Page 139 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

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)