Page 134 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

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 elec-
tron 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 exten-
sively 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 elec-
tron 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 tran-
sition 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 non-
linear 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.