Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

116 J. P. MALRIEU AND J. P. DAUDEY

In our opinion it would be better to avoid the HF step, and to start the CI process from any unbiased function, symmetrical for as are the Huckel MOs. We think that it is risky to study the existence and amplitude of a physical symmetry-breaking phenomenon through a computational sequence involving a symmetry-broken wave function at an intermediate step, since the use of this function introduces a prejudice and may result in an overestimation of the geometrical symmetry breaking. In that case the singlet symmetry breaking goes through an overestimation of the pairing of electrons into bonds (bondcentered charge density waves) as previously discussed and this overpairing, evident for

necessarily acts for constraining the bond alternation. The approximate CI cannot repair the defect of this starting point [51].

4. Final comments

Even if symmetry-broken wave functions are difficult to use for higher levels of computation, their physical content is always instructive about the physical trends of the problem under study and they deserve interest. Their appearance and the more physical geometrical symmetry breaking are internally (but not strictly) related. Since they represent catastrophes on the wave function and/or the energy (or energy derivatives) they should be studied with attention and our ultranumericist discipline has not paid enough attention to these critical behaviours. This neglect is perhaps due to some implicit philosophical

"continuism", prevailing in a domain where most instruments are based on variational procedures and optimizations. The use of computers and algorithms as black-boxes, and even the systematic plotting of the results through graphic codes using spline interpolations sometimes lead some quantum chemists of high reputation to miss cusps and intriguing features in their results [13,37]. Since qualitative explanations or pictures may be obtained from symmetry-broken wave functions and since funny behaviours are expected around conformational symmetry breaking, these problems should not be considered as teratological. Pictorial explanations and qualitative problems are both necessary to balance the unavoidable and fruitful research of numerical efficiency.

References

1.A.W. Overhauser, Phys. Rev. Letters, 4, 415 (1960).

2.D.J. Thouless, Nucl. Phys. 21, 225 (1960).

3.W.H. Adams, Phys. Rev. 127, 1650 (1962).

4.P.O. Lowdin, Rev. Mod. Physics, 35, 496 (1963).

5.G. Berthier, J. Chim. Phys. 52, 363, (1954).

6.J.A. Pople and R.K. Nesbet, J. Chem. Phys. 22, 57 (1954).

7.H. Fukutome, Intern. J. Quant. Chem. 20, 95 (1981).

8.C.A. Coulson in “Quantum Theory of Atoms. Molecules and the Solid State",

P.O. Löwdin ed., Acad. Press, N. York (1966) p 601. T.A. Kaplan and W.H. Kleiner, Phys. Rev. 156, 1 (1967).

9.W. Kutzelnigg, Angew. Chem. Int. Ed. Engl. 23, 272 (1984).

10.R. Prat and G. Delgado Barrio, Phys. Rev. A, 12, 2288 (1975).

11.for a text-book presentation, see A. Szabo and N. Ostlund, "Modern Quantum

Chemistry ", Mc Millam, London (1982) p 221.

QUANTUM CHEMISTRY IN FRONT OF SYMMETRY BREAKINGS

117

12.M.B. Lepetit, J.P. Malrieu and G. Trinquier, Chem. Phys. 130, 229 (1989).

13. W.D. Laidig, P. Saxe and R.J. Bartlett, J. Chem. Phys. 86, 87 (1987).

14.M.B. Lepetit and J.P. Malrieu, J. Chem. Phys. 87, 5937 (1987).

15.K. Yamaguchi, Chem. Phys. Letters, 68, 477 (1979).

16.M. Bénard, J. Chem. Phys. 71, 2546 (1979).

17.R. Wiest and M. Bénard, Theoret. Chim. Acta, 66, 65 (1984).

18.M.M. Goodgame and W.A. Goddard, Phys. Rev. Letters, 48, 135 (1982);

J. Phys. Chem. 85, 215 (1981)

19.M.B. Lepetit and J.P. Malrieu, Chem. Phys. Letters, 169, 285 (1990).

20.R.J. Harrison and N.C. Handy, Chem. Phys. Letters, 98, 97 (1983) and references therein.

21.G. Trinquier, private communication.

22.J. Cizek and J. Paldus, J. Chem. Phys. 47, 3976 (1967).

23.J. Paldus and J. Cizek, Phys. Rev. A 2, 2268 (1970).

24.M.H. McAdon and W.A. Goddard, J. Chem. Phys. 88, 277 (1988).

25.M.H. McAdon and W.A. Goddard, J. Phys. Chem. 92, 1352 (1988).

26.M.B. Lepetit, J.P. Malrieu and F. Spiegelmann, Phys. Rev. B 41, 8093 (1990).

27.M.H. McAdon and W.A. Goddard, Phys. Rev. Letters, 55, 2563 (1985).

28.M.B. Lepetit, E. Apra, J.P. Malrieu and R. Dovesi, Phys. Rev. B, 46, 12974 (1992).

29.R. Dovesi, C. Pisani, C. Roetti, M. Causa and V.R. Saunders, Crystal 88, Program N°577, QCPE, Indiana University, Bloomington, 1989.

30.L.C. Snyder, J. Chem. Phys. 55, 95 (1971).

see also, P.S. Bagus and H.F. Schaefer, J. Chem. Phys. 56, 224 (1972).

31.A. Denis, J. Langlet and J.P. Malrieu, Theoret. Chim. Acta, 38, 49 (1975).

32.L.S. Cederbaum and W. Domcke, J. Chem. Phys. 66, 5084 (1977).

33.G. Durand, O.K. Kabajj, M.B. Lepetit, J.P. Malrieu, J. Marti, J. Phys. Chem. 96, 2162 (1992).

34.V. Bonacic-Koutecky, P. Bruckmann, J. Koutecky, C. Leforestier and L. Salem, Angew. Chem. Inst. Ed. Engl. 14, 575, (1975); L. Salem and P. Bruckmann, Nature (London) 258, 526 (1975).

35.For reviews see J.P. Malrieu. Theoret. Chim. Acta, 59, 281 (1981). and J.P. Malrieu, I. Nebot-Gil and J. Sanchez-Marin, Pure & Appl. Chem. 56, 1241 (1984).

36.A.D, McLean, B.H. Lengsfield, J. Pacansky and Y. Ellinger, J. Chem. Phys. 83, 3567 (1985).

As other examples one may quote the symmetry-breaking of the CASSCF (4e in

4MO) calculation of the twisted excited state of ethylene (G. Trinquier and J.P. Malrieu, in : "The structure of Double Bond". Patai ed., John Wiley (1990) p 1, or

the symmetry-breaking in electron transfer problems (A. Faradzed, M. Dupuis, E. Clementi and A. Aviram, J. Amer. Chem. Soc. 112, 4206 (1992).

37.C.W. Bauschlicher, and S.R. Langhoff, J. Chem. Phys. 89, 4246 (1988)

38.A. Sanchez de Meras, M.B. Lepetit and J.P. Malrieu, Chem. Phys. Letters, 172,

163 (1990).

39.R. Krishnan, M.J. Frisch and J.A. Pople, J. Chem. Phys. 72, 4244 (1980).

40.M.R. Nyden and, G.A. Peterson, J. Chem. Phys. 14, 6312 (1981).

41.N.C. Handy, P.J. Knowles and K. Somasudram, Theoret. Chim. Acta, 68, 87 (1985).

42.P.M.W. Gill and L. Radom, Chem. Phys. Letters, 132, 16 (1986).

43.M.B. Lepetit, M. Pelissier and J.P. Malrieu, J. Chem. Phys. 89, 998 (1989).

118	J. P. MALRIEU AND J. P. DAUDEY

44.M.U. Bohmer and S.D. Peyerimhoff, Z. Physik D, 3, 195 (1986).

45.M.T. Bowers, W.E. Palke, K. Robins, C. Roells and S.Walsh, Chem. Phys.

Letters, 180, 235 (1991).

46.J.M. McKelvey and G. Berthier, Chem. Phys. Lett ers, 41, 476 (1976).

47.D.P. Kleier, R.L. Martin, W.R. Wadt and W. Moomaw, J. Amer. Chem.

Soc. 104, 60 (1982) and references herein.

48.I. Nebot-Gil and J.P. Malrieu, J. Chem. Phys. 77, 2475 (1982).

49.H.C. Longuet-Higgins and L. Salem, Proc. Roy. Soc. (London) A 25, 172

(1959).

50.G. König and G. Stolhoff, Phys. Rev. Letters, 65, 1239 (1990.)

51.For an interesting discussion see E.R. Davidson and W.T. Borden, J. Phys. Chem. 87, 4783 (1983).

Molecular Orbital Electronegativity as Electron Chemical Potential in

Semiempirical SCF Schemes

G. DEL RE

Chair of Theoretical Chemistry, Università “Federico II”, Via Mezzocannone 4, I-80134 Napoli, Italy

1. Statement of the problem

The identification of the electronegativity of an orbital with the corresponding electron chemical potential - i.e. the derivative of the total energy with respect to the orbital occupation - is well known, and was in fact mentioned in Hinze and Jaffé’s classical paper on electronegativities [1]. That paper referred to atomic orbitals; as far as we know, the notion of electronegativity of a molecular orbital has not been extensively discussed, although an explicit expression of the electronegativity of a molecular orbital has been given in the context of a theoretical analysis of ground-state charge transfer [2]. That expression closely matches Mulliken's classical expression [3], but does derive from an explicit general equation for the chemical potential of an electron in that orbital. We describe here the derivation of such a general equation with special reference to the semi-empirical methods leading to SCF schemes, which are especially useful nowadays for treating large molecules. Probably the method of that kind that is least charged with unphysical and possibly contradictory assumptions is the BMV method, which G. Berthier developed with his collaborators Millie and Veillard [4] in 1965, and De Brouckère [5] extended to molecules containing transition metals in 1972. It is an all-valence-electron method not involving neglect of differential overlap, in which the the diagonal elements of the Hamiltonian depend of the AO populations and the off-diagonal elements are estimated so as to avoid the drastic simplifications concealed in the Wolfsberg-Helmholtz approximation. Many of the ideas of the present author on SCF schemes and their properties go back to discussions and joint work with Berthier on his method. A late development of those discussions is the question discussed here.

The analytical determination of the derivative		of	the	total energy
with respect to population	of the r-th molecular	orbital is	a	very complicated

task in the case of methods like the BMV one for three reasons: (a), those methods assume that the atomic orbital (AO) basis is non-orthogonal; (b), they involve nonlinear expressions in the AO populations; (c) the latter may have to be determined as Mulliken or Löwdin population, if they must have a physical significance [6]. The rest of this paper is devoted to the presentation of that derivation on a scheme having the essential features of the BMV scheme, but simplified to keep control of the relation between the symbols introduced and their physical significance. Before devoting ourselves to that derivation, however, we with to mention the reason why the MO occupation should be treated in certain problems as a continuous variable.

119

Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 119–126,

© 1996 Kluwer Academic Publishers. Printed in the Netherlands.

120 G. DEL RE

In an ordinary MO scheme, fractional occupation of an orbital can only be accepted as a more or less useful fiction. This is because the whole electronic state is assumed to be correctly described by a single Slater determinant. An improvement which is sometimes indispensable is provided by CI (configuration interaction), which associates different occupation schemes to a given set of orbitals. Now, as is well known, already in the simple case of the linear combination with coefficients of two Slater determinants, the expectation value of the population of an orbital

the n values denoting the (integral) occupations of that orbital in the two Slater determinants. Thus, as soon as the reference scheme becomes one of configurations over MO's, the expected occupations of the latter must be assumed to be in general fractional. Now, when we juxtapose two molecules D and A acting as a donor-acceptor pair in some redox process, a very reasonable and simple way of treat- ing the situation theoretically, in accordance with Mulliken's original formulation [7], consists in assuming that the two partners are described by (possibly SCF) MO's that are localized on either partner and enter two Slater determinants correspond-

ing to the states

For a vanishing coupling between the

two states, the requirement that the actual situation should be described by a linear combination of those two states corresponding to the lowest energy can be translated into the condition that the chemical potentials of the orbitals differing in occupation in the two states should be equal [8]. This is the foundation for a rigorous derivation of the principle of electronegativity equalization [9].

2. Expression of the variations of the MO's

We consider the equation:

where:

and

The Hermitian Hamiltonian matrix H, the diagonal matrix E, and the unitary matrix

C are assumed to satisfy the equation:

The barred matrices have the same properties as those of eqn 5; in the case of normalization to unity of the single columns is ensured by an ad hoc diagonal matrix N. As will appear below (eqn 6), if terms in of order higher than the first are negligible, N can be taken equal to the identity matrix. This is what will be assumed in the following.

We now follow the familiar procedure of perturbation theory to extract from eqn 1

the first order expression of the variations		indicated by	Let	us start	from the
normalization condition, and denote by		the j-th column of any given matrix M.
Since both	are normalized to	unity, to first	order in	as	has been

mentioned, N may be taken to be unity, and we must have:

MOLECULAR ORBITAL ELECTRONEGATIVITY AS ELECTRON CHEMICAL POTENTIAL			121
whence it appears that	must be orthogonal to	a linear combination of
all the columns of C except	itself:

We now substitute eqns 2, 3, and 4 into eqn 1, eliminate second order terms in

multiply on the left by We find, first of all:

and separate the resulting equation into two as follows.

Multiplication by the j-th row of

on the left gives:

which, since

yields the familiar expression:

Multiplication by the k-th row

on the left gives:

Since	is orthogonal to	with the same consideration as has led to eqn 10,
eqn 12 becomes:

Comparing with eqn 7, we find:

For k = j and for degenerate eigenvalues the elements of f are taken equal to zero. Let us next consider the variation of the population-bond-order matrix, which, in the orthogonal case of eqn 1, is just:

where p and b are the diagonal (population) and the off-diagonal (bond-order) parts

of P, respectively, and	is the occupation number of the j-th MO.
From eqn 7 we find:

122	G. DELRE
3. Derivation of	electronegativity

Let us now specialize the above equations for the special case when only the population of the r-th MO changes, and the reference scheme is a simple -technique

[10] applied to an extended-Hückel method, which is a highly simplified form of the

BMV procedure.

We start from a Hamiltonian	whose off-diagonal elements are assumed to form
a constant matrix and the diagonal elements depend on the net charges		of the

individual AO's according to the expression

where

is a standard atomic parameter matrix, Z is the diagonal matrix of the AO oc-

cupations, and

is a suitable constant. Finally,

is a diagonal element of the

population matrix associated to the given AO's. We adopt here the Löwdin population analysis, i.e. assume that P (and therefore p) is defined by eqn 14 in terms of

the coefficients of the Löwdin AO's associated to	If S is the AO overlap matrix,
then H of eqn 5 is given by

and therefore, in virtue of eqns 16 and 17,

This gives

where

Now,considering

as the only independent variation, and remembering that f is

an antisymmetric matrix, one gets from eqn 15

where, for the sake of simplicity, the eigenvector coefficients have been assumed to be real (as they are in molecular problems) and

Equation 22 depends on

in virtue of eqn 13, and therefore does not define

completely. However, insertion of eqn 20 into eqn 13 transforms eqns 22 into a linear system that can be solved for We write eqn 13 for our special case in the form

MOLECULAR ORBITAL ELECTRONEGATIVITY AS ELECTRON CHEMICAL POTENTIAL

123

with

where	is the eigenvector matrix of	in the original non-orthogonal basis.
With this notation, eqn 22 becomes

where

is a matrix formed by the diagonal elements of the matrices:

Equations 26 form a linear system which can be solved without any difficulty. Let us first of all divide eqn 26 by and pass to the limit, so as to work directly in terms of partial derivatives. Let us then define the matrix:

With this notation, we can write the very simple expression:

For the off-diagonal part of P we have:

with the matrices W and X defined by eqns 28 and 29, respectively. If we next define a set of matrices with elements

we can finally write

124		G. DEL RE
We are now ready for computing the electron chemical potential within the		scheme.
Since ours is a Hückel-like scheme, the total energy	is the sum of the orbital
energies multiplied by the pertinent occupations, and therefore
where Tr stands for the trace. Deriving the above expression with respect to			we
obtain:

Substituting eqn 20, eqn 30 and eqn 31 into eqn 35 we find

and

If we now apply the well known property		(the latter being the energy
of the r-th MO) and take into account that		, R being the density
matrix over the non-orthogonal MO's	, (cf.	eqn 26) obtained from	, we find for
eqn 35:

This is our final equation. A simplified form is found if the matrices W defined in eqn 28 are neglected (so that X and the matrices are ignored). This is possible, for example, in the case of large energy differences between MO's whose occupations are different. Then

4.Discussion

We have presented above the derivation of eqns 38 and 39 in great detail because it includes expressions of general utility, in particular the variation of the eigenvectors (eqns 7 and 24) of an MO problem after Löwdin orthogonalization and the resulting variation of the population matrix P. The generalization to a Hamiltonian more complicated than that of eqn 19 is possible by following step by step the above derivation.

The physical meaning of our final equation is best seen on eqn 39. The term containing is essentially the self-energy correction introduced by Mulliken in his analysis of electronegativities to account for the average repulsion of electrons occupying the same orbital. In order to get an idea of the orders of magnitude, let us apply eqn 39 to a model computation of FeCO, made to compare the CIPSI results of Berthier et al.

[11] with those of a simple orbital scheme. Consider one of the two systems of FeCO,

treated under the assumption of full localization (and therefore strict

separation)

MOLECULAR ORBITAL ELECTRONEGATIVITY AS ELECTRON CHEMICAL POTENTIAL

125

in an iterative MO-LCAO scheme using as an AO basis maximum localization hybrids [12], Hoffman’s atomic parameters [13], and Cusachs’ expression [14] for the offdiagonal elements of the Hamiltonian . Since this is just an illustration of the numerical aspects of the equations given above, we need not justify further the scheme used. The special features which make the present example especially suited for our purpose include the fact that in the case of the system the MO occupation numbers must be given the values 2, 1.5, 0 (in the order of increasing orbital energies) in order to ensure the equivalence between the two degenerate systems of our linear molecule. This feature is especially important since of eqn 28 is the sum of terms containing as factors the differences between MO occupation numbers

This fact implies that only MO's with different occupation numbers play a role in the terms by which eqn 38 differs from the simpler form 39.

Table 1: Source data and		for the average system of FeCO
a. Overlap matrix for one	system of FeCO
	1.0000	0.2783	0
	0.2783	1.0000	0.2414
	0	0.2414	1.0000
b. Ham. matrix (eV) with	correction at convergence
	-9.6177	-2.2081	0
	-2.2081	-10.9490	-2.7811

0-2.7811 -12.6638

c.Eigenvalues (italics, eV) and Löw din charge bond-order matrix

-12.7349		-11.4723	-9.4836
0.7671		0.5442	-0.0530
0.5442		-0.2747	0.0418
-0.0530		0.0418	-0.4924
d. Derivative of P with respect to	and diagonal elements of			(italics)

0.2087 -0.4060 -0.0689 -0.4060 0.7583 0.1859 -0.0689 0.1859 -0.0281

0.1502 0.8355 0.0142

The source matrices (for the sequence Fe C 0) are presented in Table 1 together with the resulting derivative of the Löwdin population matrix P. The electronegativities derived from the complete expression 38 and from the approximate expression 39 are -7.1690 eV and -7.3582 eV, respectively, thus suggesting that even in the unfavourable