Page 136 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

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 and separate the resulting equation into two as follows.
We find, first of all:

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: