Page 135 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

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 asso-
ciates 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: