel spins are kept apart because of the constraints introduced by the Pauli principle. This is not so for electrons with ant~parallel spins and such electrons constitute the main contributions to the correlation error. The most common method of introducing electron correlation, first introduced by Hylleraass8 and Boys,s9 is to express the total wavefunction as a combination of determinants, i.e. as a superimposition of states or configurations, a method known as configuration interaction (CI). Applying this method the total wavefunction is written as = a0~0 + a,~, + a:~: +

(48)

By first optimizing the orbitals in each • and then variationally selecting the coefficients ao, al, a2. . . . a solution better than O0 can be obtained and an exact solution can be obtained by a sufficiently long expansion in (48). However, this method often suffers from the disadvantages of slow convergence~° and so use of multiterm expansions of • can become very expensive. An alternative is the multiconfiguration (MC) SCFLCAO-MO method in which a linear combination of determinants is again considered and the total energy is simultaneously minimized with respect to both the coefficients connecting the conl~igurations and the expansion coefficients in the orbitals from which the configurations are constructed. Ii~the 2n electrons of a closed shell system are distributed over a set of n doubly occuped MO's (q~l... q~,) then extra c o n f i lnorsanica Chim~ca Acta

13

Molecular Orbital Theory ot Transition Metal Complexes

gurations can be constructed by considering all possible excitations from this set to the set of virtual orbitals (q~,+t .... q~,,). This yields n(w-n) separate configurations and this method is known as the complete MC-SCF technique. If all possible excitations are not considered the method is referred to as the incomplete MC-SCF technique. Although CI methods are not yet in widespread use in correlations on large systems it has been found that in smaller molecules electron correlation has a significant effect on molecular properties? ~ The difference between the exact experimental energy of a system and the HF (i.e. single determinant) energy corrected for zero point energy and relativistic effects is known as the correlation energy. If results of chemical accuracy for energy quantities, such as binding energy and electronic spectra, are to be obtained, such correlation has to be taken into account. General reviews of the correlation problem and the various possibilities of combating it have been given by Lowdin, 62 Sinanoglu ~ and Nesbet. a In summary it may be said that CI methods use the variation method to obtain an approximation to the true molecular wavefunction. The Roothaan method, on the other hand, uses the variation method to obtain an approximation to the true HF wavefunction which is itself an approximation to the true molecular wavefunction.

F.

KOOPMANS'THEOREM

The orbital energies e, of equation (29) may be given physical interpretation by Koopmans' Theorem 6s which states that the ionization potentials of closed shell molecules may be identified with the one-electron HF energy eigenvalues of the occupied orbitals from which the electrons are ejected. The theorem does not take account of geometrical re-organisation upon ionization and so orbital energies are better correlated with vertical rathei" than adiabatic ionization potentials. Vertical ionization potentials are commonly 0 to 0.3 ev larger than the corresponding adiabatic values and since this is within the error limits of current theoretical methods orbital energies can be usefully correlated with adiabatic ionization potentials as well. ~ The theorem has many inherent deficiencies and errors arise from orbital rescaling, correlation and relativistic effectsY '~ In deriving the theorem the orbitals of the ion are assumed to be the same as those of the molecule. The true ionization potential is the difference between the total energy of ion and that of the molecule. The fictitious ion that results if the orbitals of the ion are the same as those of the molecule is at higher energy than the ground state and so the ionization potential calculated neglecting this re-organisation energy will therefore be too large. Failure to include the correlation energy correction, on the other hand, leads to one-electron energies that are smaller than the true ionization potentials since the correlation energy of the ion is smaller than that of he molecule. Thus a substantial cancellation of Reviews 1972

different errors may in fact occur and is such that Koopmans' estimate is usually too large but not in a uniform or predictable fashion? 7 A 10% discrepancy between calculated and experimental ionization Fotentials is common. In view of the limitations on Koopmans' theorem it: is preferable to obtain ionization potentials by SCF calculations on the molecule and ion separately and by subtracting their total energies. However, apart from the extra computation involved, treatment of the ion may require the use of open-shell techniques and so this method has not been widely used, reliance being placed on Koopmans' method. It is to be expected that the latter method should correlate well with semi-empirical MO theories which make realistic use of atomic or molecular spectra and ionization potential data. ~ The experimental measurement of iomzation potentials usually involves mass spectral methods which give the ionization potential from the least stable MO cnly or the newly developed photoelectron spectroscopyn which can give information concerning the energy and degeneracy of inner as well as outer molecular levels.

G. THE CALCULATION OF ELECTRON TRANSITION ENERGIES

While MO theory offers a convenient interpretation at electronic spectra as involving electron movement between filled and unfilled orb~tals the actual calculation of electronic transition energy is a complicated task and is not simply the difference between the energies of the one-electron functions between which the electron moves. For example, if an electron is excited from a non-degenerate MO 9, to a non-degenerate MO tp., then four wavefunctions have to be simultaneously considered for the excited state: 9

~t"~l ~ . . . . .

~',_~,_~

e9",,,9~,.,

9.,'q0.~ (4qJ

These wavefunctions give rise to a singlet and a triplet state '@,,~ and 3@.~ and Roothaan 9 has shown the average excitation energy to be E(* 30,.)--E(10o)= e.-c,--(l,~--K,D ± K~

(50)

where the plus sign holds for the singlet and the negative for the triplet The coulomb and exchange integrals |,~ and K,,, are as definide in equations (14) and (15). Clearly the excitation energy is not simply ca-e,, indeed far from it, since J,a integrals are usually in the range 5-10 ev for transition metal complexes and K,~ integrals are of the order of 1 ev. This fact has often been overlooked, especially in some semiempirical calculations where the electronic transitions have been taken as simply e,---e,. The above treatment of Roothaan lgnorc~ the fact that the excited state is genelally an open shell system

14

DAVID A. BROws, W.I. CHAMBERS, and N.l. FITZPATRICK

and so may demand the use of a multi-determinantal representation in order to become an eigenfunction of S2. Furthermore to correct for correlation effects additional configuration interaction functions should be used. The derivation of an accurate expression for the excitation energy depends on the formulation of a MO method in which self-interaction of the electron is not permitted? In many current semi-quantitative and semi-empirical methods this condition is not satisfied and so it becomes no longer clear how to correct for electron-electron interaction on excitation so that unambiguous transition assignments for transition metal complexes are not possible on the basis of these methods. As pointed out by Fanske, s the ability of a given MO method to obtain agreement between observed and calculated transition energies is no indication of the accuracy or reasonableness of such a method. There are many examples where the electronic structure of a given molecule has been studied using various MO methods, all achieving agreement with observed transition energies but giving completely different bonding pictures? Even when the same basic calculational model is used different values for the electron transition energies can be obtained depending on whether or not CI is invoked. For example, R.D. Brown et al. 71 have shown that in MnO4- the predicted energy of the "tl--~2e", IAt--~lT2 transition is 11.31 ev, 5.76 ev or 3.50 ev according as orbital energies, spectroscopic configuration function energies of CI function energies are used. A closely related problem is the estimation of the crystal field splitting parameter 10Dq. Within the HF scheme 10Dq may be defined as the difference between two independently calculated N-electron states, a formulism used by some authors, n'73'~ However, there are disadvantages to this definition since (i) it cannot be used with approximate methods in which total energies are not obtainable and (ii) the scheme suffers in not being based on orbital energy differences since it is an orbital picture which is most completely compatible with ligand field theory itselffl A more usual procedure has been to take lODq for a molecule of octahedral symmetry as the difference in orbital energy between the appropriate eg and t~. MO's and between the appropriate e and t2 orbitals in tetrahedral and square planar symmetry which are assumed to be largely localised on the transition metal. However, this definition is incorrect unless the orbital energy is properly derived in terms of the appropriate open--or closed-sheU Fock operator as in the method developed by Offenhartz. 7s It would seem, therefore, that values of 10Dq derived from most semi.empirical and some semi-quantitative methods are of dubious worth, and that the accurate simulation of ground state properties should be the major goal of any semi-quantitative or semi-empirical MO method.

IV.

THE SCATTERED-WAVE MODEL

In recent years there has been some study of the

chemical bond based on various methods of theoretical solid-state physics, e.g. the quantum-defect method of Anderson ~ and the macroscopic dielectric model of Philips. n One notable method is the multiple-scattering technique of lohnson 7s which is closely related to the scattering method of energy band theoryfl This latter method does not involve the LCAO approximation and so the problem of evaluating large numbers of multi-centre integrals is avoided. The method gives an exact solution of a model HF hamiltonian for molecules of arbitrary geometry. The development of the hamiltonian is based on the partitioning of the molecular space into three contiguous regions: (i) Atomic: the x region within non-overlapping spheres, each centred on one of the constituent atoms; (ii) lnteratomic: the region between the inner atomic spheres and an outer sphere (the 'Watson sphere') centred on the central metal atom and surrounding the entire cluster; (iii) Extramolecular: the region outside the Watson sphere. At any point in the cluster a molecular potential is set up as a superimposition of the potentials of the constituent atoms of the molecule. The determination of the potential due to each atom follows a statistical method due to Slater. ~ The superimposition is spherically averaged in regions I and III and volume averaged in region II. For ionic species the electrostatic stabilising effects of a crystal environment can be simulated by introducing a spherical shell of appropriate radius at the outer sphere radius. The partitioning of the molecular space into bounded regions of spherically averaged potential allows the introduction of a rapidily convergent partial wave representation of the MO's. Within each sphere the one-electron wavefunction is expanded as a series ~f partial waves and the resultant composite wavefunctions and their respective first derivatives are required to be continuous across the adjacent spherical boundaries. This is achieved by means of scattered wave techniques and allows the formation of a secular equation as described in the Appendix of reference 78. The starting point of a SCF-SW calculation is the generation of a set of occupied MO's and energies for a model potential.- This initial orbital set leads to an electronic charge density which is used to build up a new potential. This potential can then be spherically averaged and constitutes a model potential for the first iteration. The process is then repeated until self-consistency is achieved. The great advantage of the SCF-SW model is its computational simplicity. The problem of multicentre integralsmthe bottleneck of ab-initio LCAO calculations---is removed while agreement with experimental results is as good as that achieved with the latter methods. It has been shown by Slater 81 that the difference between the total energies of an optical transition are equal, to a good approximation, to the difference between the SCF-SW one-electron energies of orbitals whose occupation numbers are half-way Inorganica Chimica Acta

Molecular Orbital Theory o! Transition Metal Complexes

between those of the initial and final states. To determine these orbitals, called "transition states", it is necessary to perform a complete SCF-SW calculation for each pair of levels involved in the optical transition, removing one half of a unit of electronic charge from the initial orbital and adding one half a unit of charge to the final orbital This is a much easier proposition than the use of CI in LCAO methods to obtain the required transition energies. A difficulty of the method is the choice of suitable radii for the atomic spheres. These are usually based on atomic or ionic radii and in earlier applications of the method were treated as adjustable parameters Y

V.

A.

8EMI-OUANTITATIVE METHODS

THE NEED FOR APPROX1MATF MO METHODS

The major part of an ab-initio MC-SCF-LCAO-MO calculation is the evaluation of time-consuming electron repulsion integrals whose number is proportional to N * for a basis set of size N. Until recently any SCF calculations within the LCAO framework on moderately large molecular systems were forced to incorporate computer time and core saving approximations. However, with the recent development of computers with average instruction times of 10 -6 sees or faster, programs have become available capable of performing such calculations with exact evaluation of all the necessary integrals using either a STO 83 or, more commonly, a G T O 3s basis set. Molecular wavefunctions derived by such ab-initio methods and the corresponding orbital energies are invariant with respect to choice of local axes systems and to hybridization of the AO basis functions, u'~ While some transition metal complexes have been studied by such ab-initio methods nevertheless the majority of such calculations on these compounds have used a single determinantal approximation together with various other approximations to simplify the integral computation of the HF matrix elements while retaining, as far as possible, the invariance properties of the full SCF theory. The nature of these approximations is discussed in this and the following section. Semi-quantitative and semi-empiileal methods are treated separately. In this paper semi-quantitative methods are taken to be those in which electron-electron and electron-nuclear interactions are explicitly considered in the form of the hamiltonian. In the semi-empirical methods, on the other hand, the exact nature of the hamiltonian operator is not considered.

atoms of the molecule, say B. Thus the orbital product X~^(1) XbB(I) can be replaced by X.,(1 )X~.(t) = V2 ~ [ Sbt(X.,(t)X~A(t )) + S~,(X,.( l )X~( 15]

k-i

THE MULLIKEN/RUEDENBERG METHOD

One of the major aims of any approximate MO theory within the LCAO framework is to simplify the calculation of the atomic many-centre integrals (abl cd). Ruedenberg ~ has shown that in a molecule any orbital on atom A may be expanded in a complete orthogonal basis set of AO's on one of the other Reviews 1972

(51)

where XkA and Xku are members of complete orthogonal basis sets on centres A and B respectively. Thus two-centre distributions are expressed in terms of one-centre distributions. For non-infinite basis sets equation (51) is an approximation and is referred to as Ruedenberg's approximation. If the Ruedenberg expansion is simplified by neglecting a number of terms in (51) then the Mulliken s7 approximation (52) is obtained. x,^( 1)Xb~(! ) = I/2Sob[ X~^(1) X,^(t ) + Xb.(l )Xb~(l ) ]

(52)

This is considered ~7 to be a good approximation if A and B are close to each other and ×~A and Xbe are identical except for their respective centre points. However, for typical distances of 1-2 .~ it is not cleat" how good the convergence to (52) is and which terms are most important. Under certain circumstances Ruedenberg ~ also suggested that the expansion (53) might bc preferable to (51). x.(1)xb~(2) = I/2 ~ [ S.(X~(1 )x~(2)) + sb~(x~( 15x~(255]

k-i

(535

The largest term in this expansion is when a ~ b and neglect of the other terms gives (54) X,A(l)Zba(2)= V2S~b[×,^(I)×~^(2)+XbB(!)XbB(2)]

(545

Equation (54) rather than (51) is sometimes referred to as the Ruedenberg approximation. The use of these approximations allow the esumation of three--and four--centre integrals as well as many two-centre integrals. For example, apphcation of Mulliken's approximanon allows ( a b [ c d ) to be expanded as in equation (55) and the individual members of this expansion are more amenable to calculation. (ablcd)-- I/4S,bS~d[(aalcc)+(aaldd)-t-(bblcc)+(bbldd)]

(55)

It has been shown by Richardson and Rundle 88 as well as Berthiers9 that the HF hamiltonian operator can be written as in equation (56) by systematic application of the Mulliken/Ruedenberg methods. F=--V2 V2+EV,

(555

where

v^=.ze.{2( B.

15

fx.Ax~5--~ x.f xo~51 zA r

(565

where the sum is over the a orbitals of centre A and P, is the population of orbital a as determined by the Mulliken population analysis scheme. VA describes the potential due to atom A The first term on the right hand side of (56) gives the contributions of the electrons of A to the effective field acting on

16

DAVID A

BROWN, W.l.

a given electron and -ZA/r gives the contribution of the core of atom A to this field. Many authors have used a hamiltonian of the form (56) to develop MO treatments of transition metal complexes so it is worthwhile considering how the elements of the secular determinant can be developed using this approximate form of the hamiltonian. The following treatment due to Hillier 9° is typical The diagonal terms are given by

F.~=(X~,l--t/2 V~+ V,lX,^)+ (X,^l B~A x v,IX~^)

(57)

Using the Goeppert-Mayer-Sklar 9t approximation, the orbital X.^ is taken to be an eigenfunction of (-t/2 ~72+ V^) and, if a point charge approximation is used for VB, then Fa~ is given by (58) F.. = e.,(q,)- X q,(X,l--~-IX,)

(58)

B~r£

where e~(q^) is the energy of orbital X..~ on centre A whose atomic charge is q^. Some workers 92'93 have evaluated this energy from spectral data often using valence orbital'ionization potentials (VOIP), quantities to be discussed in Section F This procedure has been criticised by Fenskc and Radtke 94 since it has the undesirable feature that changes in the size of the basis set cause no variation in the orbital energy. These latter authors have evaluated e~^(q^) from atomic HF calculations The VOIP method has also been criticised by Orchard 95 because it neglects selfrepulsion terms. The second term on the right-hand side of (58) represents the potential due to charge distributions on atoms surrounding A, a "ligand field" correction term analogous to the Madelung correction term of Jorgensen. z This type of nuclear attraction integral can be easily calculated,97 although some workers have chosen to use an approximate value. For example, T.L. Brown and Schreiner 92 estimated it as a classical charge interaction between spherically symmetric nonoverlapping charge distributions, modified by a factor 13 which corrects for interpenetration of electronic charge distributions and w r o t e

CZqB

F,.=e.A(qD--- T. ~^.

TAB

B~A

(59)

where [ is the electron charge. The 13 factor was evaluated from relative overlap considerations. Again following Hillier9° the off-diagonal elements may be developed as follows: F.b= (X.,l--t/~V'+ V,+Vn+ X V,lXb,) C~A B

(60)

F,~=(x.~l-v2v'+v~-t/2v~+v.lx,,)+(X~l x v, Ix~) ---(x,A[-v2x7:[xb~) r+,, (61) F,b = S.b[e,A(qD + e~.(q.) ] +(X.,l X V,IXb.)--T,b C~,A

B

where Tab is the kinetic energy integral.

(62)

Again a

CrlAMBEIIS,and N.J. FITZPATRICK

point charge approxtmation can be invoked to estimate the Vc terms according to equation (63)

(X.lc.

q, ×,.1¼l x o)

(63)

The evaluation of these three-centre nuclear attraction integrals has been discussed by Fenske 98 who used Sbavitt's method 99 and by Hillier 9° who invoked the Mulliken approximation to reduce the three-centre integrals to two-centre integrals which were then evaluated by a point charge approximation, a procedure recently used also by Fenske et al) °°'t°t The evaluation of T,~b is straightforward3 t Somc authors who have taken pains to include some form of electror~electron and electron-nuclear interactions in the diagonal terms of the HF matrix elements have used very approximate forms of the off-diagonal elements. For example, T.L. Brown and Schreiner, 9-' whose semi-quantitative method for the diagonal term was mentioned earlier, use a form for the off-diagonal elements due to Wolfsberg and Helmholz t°z the limitations of which will be discussed in Section VI. The use of such approximations for the off-diagonal terms would seem to render unnecessary the more rigorous treatment of the diagonal elements. The Mulliken/Ruedenberg method is open to criticism on a number of issues. As pointed out by Nicholson ~°3 there is an important class of non-zero integrals which are taken equal to zero in this method because the associated overlap integral is zero e.g. the one-centre exchange integrals (X^Y^ I X^Y^), ( X # Y). Since lnterelectron forces can be quite high the neglect of such integrals render suspect resultant charge distributions and other calculated quantities. The Mulliken approximation is also inappropriate for usc m exchange integrals,4,~°3 while the Ruedenberg approximation is unsuitable for coulomb integral evaluation) °3 Furthermore, the use of the Mulliken approximation for coulomb integrals and that of Ruedenberg for exchange integrals can lead to serious inconsistencies ~°3 and may result in an electron interacting with itself: :°~ While use of the Ruedenberg approximations to evaluate multicentre integrals gives results which are invariant under rotation of the local axes and which are independent of choice of hybrids ~°~'t°s this is not true of the Mulliken approximation, t°4 The removal of such obstacles can be a formidable task. To apply the Mulliken approximation in a rotationally invariant manner to the integral (axbalecdo) requires the transformation of the orbitais an and ba to axes a and ~ with respect to the A - B axis, t°3 with an analogous transformation for Cc and dD. An alternative but less accurate method is the use of average values for the one- and two- centre integrals causing the lack of invariance e.g. replacing two-centre nuclear attraction integrals by a point approximation 1~ and evaluating coulomb repulsion integrals of the type (a^a^lbBbB) by treating the orbitals aA a n d bB as being angularly symmetrical,s5 A compromise solution is to average where this revolves little loss in accuracy and to transform to local axes for the critical integrals )03 inorgamca Chtmica Acta

17

Molecular Orhttal Theory o/ Transtlion Metal Complexes

C.

ZERO DIFFERENTIAl. OVERLAP (ZDO) METHODS

in the ZDO aproximation the following set of simphfications as detailed by Dahl and Ballhausen, * are introduced: (l) All overlap integrals are neglected; (2) The core integrals Hmn are set equal to zero unless X= and Xo belong to the same centre or are nearest neighbours; (3) The distribution Xm(1)X~(l) is neglected in evaluating two electron integrals if X~ and ×~ belong to different centres. The above set of simplifications differ somewhat from those given by Pople et al. 8s In the latter scheme the core integrals Hmo which involve an overlap distribution are treated in a semiempirical manner and electron repulsion integrals involving the overlap distribution X~(l) ×~(1), ( m # ~) are ignored, i.e. (mn]uv) = (mmluu) 8=.8.~

(64)

Either set of approximations greatly simplify Roothaan's equations. Using the former set of approximations(:

expansion. The above version of the ZDO method in which differential overlap is neglected for AO's on different atoms only, orbital products for different orbitals on the same atom being retained, is known as the NDDO method (Neglect of Diatomic Differential Overlap). as A more approximate method is to extend ZDO condition (3) so as to neglect all integrals involving products of different orbitals on the same centre. This is known as the CNDO method (Complete Neglect of Differential Overlap) a title first used by Pople et al. ~ Since this method has seen much service in the study of transition metal complexes it is useful to develop the elements of the HF matrix using the CNDO approximations. Reference to equations (30) and (32) allow these elements to be written as

F=. = H=. + ZZP., [ (mnJuv)--=/2(mvlnu) ]

Denoting (mmlnn) by "rm~ and applying the CNDO approximations the diagonal ( m = n ) terms become F===H==+I/2P=,.Ym,.+ Y- P,,.,Tmo

(3) All three- and four-centre integrals vanish. It is possible to justify the ZDO method by treating it as an approximation to a calculation in a symmetrically orthogonalised basis in the case of a single orbital per atom using Lowdin's procedure ~°7 where the orthogonalised atomic orbital (OAO) basis ~. is related to the normalised AO basis X by the relation (in matrix form): X=XS -'~

(65)

where S is the overlap matrix. Attempts to generalise this' result for calculations in which more than one orbital per centre is to be treated often employ the S-expansion technique t°8 in which the matrix S -'~ connecting the OAO basis to the normalised AO basis is expanded in terms of a power series in the matrix d S-~'=(I +d)-'~ = l--1/2d+-{--d2-

(66)

where 1 is the unit matrix and d is the overlap matrix but with zero diagonal elements. However, it has recently been pointed out t°9 that such a series does not converge for the general case. Dahl and Ballhausen ( have shown that if a Schmidt orthogonalisation procedure It° is used instead of the L6wdin technique it is still not possible to give a justification for the ZDO approximation. However, a justification has recently been given by Roby m which is not based on the S-expansion technique but rather on the Ruedenberg Reviews 1972

(68)

=*m

and the off-diagonal terms become

(1) all overlap terms between non-identical orbitals vanish;, (2) all two-centre two-electron integrals take on the form (axbxlcvdv);

(67)

F=.=H=.--I/2P..Y=°

(69)

In the formulation of Pople et al ss invariance of the results to transformations which mix any AO's on a given atom is achieved by assuming that the electroninteraction integrals Ymn depend only on the atoms to which the orbitals ×m and Xn belong and not on the actual type of orbital Thus if Xm belongs to atom M and ×, to atom N then "f~ is replaced by "rMN. From may now be written as F===H=m--)/aP~.~.TMM+P~,uYu=+

5: P~.~YuN

(70)

NeM

The Fmn terms may be written as F=.= Hm,,--I/2P=.YM~

(71)

The diagonal core matrix elements for orbital Xm on centre M can be developed as follows: Hmm=(X=l--)/2V 2 V=[X=)-= Z (X=[VA[Xm)

(72)

A~M

or Hmm=Umm -

~ A#M

(×=lv~lx~)

(73)

where --VA is the potential energy in the field of the core of atom A and Umm is the diagonal matrix element of ×m with respect to the one-electron hamiltonian containing the core of its own atom This is essentially an atomic quantity and can be evaluated from spectral data making appropriate adjustment for repulsions between the electron in question and the other non-core electrons ~ Again, to preserve invariance under local axes rotation and hybridisation it is necessary to replace interaction of the distribution X,, X,1, with the cores of

lg

DAVID A. BROWN, W I CHAMBERS, and N l FITZPATRICK

the other atoms A by an average value which is taken to be the same for all orbitals on M, thus (x,Iv~tx,,)=v ~

(74)

For m # n the core matrix elements are given by H,..=U=~-

X (X=[VAIX.),X=,X.

A#M

on M

(75)

H . . = (X,,I-I/2 V 2-V=.-V.IX.)---. X A#(M N)

(X=lv~lx,),x~ on M,X° on N

(76)

The interaction of the distribution XmXn ( m # n ) with the A cores is neglected and Hn,. is referred to as a 'resonance integral' and taken as being proportional to the overlap integral H=.= f3o~S=.

(77)

where 13°UN is a parameter depending only on the nature of the atoms M and N and is given by I3*.N= I/2(13°.+ B*.)

(78)

Clearly if Xm and X. belong to the same atom then Hmn = O , since Stun = O. The atomic 130 parameters can be obtained from analysis of empirical data or results of ab-initio calculations. While strict application of the CNDO approximations would require that the Hm, terms be set equal to zero, Dewar" has shown that the obtaining of realistic molecular energies while neglecting overlap in normalisation of LCAO orbitals requires the use of a non-zero value for Hm,. The two-centre coloumb repulsion integrals yMN in equation (70) are evaluated as the coulomb repulsion between valence s orbitals on atoms M and N3 s In iustification of this procedure Sichel and Whiteheadm have claimed that "fun for two valence s electrons on atoms M and N represents an upper bound for such coulomb repulsions involving valence AO's. The nuclear attraction integrals VMN can be evaluated in terms of coulomb repulsion integrals" and the local core matrix elements from coulomb repulsion integrals, ionization potentials .and electron affinity data.~ In its application to the study of transition metal complexes various authors have invoked various modifications. The method of Allen and Clack n3 is essentially identical to that proposed by Pople with the exception that one- and two-centre repulsion integrals involving the transition metal orbitals are assumed to depend on the principal quantum number of the orbital and so invariance to hybridisation and local axes rotation is sacrificed. In the CNDO method of Perkins TM one-centre coulomb repulsion integrals are evaluated using separate integrals for s, p and d orbitals, invariance of the calculation to orthogonal transformation thus again being abandoned. The two-centre repulsion integrals are obtained from the theoretically calculated onecentre integrals by a scaling procednre due to Ohno. It5 In the ZDO method of R.D. Brown and Roby~°*

the expressions for the HF core matrix elements are taken to refer to a Lowdin orthogonalised basis rather than, say, a non-orthogonal Slater basis. Accordingly the core hamiltonian elements are first calculated theoretically in a full overlap basis and the resulting matrix is transformed to a L6wdin basis by a suitable transformation. This procedure is considered superior to direct evalution in an orthogonal basis. Its The procedure of using coulomb integrals involving s orbitals only is not employed. Instead a technique of using scaling factors to convert integral values derived from STO's with exponents derived from rules given by Burns m to corresponding values derived from approximate HF functions is used. Nuclear attraction and coulomb repulsion integrals are then determined as van average weighed for the numbers of each integral that arises. The CNDO scheme of Ballhausen, Dahl and lohansen4'lls follows the scheme of Pople et al in considering the AO's to be s-like in their angular dependence in evaluating repulsion integrals. However, these workers use an orthogonalised basis and refrain from using empirical parameters and evaluate integrals theoretically using the DIATOM programs of Corbato and Switendick.ll9 The valence AO's are divided into two groups, group one consisting of the metal 3d orbitals and the ligand 2s and 2p orbitals and group two containing the metal 4s and 4p orbitals. The AO's within both groups are made mutually orthogonal. For integral evaluation the following special cases of Lowdin's overlap charge expansio# 9 are then employed (assuming orbitals with subscript ~< k in group one and those with subscripts > k in group two): X.(l)Xb(1)• 1/2S.b[X.(l) X.(l)+Xb(l)Xb(l)],(a,b~k)

(79)

X,( l )Xb( 1) = S=bX.( l ) Xo( ! ),(a ~ k,b > k)

(80)

Although not used for integrals involving the differential kinetic energy and core operators these approximations are employed to simplify integrals involving more than two centres. Although computationally simple and attractive the general CNDO model has important defects. The use of average s-like functions in evaluating the coulomb integrals may be a large over-simplification for use in transition metal complex calculations since such integrals between d-orbitals are quite different in magnitude from those involving s- and p- orbitals. Furthermore, the omission of monocentric exchangelepusien integrals is also serious since it neglects quantities of the order of 4 - 5 cv and may result in the failure of the calculational method to give singlettriplet splitting in the calculated electronic spectra, t~ However, it is noteworthy that the neglect of orbital products for different orbitals on the same atom, despite the numerical inaccuracy of the approximation, is at Icast consistent with the ZDO framework.4 Such monocentric integrals are not omitted in the more elaborate NDDO method but .q new difficulty of calculating a vast number of new repulsion integrals is inroduced. Rather than use the NDDO technique R.D. Brown and RobyI°* have suggested a Inorgamca Chtmtca Acta

Molecular Orbital Theory ol Transition Metal Complexes

modification of the CNDO scheme, the many-centre ZDO (MCZDO) method. This model consists of (1) not using ZDO approximations for core integrals nor for monocentric repulsion integrals; (2) using the ZDO approximation on all manycentre repulsion integrals, i.e. (ablcd) = O unless a -- b a n d c = d; (3) defining an average two-centre coulomb repulsion integral per set of orbitals on each centre. Thus average coulomb repulsion between two centres occur as Y,, Yd', T~, Y~, Ya,f, etc. (the bar indicating an average value while absence of a bar implies only one such integral exists). This method eliminates the more difficult integrals of the NDDO approximation while retaining the important monocentric exchange-repulsion integrals, and thus computational ease without a serious loss in accuracy is promised. The relationship between the ZDO methods using an orthogonal basis and the Mulliken and Ruedenberg methods using a full overlap basis has been traced by many workers? °~'~ The Mulliken method emerges as a less approximate version of the CNDO method and the Ruedenberg method as the counter-part of the NDDO method. Vl. SEMI-EMPIRICAL METHODS (THE SCCC-MO TECHNIQUE)

The methods considered in this section differ from the quantitative and semi-quantitative methods considered earlier in that the exact form of the oneelectron hamiltonian is not explicitly stated and is replaced by an effective operator FaL Empirical parameters and spectral data are used to evaluate the required integrals in the hope that they may compensate for the electron correlation deficiencies of the HF method. A theoretical background of these methods can be obtained by taking equations (58) and (62) as starting points:

Clearly in a transition metal complex with largely covalent metal bonds the charge on any ligand, qL, will be small and so the diagonal metal terms are approximately given by (8 l)

The value for e,A(qA) is usually taken as the appropriate valence orbital ionization potential (VOIP). The VOIP has been defined as the free atom or ion average of configuration ionization energy in an orbital at the computed charge and configuration, m Data for atomic non-integral configurations and partial charge are obtained by fitting the data for integral configurations to a function quadratic in atomic number or, more commonly, in atomic charge. Reviews 1972

(83)

re

F.b=S.b[e,~(qA)+ebe(q.)]+(X.AIC~A,B Z VdXba)---T.b

F. = e,.,t(qA)

(82)

Values of A,B,C, for various orbitals in various configurations have been given by Basch, Viste and Gray m when R represents the atomic charge and by Anno n' when R represents the atomic number. Both sets of coefficients are derived from spectral data, principally Moore's Tables m of atomic states. Unfortunately because of incomplete atomic spectral data the above tables are only available for elements down to and including the first transition series. However, it is possible to extrapolate to the lower two transition series and apply the same type of formula. Fortunately no such ambiguity exists in the actual form of the metal radial functions to be employed in these calculations since as already mentioned the analytic function of Basch and Gray -'s have been shown 3° to reproduce quite accurately the main features of more elaborate functions where available. The use of charge dependent energies implies that some account is being taken of interelectron repulsion since an electron at a relatively negative site can be assigned to a higher energy than when at a more positive location. For ligand diagonal terms the expression of equation (81) may not be a good approximation since the summation on the right-hand side of equation (58) now contains qM, the charge on the metal which may be relatively high. The usual procedure '24'''~''2° is to use diagonal terms appropriate to the neutral ligand atom or molecule with no adjustment for the common ligand negative charge. Since F,,I:, elf represents the energy of an electron occupying the overlap cloud between orbitals X~A and Xb8 and moving in the field of the core and the noncore electrons it is to be expected that F,bat should depend on the magnitude of the overlap cloud and some mean of the energies of X,A and ×bs. In the first attempt to solve the secular determinant for an inorganic complex, Wolfberg and Helmholz '°2 approximated these off-diagonal terms in Mn04- and CrO 2- by F.b'"= V2KS.b(F,'"+ Fbb") "~ l/2KS.b(e.+ eb)

F . = e.,,(q,,)--. Y. q.(X.,I I - - I X . , , ) ~,,A

VOIP=AR2+BR+C.

19

where e, and el, are orbital energies which were not corrected for ligand field effects. The values of K were taken as K, = 1.67 for or overlaps and K,. = 2.00 for ~ overlaps, values derived from energy calculations on homonuclear diatomic molecules. The WH approximation is based on writing F~b~" as the sum of a kinetic energy term and potential energy term and applying Mulliken's aproximation to the latter F.b'"= --T,bWV'.b

(84)

v'.b = 1,~s.b(v'..+ v'~)

(85)

The potential energy term Vxx' is approximated as the orbital energy ex and the multiplicative factor K is taken to include kinetic energy effects. By comparison with (62) it is seen that neglect of the last two terms in the later equation gives the formula

20

DAVID A. BROWN, W.I. CHAMBERS, and N.I. FITZPATRICK

F.b= t/2KS.b(e.+ eD

(86)

with K = 2 which corresponds closely to the WH approximation. The second term on the right-hand side of (62) can be evaluated as a point charge approximation, i.e. it is a ligand-field off-diagonal correction term, so that for molecules with little charge separation the major deviation from the WH formulism occurs via the two-centre kinetic energy integrals. For molecules with a large amount of ionic character the ligand field correction term will be of greater importance. Ballhausen and Gray, p' in a study of the vanadyl ion, advocated a geometrical mean recipe for the offdiagonal elements, especially when there is a large difference between F,,'" and Ebb"tf (87)

F.b" = KS.b'V'Fd"Fv,,'" .---KS.b

and took K, = K~ = 2.0. Yeranos ~ suggested the use of equation (88) F.t,'" =

KS.b(-

'9 F,,'trFbb "ff

F.'" + F~'"

)

(88)

while a simple direct proportionality between Fabelf and S~b has also been employed,tz7 F,b'" = KS.b

(89)

Cusachs m.~ has suggested that the off-diagonal terms be given by: F.J" = 1/2(F.'" + F~,'")S.b(2--IS.b[)

(90)

This formula involves no empirical parameters, is dependent on the square of the overlap term as is the two centre kinetic energy integral Tab for some orbitals ~3° and has a potential term of the expected form. In their calculations on the transition metal porphyrins Zenner and Gouterman TM proposed that F.b'" = V2(F , ' " + Fbb'")S,~[k+ (l--k)S.b]

(91)

and took the value of the parameter k to be 1.89, a value derived from fitting the differences between filled and empty MO's to the observed transition energies. In the original calculation of Wolfsberg and Helmholz ~' the chosen orbital energies were varied by trial and error until the first and second excitation energies were in reasonable agreement with experiment and until the resulting charge distribution was close to the assumed one. Subsequent work within this type of MO scheme has maintained the self-consistent framework. Neutral ligand energies are usually used for the ligand diagonal terms and the calculation is repeated until self-consistency with respect to charge and configuration on the metal is achieved. The Mulliken population analysis scheme is invariably used in such semi-empirical methods which are generally classed as the SCCC (Self-Consistent Charge and Configuration) technique. Because of their approximate nature semi-empirical

techniques invite and have received much criticism. It has been pointed out by Harris m that iterative solution of the equations IFd"--S,e. [= 0

(92)

to obtained consistency between the charge-dependent parameters and the distribution which they produce does not necessarily result in a MO set corresponding to an energy minimum, the condition upon which the Roothaan equations are based. The reason for this is that the charge dependence of the parameters must be included in the differentiations which define the minimum energy condition and thereby the MO's. The true operator F is related to F ef~ by F.b= F.b'"+ XXP=."*

0F= '" " 0P,b

(93)

where Pro, is an element of the bond order matrix as defined earlier and the sum Xr,X, is over the AO basis set. Thus the use of Felf in the secular determinant has no firm theoretical justification but it is not clear whether the use of this operator instead of F will make a large practical difference.133 An obvious failing of the WH technique is the reliance on scaling parameters to obtain the desired agreement with experiment. Cotton and Haas TM have shown that in a series of ammine complexes K, had to be varied from 1.82 to 2.30 in order to obtain agreement with experimental values of lODq. Basch, Viste and Gray, 1~ in their study of halide and oxide complexes, similarly used K~ = 2.10 with K¢ varying from 1.53 to 1.81. Also for complexes ot high negative charge neglect of the compensating potential of the accompanying cation may result in a positive value for ea98, which, when used with a negative value for eb in the geometrical mean method, results in imaginary values for Fabelf. Fenske and Radtke ~" have demonstrated that the addition of a ligand field correction term to the diagonal elements to ensure that all orbital energy values are negative requires that this potential be also included in the off-diagonal elements and is merely the inclusion of another empirical parameter. The parameter variations demanded in the application of these methods is not surprising since, as pointed out earlier, use of the Mulliken/ Ruedenberg approximations in developing the offdiagonal terms predicts deviations from the W H formulation for both ionic and covalent complexes. In order to retain derived invariance to hybridisation and choice of local axes Fabeft should be independent of the type of orbital X~^ and Xbn and should only depend on the nature of the atoms A and B.*s Clearly this is not so in the case of the WH/SCCC techniques discussed above even though rotational invariance can be achieved with the Cusach's approximation if properly used m The differences between results obtained using hybridised basis sets have been highlighted by Fenske) ~ For example, if a calculation is ma'de for an octahedral complex MI~ and it is assumed that only the 2s/2p hybrid ligand orbital which points towards lnorganica Chimica Acta

Molecular Orbital Theory o] Transition Metal Complexes

the metal interacts appreciably with the metal d orbitals then a serious error may result since the "nonbonding" orthogonal hybrid may in fact also overlap appreciably with metal orbitals. Dahl and Bailhausen4 have shown that another inherent source of error in the WH/SCCC methods is that the empirical recipes for the off-diagonal elements fail to prevent electron self-interaction, an error that may be as large as 5 ev. The rationale for using equation (81) revolves on the assumption that the system being studied is largely covalent in character, e.g. neutral metal carbonyl complexes. This neglect of ligand-field correction terms is not justified for more ionic species and leads to the prediction of excessively high covalencies in such systems. Cotton and Harris 1a7 found that modification of Fa,e" to include ligand field effects (estimated by a point-charge aproximations3) resulted in a MO model for PtCh2- with a degree of ionicity in the Pt-CI bond consistent with NOR data. lorgensen et al TM have shown that for TiCI4 the inclusion of these ligand field or Madelung corrections results in an increase in the metal atom charge from +0.39 to +2.0, a charge which is probably too large due to overestimation of the ligand field contribution to the off-diagonal elements.93 However, it has been suggested by Gray TM that the SCCC procedure of using ligand orbital energies maintained at their neutral values helps to cotlnteract the covalent producing treatment of the metal diagonal terms. Finally we should remind the reader again that SCCCMO calculations carried out on complexes of second and third-row transition elements involve some ambiguity in the choice of coulomb terms for the central metal atom and especially in the construction of VOIP formulae because of insufficient atomic spectral data.

Vll

A.

RESULTS AND DISCUSSION

SOME GENERAL'COMMENTS

It is clear from the discussion in Sections V and VI that all semi-quantitative and semi-empirical methods are open to criticism for the approximations they employ and the foundations of the latter methods are especially weak. Nevertheless, such methods, especially the semi-empirical techniques, have shown themselves to be good simulatory models of many intramolecular electronic processes and consequently have some inherent predictive value which should be tested by experiment whenever possible. In addition such methods give some feel for the nature of the chemical bonding m the systems studied although the quantitative details should always be treated with a healthy caution. In the following sections illustrative results obtained from all the methods considered in the earlier sections are discussed.

B.

THE TRANSITION METAL FLUORIDES

In the development of the electronic theory of Reviews 1972

21

transition metal complexes the fluorides of the transition metals and fluoride clusters occupy a special place. Because of the ionic nature of such complexes various approximations may be invoked to simplify the secular equation and some of these, for example point-charge approximations, have been carried over for use in semi-quantitative calculations on more covalent systems. Transition metal clusters have long been of great interest because of the large practical and theoretical importance of magnetic-impurity effects in crystals, the clusters being considered as being representative of the crystal. They may also be considered the "monomers" for extensively studied ferromagnetic and anti-ferromagnetic lattices. "~ A system of special interest has been the antiferromagnetic compound KNiFe. In 1963, Sugano and Shulman, '4° in one of the first detailed, non-empirical MO treatments performed a calculation on KNiF3 by aproximating the crystals as an open shell octahedral NiF~- cluster m the external field (assumed uniform) of the rest of the crystal. The octahcdral approximation was used despite the fact that KNiF3 has a perovskite lattice and so individual fluorides are not associated with a particular cluster but are shared between two such clusters. The basis set used corresponded to the wavefunction for the free ions without overlap and this approximation enabled a convenient ionic-effective hamiltonian to be developed Overlap was not neglected in solving for the MO's The authors used no empirical parameters, evaluated two-centre integrals exactly using mainly the DIATOM programs of Corbato and Switendick"9 and used reasonable approximations for three-centre integrals, often revoking point-charge potentials. Because an ionic model was assumed no SCF iterations were considered necessary Excellent agreement with the lowest optical transition energy was obtained. The method was augmented and refined by Watson and Freeman,/4t Simanck and Sroubek 73 and Sugano and Tanabe, 142 to eliminate certain non-orthogona!ity problems and to estimate the effects of achieving a SCF hamdtonian. These corrected calculations disagreed with experiment. Offenhartz)43 from similar calculations on FeFe3- and CrF6~- concluded that the SCF-MO method is ineffective when the basis functions are restricted to the atomic orbitals of the separated ions. However, Richardson et al. TM have recently examined the clusters TiF:,3-, CrF~3-, FeF64- and NiF64- using a free-ion STO basis. All one- and two-centre integrals were obtained exactly and three-centre and fourcentre integrals were obtained from them using the Mulliken approximation Integrals containing products of two AO's on different fluoride ions were neglected. By use of a limited CI treatment good agreement between experimental and calculated spectral transition energies was obtained. More recently Richardson and Kalman ~45 have used this method to study and compare CoF63- with the isoelectronic Co(NH3)~3+, again with excellent agreement between theory, and experiment. Ellis, Freeman, and Ros) ~ in their study of KNiF3, also obtained reasonable agreement with experiment using a special one-centre STO basis A less pleasing

DAVID A. BROWN,W.J. CHAMBERS, and N.J. FITZPATRICK

22

picture emerges from the all-electron ab-initio calculation on NiF~- by Basch, Hollister, and MoskowitzTM using a GTO basis set. The value of 10Dq obtained from calculations on a single determinant ground state configuration and a single determinant excited state was four times too large. A similar calculation using a better-than-minimal GTO basis by Veillard and Gladney~ resulted in a 10Dq value 35% smaller than that observed. This disparity between theory and experiment may be due to the use of too small a basis set, insufficient configuration interaction or to the breakdown of the cluster model. The experimental and calculated values of 10Dq for NiF~- are given in Table I.

Table I.

Experimental and Calculated values of lODq(cm-~)

for NiF~-.

10Dq(cm-~) Experimental Richardson et al. Ellis, Freeman and Ros Basch, Hollister and Moskowitz Veillard and Gladney

7250 7125 7210 10800 29194 4670

Ref. 191 144(a) 144(b) 146 74 139

R.D. Brown and Burton t47 have used the MCZDO method to study charge densities and spin properties in Cs2MnF6, K2NaCrF6, I~MnF6, K2NaFeF6, KMnFj, RbMnFj, and KNiF3 treating each as a MF~- cluster and including the lattice effects by the electrostatic method of R.D. Brown, O'Dwyer, and Roby.z'~ The basis set consisted of metal 3d, 4s, 4p, and fluorine 2p orbitals. Single exponent STO's were employed using the Burn's Rules "7 for exponent evaluation. Scaling factors were introduced for all transition ion monocentric integrals to take account of the inaccuracy of the Burns functions in near-nuclear regions. Where the octahedral cluster approximation is crystallographically accurate (isolated octahedra occur in the first four crystals of the above list) reasonable charge distributions were obtained together with good correlation with experiment.ally measurable spin parameters (f,-L), the difference in the spin densities of the fluorine 2p~ and 2p~ orbitals. For the three other fluoroperovskite structured complexes the results, while satisfactory with respect to charge distribution, did not give good agreement with spin properties. This was interpreted as a measure of the inaccuracy of the octahedral cluster approximation in such cases. The CNDO method of Allen and Clack m has been used to,investigate the electronic structures of MnF~-, FeF64-, CoFP-, NiFo~-, and CuFf-. m A valence single-[ STO basis set was employed with the 4p orbital exponent set equal to that of the 4s for a given metal. Exponents were fitted to reproduce overlap integrals using the extended basis set of Watson.n To predict the stereochemistry of the complexes, NiF~- was exhaustively studied for a number of conformations. The variation of the total energy of the ion suggested a regular octahedral symmetry aro-

und the central metal ion and this geometry was era. ployed for the ions MnF6~-, FeF~-, and CoFfl-. For the d9 ion CuF64- calculations indicated that both the elongated and compressed octahedral structures should be slightly more stable than the regular form, in agreement with the ]ahn-Teller theorem. The order of the orbital scheme for both cz and 13 spin corresponded to the order expected from crystal field theory for both filled and virtual orbitals. The covalency of these ions was also discussed in terms of the MO's involving the d-orbitals of the central metal ion. The Racah parameters B and C, a measure of the coulomb repulsion between electrons, are expected to depend on a~2a~2, the product of the squares of the coefficients of the metal d, and d~ orbitals in the molecular symmetry' orbitals.~ In the calculation being considered, close agreement was obtained between a,2a,.2 and the spectroscopically determined nephelauxetic ratio 13. A general increase in covalency across the row of transition metals was obtained, also reflected in a metal charge variation of +0.165 in MnF~- to---0.160 in CuFf-. An early calculation by Fenske et alfl on MF~where M = Ti, V, Cr, Fe, and Co employing a valence STO basis which included metal 4d orbitals used empirical parameters and a point charge approximation to evahmte some electron repulsion integrals. Good agreement between (f,,--f,) and NMR and EPR results was obtained and between experimental and calculated 13 and B values but a~2a~z did not correspond to the experimental 13 values. This was attributed to a lowering of B due to a charge reductton on the metal. Other recent calculations on fluoride systems include ab-initio studies of CuF2,t~ the non-existant TiH3FtSa and the diatomic ScF. m

C.

AB-INITIO CALCULATIONS

One of the few ab-initio calculations on organometallic systems published to date is the study of tetrahedral Ni(COh of Hillier and Saundersfl A minimal STO basis set was expanded in a contracted Gaussian basis using three to four GTO's per STO. A Mulliken population analysis of the SCF eigenvectors indicated a charge of --,0.5 on the Ni atom. The change in electron distribution of the CO ligand on coordination occurs mainly at the carbon atom whose 2s population decreases by less than 0.1e (* donation) while that ot the 2p increased by more than 0.2e (~t back-donation). The highest filled orbitals were found to be 9h and 2e of mainly Ni 3d character. The corresponding orbital energies of 11.7 ev and 13.5 ev do not compare well with the experimental photoelectron values of 8.9 ev and 9.8 ev m but the experimental degeneracy order was reproduced by the calculation. More recently, these same authors reported the most detailed ab-initio calculations to date for Ni(CO)~ and Cr(CO)6.m The basis sets of contracted Gaussian typ~ functions were the largest which could be employed on the ATLAS computer and for Cr(CO)6 integral evaluation time was 100 hours and the SCF procedure took 15 hours. lnorganica Chimica Acla

Molecular Orbital Theory ol Transition Metal Complexes

In view of such elaborate calculations, it is most depressing that the calculated C-O overlap populations of complexed CO were found to be greater than that of free CO a result which is at complete variance with experiment. It is obvious that ab initio calculations have some way to go before their results can be compared with experimental trends in the same way as can results from cruder theoretical methods, e.g. SCCCMO. Veillard L~'m has studied the electronic structure of bis-(~-allyl)-nickel in a minimal GTO basis obtaining a nickel charge of + 1.92 and an ionization potential of 9.61 ev. The four outermost orbitals were predicted to be non-degenerate and made up of mainly pure ~-orbitals of the allyl group with the excepuon of the third level which was a mixture of tr-allyl orbitals and nickel 3d orbitals. These results are not in good agreement with those obtained from photo-electron spectroscopyI~ which gives an ionization potential of 7.85 ev and the band intensities suggest that the five highest-lying occupied MO's are closely spaced wlh degeneracies 2,2 and 1. The band energies seem to be more consistent with outer orbitals localized mainly on a metal atom of charge close to zero, and so the calculated charge distribution involving essentially Ni n is inconsistent with the largely covalent nature of the complex. A g~ound-state single-determinantal molecular wave-function has been calculated for the TcH~- ion in Dab symmetry by Basch and Ginsburg. I~ A minimal GTO basis set was used and a Mulliken population analysis yielded the metal electronic distribution 4ds ~ 5s°6u 5p I~s with the 4d and 5p electrons distributed rather uniformly among the individual orthogonal 4d and 5p orbitals. The absolute magnetic shielding at the protons and the anisotropy in the strictly diamagnetic part of the bulk susceptibility was estimated from the calculated MO's. The calculated shielding was in good agreement with the experimental value and it was concluded that the hydrogens in TcH~- are almost identical with respect to electronic environment. The lowest lying band positions in the electronic spectrum were estimated by the method of Roothaang~together with band polarizations and intensities. Demuynck, Veillard, and Vinot m have investigated square planar Ni(CNh 2- in its ground and first excited states in a minimal GTO basis. The six outer orbitals in the energy range --4.35 ev to -3.27 ev were found to be mainly ligand ~-orbitals with small admixture of metal d. The relative molecular energies occurred in the order d = c r < ~ which is the opposite to the order obtained in the semi-empirical calculations of Mason and Gray) s9 Population analysis indicated a Ni charge of +0.46 which implies a charge transfer of 1 54 electrons from the ligands to the metal on complex formation. This donation arose principally from the 5o" orbitals of the cyanide ion which were mainly carbon 2s in character. Charge transfer due to back bonding was found to be very small, the total ~ population of the cyanide ligand rising from a free ion value of 4.00 to acomReviews 1972

23

plex value of 4.03 electrons. These results are in accord with traditional views of the cyanide group as a poor "n-aceptor.I~° The authors computed and assigned electronic transition energies by solving the Roothaan SCF equations for open shell systems for each excited state keepmg the geometry used for the ground state. A similar type of calculation has been performed by Demuynck and Veillard TM on square planar CuCI42-. A minimal GTO basis set and Roothaan's open-shell method ~7 were employed. The outer unfilled orbital was found to be predominantly a Cu 3d and this was followed by a set of MO's of predominantly C1 3p character. A Mulliken population analysis suggested a Cu charge of +1.28 and a charge o f - 0 . 8 2 on each chlorine ligand. Hillier and Saunders ~u have studied the electromc structures of permanganate and chromate ions. All core orbitals were included in an STO basis and expanded in a GTO basis by the least squares method. The ordering and form of the orbitals were found to be ramer similar for both ions. The highest filled orbitals were found to be the lh, 6h and 6a~ orbitals, mainly oxygen 2p in character while the first virtual orbitals (2e, 7h) had considerable metal 3d and oxygen 2p components. CI calculations on the excited states were performed considering all singly excited configurations constructed from twelve filled and nine virtual orbitals. A reasonable correlation with the experimental spectra was achieved, including the shift to higher energy observed for CrO~2- compared with MnOc. The recent SW-SCF calculation of lohnson and Smith 163 on MnO~- yielded an outer filled and virtual orbital ordering in agreement with the above abinitio calculation of Hillier and Saunders ~62 and with that expected from simple crystal field theory for tetrahedral geometry. A satisfactory interpretation of the optical absorption spectrum of MnOc on the basis of energy differences between initial and final SCF "transition state" orbltals was obtained. The virtual orbitals were found to have real negative energies in contrast to the unphysical positive energies, probably due to an in sufficiently large basis set, of the LCAO ab-imtio results. It has been suggested 16~ that the binding energy for the permanganate ion, defined as the difference between the total molecular energy and the energy of free atoms, should be of the order of 19 ev since the ion contains four strong Mn-O bonds. The work of Hillier and Saunders ~62 has been shown ~64 to predict a bonding energy of 8 ev Dacle and Elder ~ have performed an ab-initio LCAO-MO calculation on the permanganate ion in a GTO basis and employing an orbital-product expansion technique for evaluating the many-centre two-electron integrals.~ This work gave a bonding energy of 14 ev and a manganese atom charge of + 1.43. The form and ordering of the orbitals closely resembles that of Hdlier and Saunders lu and of lohnson and Smith 163 but differs in making the 6a~ orbital more stable than the 6h

24

DAWD A Baown, W I. CHnMBERS,and N.I. FITZPATRICK D. CALCULATIONS USING THE MULLIKEN/RUEDENBERG METHOD

This method has been applied by Hillier and Canadine4~ to the study of bis ~r-allyl palladium, ferrocene, and dibenzene chromium using a valence STO basis set and so implicitly including the o-framework of the co-ordinated organic ligand. Orbital energies were evaluated from spectral data. The results gave chemically reasonable charge distributions and good correlation with the photoelectron spectrum of ferroceneJ ~ The calculations suggested that the C~ isomer of bis ~-allyl palladium is more stable than the C2h isomer. When applied to the permanganate ion93 the method gave a chemically reasonable charge distribution of qu, = +1.85 and qo = ---43.71. An experimental estimate of +1.28 for the manganese charge has been estimated from the metal K-edge Xray spectrum?~ Fenske and Radtke94 have studied the tetrahedral chloro-complexes TiCI4, VCh, FeCI,-, MnCl4Z-, FeC142-, CoCl~-, NiCh2- and the octahedral CrCI63-, TiCl3-, VCI~-, and FeC163-. A better-than-minimal STO basis set was employed, metal 4d orbitals being included. The 4s, 4p, and 4d metal orbitals were constructed using a criterion of maximum overlap with the ligand p functions as distinct from Richardson's criterion of minimum free-ion energy,z7 No empirical parameters were employed, all orbital energies being theoretically calculated. Satisfactory correlation between experimental and calculated 10Dq values (measured as differences in energy between the t, and e orbital levels) was obtained but in some cases even the occupied orbitals had physically unreal positive energies reflecting the need for the stabilising potential of the cauon. A qualitative trend between overlap populations and M-CI stretching frequencies was found and a bond order analysis revealed an increase in covalent character with increasing atomic number for different transition metals in the same oxidation state. However, the results have to be treated with some caution since the authors have treated open shell sysems using a closed shell approximation In addition the use of an unbalanced basis set which employs a metal 4d but not a 5s or 5p orbital has been criticised by Becker and Dahl) 67 Caulton and Fenske~ have applied the same method to the study of the bonding in the isoelectronic organometallic series V(CO)~-, Cr(CO)~, and Mn(CO)6 + and the CO ligand itself. It was found that backbonding to the carbonyl ~* orbital decreased with increasing metal oxidation state. Furthermore, overlap population analyses indicated an invariant o" bond strength in the C-O segment of these complexes, Ihe variations in the C-O stretching frequencies depending upon the ~: overlap population. The metalligand overlap populations show a decrease with increasing metal oxidation state, in agreement with the trend of the observed stretching frequencies. This unusual effect was found to be due to decreased metalcarbon 5d~-2p, interaction. The first nine ionization potentials, obtained by applying Koopmans' theorem, correlated to within 5 per cent of those measured by photoelctron spectrogcopy.

There is good agreement between the results of Fenske and Caulton ~ for Cr(CO)6 and those obtained by Hillier ~ by a calculational method similar to that of Fenske but using spectral instead of calculated orbital energies. Both methods give the highest filled orbital as 2tzs and 4hu with a separation of about 6 ev which is in reasonable agreement with the photoelectron spectrum. The chromium atom charge was calculated to be -1.02 by Fenske and Caulton ~u and -0.96 by HillierJ ~ The charge distribution obtained by the above workers is considerably different from that o| T.L. Brown and Schreiner92 who, while including ligand field corrections in the diagonal elements of the HF matrix used the WH formula for the offdiagonal terms and obtained qcr : 0.63. This letter result is more in ac~rd with a chromium charge of +0.4 m Cr(CO)0 estimated by Barinskii ~69on the basis of metal K-edge X-ray measurements. SCCC calculations on Cr(CO)6 have also predicted a small positive charge on the chromium atom. ~n'~a The results obtained for the iomzation potential and the metal charge in Cr(CO)6 by the various calculational methods are included in Table 1I. Fenske and DeKock ~°° have studied the C~v series Mn(CO)sL (L = Ci, Br, I, H) and found that the highest occupied orbital for the halogen compounds was predominantly halogen in character and good agreement with the photoelectron levels was obtained. It was found possible to rationalise trends in the Cotton-Kraihanzel force constants for the carbonyl groups cis and trans to the halogen in terms of localised carbonyl orbital populations via a Mulliken population analysis Overlap populations suggest little difference in the M-CO,s and M-COt.... bond for the series but the M-COt .... bond is predicted to be stronger on the basis of ~ overlap populations. The metal-halogen bonds appeared to be completely o" in character and of constant strength for the series, in agreement with simplified force constants calculated by including the carbonyl groups in the effective mass of the metal. A charge asymmetry about the halogen was predicted due to direct donation from the halogen p~ orbital to the equatorial carbonyl ~x orbitals while the halogen p~ orbitals remained completely filled. The same calculational scheme has been employed1°~ to investigate the bonding characteristics of carbon monoxide and dinitrogen as ligands in transition metal complexes. A comparative calculation was performed on Cr(CO)5 and the non-existant Cr(N2)6, the results suggesting that dinitrogen is a poorer tr donor and ~ acceptor than carbon monoxide. The ligand bonding characteristics of carbon monoxide has also been compared with those of the cyanide ion m via calculations on the isoelectronic series Mn(CO)6 +, Mn(CO)sCN, Mn(CN)sCO 4- and Mn(CN)~-. Within the series the trends in bonding properties of the carbonyl group could be interpreted primarily on the basis of alterations in the degree of ~z* orbital participation. The cyanide group, on the other hand, behaved m a more complicated fashi'on. When the principal ligand "environment" was' due to cyanide groups alteration within the C-N bonds was primarily due to changes in the ~ bonding. However, when Inorganica Chtmica Acta

25

Molecular Orbital Theory o! Transztion Metal Complexes Table II. Comparative Table of MO Calculations on Transition Metal Carbonyl and ~-Allyl Complexes.

Molecule

Molecular Symmetry Point Group

Exptl. l.P. (ev)

V(CO):

Ok

Cr(CO)6

O~

8.4

Mo(CO)6

Ok

8.4

W(COk

Ob

8.35

Re(CO)~÷ Mn2(COh, Tc2(COh, Re2(CO)t0 MnRe(CO),0 Fe(CO)s

Oh D~ D~ D~ C~, D~b

8.46 8.30 8.36 8.14 8.53

Fe(COh 2

Td

Co(CO),-

Td

Ni(COh

Td

Calc. I.P. (ev)

Calc. Symmetry of outermost Orbital

Cr.

/hPd

C~ C~ Ca, Cr~

AzPt

7.85

7.59

tz~ ta t~ t~ t~ t~ t~ ta t~ t~

9.65 9.43 9.47 9.42 9.5 14.43 6.8

a~ a~ al bt e not given e

11.7 10.37 7.92

tz tz h e

9.61

bl

+ 1.92

A IL

155

8.18 12.68 not given 8.16 8.31

a. at not given a~ a.

+ 0.03 ---0.30 ---0.32 ---0.09 ---0.34

SCCC M/R M/R SCCC SCCC

186 48 48 186 186

whereas AlL

refers to

*M/R refers to calculations based on a method using the Mulliken/Ruedenberg approximations, calculations using the ab-initio LCAO technique.

the environment was mainly lhat of the carbonyl group and the cyanide was a minor component then bonding characteristics were dependent on changes in both ~ and ~ contributions. A htstorically important series of calculations is that of Nieuwpoort 17s on the isoelectronic metal carbonyl sequence Ni(COh, C o ( C O b - and Fe(CO)~-. For some years these calculations remained the only ones in which an attempt to evaluate all integrals was made and so they approximate most closely to the current polyatomic ab-initio calculations. A STO basis was employed, the carbon and oxygen l s orbitals together with the orbitals of the argon core on the metal being regarded as core orbitals. One- and two-centre integrals were exactly evaluated. Threeand four-centre integrals were estimated using the L6wdin expansion of equation (41) viz. X.d I g ~ ( l ) = Z.Z.~(I) ?~A(I)+)~gbB(1)gbS(I)

(41)

with ~,, and kb as variables. No independent calculations were performed for C o ( C O b - and Fe(CO)?but instead the HF matrix was modified for the change in nuclear charge and self-consistency achieved. Reviews 1972

M/R SCCC M/R M/R M/R SCCC SCCC SCCC SCCC SCCC SCCC SCCC M/R M/R SCCC SCCC SCCC SCCC SCCC SCCC M/R M/R SCCC M/R M/R M/R M/R AI L M/R M/R M/R

Reference

8.19 9.1 12.75 12.01 8.38 9.65 8.45 9.38 8.55 9.64

10.2 A~Ni

Calculational* Method

--1.90 +0.39 --1.02 -----0.96 + 0.63 +0.63 +0.42 +0.43 +0.56 +0.47 +0.59 + 0.47 ------0.49 ------0.42 +0.48 + 0.58 +0.26 + 0.40 + 0.39 +0.26, +0.36 ------0.90 +0.57 +0.42 --1.37 +0.52 --1.20 --0.33 + 0.45 ~0.94 ---0.74 --0.96

Mn(CO), +

8.9

Calc. Metal Charge

168 171 (b) 168 106 92 171 (a) 171 (b) 171 (b) 171 (b) 170 171 (b) 170 168 172 171 (b) 171 (b) 184 184 184 184 106 92 192 106 173 106 173 39 106

92 173

The highest filled orbitals in Ni(CO)4 were found to be in the order e followed by t2 i.e. the opposite to that expected from crystal field theory. The calculated ionization potential of 7.92 ev is in good agreement with the experimental photoelectron value of 8.9 ev. m However, the photoelectron band intensities indicate that the expected h followed by e sequence is the experimentally correct ordering and so Nieuwpoort's calculation is inadequate. Hillier m has also investigated the bonding in the series Ni(CO)4, C o ( C O b - and Fe(COh 2-. For Ni( C O b the calculated charge distribution closely resembles that of Nieuwpoort 173 and of T.L. Brown and Schreiner 92 but this is quite different from the corresponding ab-initio distributionfl Ionization potentials and charge distributions for N i ( C O h using various calculational methods are included in Table II. In contrast to the calculation of Nieuwpoort m Hillier's work ~°6 gave the correct outer level ordering and an ionization potential of 10.2 ev. Schreiner and Brown 92 also obtained the correct ordering and an ionization potential of 10.37 ev. As noted earlier, the ab-initio calculation of Hillier and Saunders 39 also gave this ordering and an ionization potential of 11.7 ev.

26

DAVID A. BROWN,W.l. CHAMBERS, and N.I. FITZPATRICK

The metal charges for Co(CO),- and Fe(CO),'- derived by Hillier ~06 were -1.20 and -1.57 respectively. The corresponding quantities given by Nieuwpoort were --0.33 and +0.52. The large difference in the results obtained by the two methods may be due to the fact that Nieuwpoort treated these ions as perurbations of Ni(CO),.

E.

CALCULATIONS USING Z D O APPROXIMATIONS

In early calculations on organic ~-systems the molecular integrals which remain after the application of the ZDO approximations were often treated as semi-empirical parameters. ~74 However, lack of experimental data and increased electronic complexity render difficult the application of such a method to transition metal complexes. Nevertheless, calculations in this spirit on such systems have been reported, For example, in applying the method to copper complexes Roos ~Ts,z~ evaluated two-electron integrals in terms of Slater-Condon parameters m calculated from the spectral energies of different terms and configurations of the atom being considered. Two-centre coulomb repulsion integrals were expressed in terms of the one-centre coulomb integrals by an essentially empirical recipe. The one-electron diagonal core terms were estimated from experimental ionization potentials and point-charge approximations while the off-diagonal core elements were estimated by the Wolfsberg-Helmholz ~" recipe, the parameter K being determined from the ligand field spectrum. The method was applied with configuration interaction to investigate the electronic spectra of Cu(NH~): + and Cu(H:O)2+. m Dimethylglyoxime copper has also been investigated t~s and the results suggested the existence of a strong metal-ligand "~-bond and a reasonable interpretation of the electronic and EPR spectrum of this molecule. The most popular scheme for the study of transition metal complexes which is related to the ZDO framework has been the CNDO model. Calculations using the model of Clack, m apart from those on the M F : - series already discussed, m have included a study of the hexa-aquo metal It complexes (Ti H to Cu n) and the hexa-aquo metal III complexes (Ti m to Corn). ~ The results suggested an approximately tetrahedral geometrical configuration about the oxygen atoms but equilibrium bond lengths were not well predicted. The method also predicted a ]ahn-Teller distortion for Cu(H20): +, Cr(H,O): + and Mn(H20) 3+ which have orbitally degenerate ground states. When applied to the permanganate ion ~ the method gave an ordering of the outer levels corresponding to that predicted by crystal field theory with the central metal orbitals lying at highest energy and a charge of +2.09 on the manganese atom. The only CNDO calculation on a transition metal carbonyl complex is that of Perkins, Robertson, and Scott n° on ~-butadiene iron tricarbonyl. The exponents of a valence STO basis set were derived from the rules of Burns n~ and configuration interaction was als6 invoked. The iron atom was

calculated to have a negative charge of --0.45, drawn mainly from the carbonyl carbon atoms. The butadiene moiety was predicted to be essentially unchang. ed in the complex. Charge distributions within the complexed butadiene fragment corresponded closely to those calculated for two excited states of the butadiene molecule and it was inferred that the complexed diene is, to some degree at least, in an excited state. Elongation of the butadiene-iron distance produced no change in these effects and the authors predicted that in a substitution reaction involving replacemen of the butadiene, the latter would be released in a neutral excited state. The CNDO mehod of Ballhausen, Dahl, and lohan. sen4'm has been applied to the study of the tetra. hedral dO series Mile,-, CrO42-, VO43- and TiCh. m A valence STO basis set was employed and in all cases the pattern of the filled orbitals was found to be lat, 2a~, lh, 2h, 3h, le, lb. A characteristic feature of the results was the heavy involvement of the 4s and 4p metal orbitals in bonding compared to the relatively small contribution from the 3d orbitals.' Approximate one.electron transition energies for the allowed transition ~Ar--~T2 were obtained from the equation AE,., ~= eh--e,--I ~k

(94)

were J,k is the usual coulomb integral, the less important exchange contributions in equation (50) being neglected. This procedure has been criticised by R, D. Brown et al. m on the grounds that the exchange contributions cannot be ignored and spectroscopic configuration functions were not used. Nevertheless, good correlation was obtained between the trend ia, AE and the experimentally observed trend in absorl~. tion maxima. In these calculations the order of th$ highest occupied and lowest empty orbitals changed in going from one system to another. In VO, 3- and TiCI~ the 4h level fell below the 2e level, the difference being greatest for TiCh. This inversion of the order expected from crystal field theory was not o1~ served in the calculation of Fenske and Radtke ~ an~ was thought to be due to a poor choice of radial functions. However, a reinvestigation of TiCI, by Becket and Dahl ~81 which employed extensive variations of the metal and ligand valence orbitals to minimize the total energy also found this unusual ordering. More recently these attthors have investigated the electronic structure of open shell VCh ~67 using an approximate form el Roothaan's open-shell method. ~7 A regular tetrahedral structure was assumed since there is no experimental evidence for a static ]ahn-Teller effect in the vapour or solution phase. Again a minimal STO basis set was used whose components were systematically varied to minimize the total energy. Good quantitative agreement with the results of Fenske and Radtke ~ with regard to ordering of filled levels and cigenvalue magnitudes was obtained. A ~I', ground state was tentatively assigned, the unpaired electron going into the 4h level. However, a more recent calculation by Copeland and Ballhausen) '~ using more elaborate chlorine basis orbitals and a more rigorous open-shell calculatlonal method, placed the lnorganica Chimica Acta

Molecular Orbital Theory o] Transition Metal Complexes

lone electron in the 3at level and so predicted a 2At ground state configuration. The permanganate and chromate ions have been studied tt6 by the CNDO technique of R.D. Brown and Roby )m A near valence STO basis with Burns exponents was employed, the ligand 2s electrons being regarded as part of the core. Configuration interaction calculations based on the ground state wavefunctions and all smgly excited configurations were also performed. A series of comparative calculations to investigate the effect of different paramaterizations on the numerical results was carried out. The calculation which gave the most acceptable results suggested a very low charge of + 0 . 1 7 on the manganese and a charge of -0.81 on chromium. On the basis of intensity calculations and a theoretical analysis of the magnetic circular dichroism data all bands in the visible/near UV spectrum were assigned to symmetryallowed IAr-*tT2 transitions. In an effort to improve the theoretical treatment of their lower excited states R.D. Brown et al. m also investigated the electronic structures of permanganate and chromate ions using the MCZDO ~°Bmethod which includes evaluation of one-centre exchange integrals. However, no improvement over the CNDO spectral energies was reported even though a more reasonable

Table III. Experimental and Calculated Metal Charge in the Permanganate Ion. Method

q~

Ref.

Experimental Ab-initio LCAO - SCF Ab-initio SW - SCF

+ 1.28 + 1.43 +0.64

Mulliken - Ruedenberg

+ 1.85

MCZDO CNDO CNDO SCCC

+ 1.22 +0.17 +2.09 + 0.33

166 164 163 93 116 116 179 93

27

charge distribution was obtained (qM, = + 1 22, qcr = +0.12). The authors have emphasized the sensitivity of their CNDO and MCZDO results to values of two-centre integrals and the need for more accurate values for these integrals using functtons more claborate than the approximate HF or Burns wavefunctions. The permanganate ion has clearly been the subject of numerous ab-initio and semi-quantitative MO caitulations. The results are marked by a diversity of calculated transition energies, spectral assignments and charge distributions, a situation evident from Tables III and IV. It has been suggested 164 that the form of the e orbitals is quite sensitive to the basis set used, and, since many bands have been assigned to transitions to the virtual 2e orbital, this may explain the difficulty in providing a good theoretical interpretation of the electronic spectrum.

F.

SEMI-EMPIRICAL CALCULATIONS

As an example of the good correlation with experiment obtainable with semi-empirical methods reference may be made to a recent series of SCCC calculations on transition metal carbonyl and x-allyl complexes performed in this laboratory using the Cusachs formula for the off-diagonal Fabcrf terms, t~,ts~,~.ts7 The nature of the metal-metal bond in a series of D,a binuclear metal carbonyls MtCOh0 (M2 = Mn2, Te2, ReD and C~v MnRe(CO)10 has been investigated tu and good correlation obtained between calculated orbital energies and Mulliken overlap populations and experimental photoelectron spectra and bond dissociation energies. The calculated and mass spectral ionization potentials for the series agreed to within 15%. In the case of Mn2(COh0 the symmetry ordering of the outer levels closely matched that deduced from relative band intensities in the photoelectron

Table IV. Experimental and Calculated Electronic Transition Energies (ev) for the Permangante Ion. Experimental Transition Energy m (ev)

Calculated Transition Energy (ev)

Calculational Method

Assignment

Reference

2.27

LCAO - SCF SW - SCF MCZDO CNDO CNDO

ltr-*7h lh--*2e 6tr-~2e 1h-*2e 6tr--~2e

162 163 183 116 4

3.5

3.81 3.3 6.98 4.13 3.99

LCAO - SCF SW- SCF MCZDO CNDO CNDO

7tr--*2e 6tr-~2e lh--~7t~ I tc--*2e 6tr--*2e

162 163 183 116 4

4.0

4.24 4.7 10.08 5.83

LCAO - SCF SW- SCF MCZDO CNDO

ih---~2e Ih--*7t2 6tt--*2e i h--~.7h

162 163 183 116

5.5

6.29 5.3 13.35

LCAO - SCF SW - SCF MCZDO CNDO CNDO

lh-~2e ltc--~7t: 6tr-*7h 5tr-~2e Ih--~St2

162 163 183 116 4

2.3

3.42 2.3 5.66 2.16

7.29

5.45

Reviews 1972

28

DAVID A. BROWN,W.l. CHAMBERS,and N.I. FITZPATRICK

spectrum. Both M-C axial ('axial' refers to an elongation of the metal-metal axis) and M-C equat ('equat' refers to an axis perpendicular to the metal-metal axis) total overlap populations paralleled the M-C stretching frequencies and, in all cases, the M-C axial bond was predicted to be the stronger, in agreement with force constant data. Overlap populations suggested that, at least for Mn2(COh0, the interaction between a given metal and the ligands on the other metal constitute an important contribution to the formation of a reasonably strong metal-metal bond. In a similar calculation tSs on the C4,. Mn(CO)sX and Re(CO)sX (X : Cl, Br, I) calculated and experimental ionization potentials correlated to within 3%. The M-C axial bond was predicted to be much stronger than the M-C equat, bond in all cases and the M-C bends in the rhenium compounds were predicted to be stronger than in the corresponding manganese complexes. In general, for a given metal, both the calculated M-C axial and M-C equat, overlap populations decreased in the order of C l < B r < I although this order was not found for Re-C equat. These halogenpentacarbonylmetal compounds undergo carbonyl mono-substitution by a first-order dissociative mechanism involving the M-C equat, bond. The trend in rate constants tu parallels the trend in M-C equat. overlap population and the observed lower reactivity of the rhenium compounds is in agreement with the higher Re-C equat, overlap populations. C-O stretching force constants correlated well with the electron population of the ant|bonding 5tr (the main ~-donor) and ~* (the ~-acceptor orbital) of CO. In addition the results suggested the M-X bond to be stronger than the M-C bonds which is in agreement with mass spectral patterns which show a strong tendency in those complexes for a retention of the halogen in the metal-containing fragment. In a study of the electronic structures of bis-~-allyl metals (A2M, M = Ni, Pd, Pt,) D.A. Brown and Owens t~ have been able to correlate the M-C overlap population with the observed stability sequence Ni-,P d < P t and also achieved good agreement with the photoelectron spectrum for A2Pd and A2Ni. The charge distributions suggested that the metal is essentially uncharged in the Ni and Pd complexes, for example in A2Pd the metal charge was calculated to be -0.09. The corresponding charge in the semiquantitative calculation of Hillier and Canadine" for the molecule was -0.30 Extent|on of the method t~ to the ~-allyl metal chloride dimers ((AMCIh, M = Ni, Pd, Pt) resulted in a greater M-C overlap population than for the A2M series, especially in the case of the nickel compounds. This is in agreement with the general enhanced experimental stability of (ANiCl)2 and (APdClh over the corresponding A2M complexes On the basis of its calculated high M-C overlap population (APtClh was predicted to be the most stable of the series although experimental evidence for its existence is still lacking. In general terms it was concluded that the ~-allyl ligand is primarily electron donating in character and, in the case of the (AMCIh sequence, the presence of electron-attrachting groups make it function almost exclusively in this way.

VIII.

CONCLUSION

The last decade has seen remarkable progress in the field of molecular computations. Based on the foundations then laid the stage now seems set for comparatively large-scale ab-initio calculations in the 1970's. Increased computing power and more streamlined computing algorithms will permit the more accurate study of small systems coupled with experimentation with basis sets to reduce the amount of computation involved in building accurate wavefunctions for large systems. It may also happen that methods such as the SCF-SW technique with their modest demands on computer time will prove superior to the traditional LCAO techniques. In addition the accurate treatment, of transition metal complexes which contain atoms of high atomic number may demand the inclusion of relativistic effects~ and, in recognition of the fact that the Born-Oppenheimer approximation is merely a limiting case of the vibronic motion of a molecule, adiabatic hamilton|an potentialsfl°

IX.

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Metal

Complexes

29

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30

DAVID A BROWS, W.J. CHAMBERS, and N.J. FITZPATRICK

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lnorganica Chimica Acta