Page 137 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

122 G. DELRE
3. Derivation of electronegativity
Let us now specialize the above equations for the special case when only the popu-
lation 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 popula-
tion 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