Page 198 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

HARTREE-FOCK MODEL TO THE LOWEST SINGLET AND TRIPLET EXCITED STATES 181

Finally, a restricted overlap between the two determinants limited to the nonorthog-
onal orbitals is defined:

In order to deduce the new pseudo-eigenvalue equations (22), we have to distinguish
the two possibilities:
When the Brillouin Theorem equation (20) is reduced to:

where the cross energy term, between the two Slater determinants, takes the form
of a simple repulsion integral:

In contrast, when the following expression is found:

From equations (33) and (35), a general HPHF Fock operator for determining the
orbitals of excited states can be extracted after some straightforward transformations:

Since equation (36) is not symmetric, it is symmetrized by addition of the ad joint
of the asymmetric part. We obtain the new expression:

A similar equation can be deduced for the b i orbitals.

3. Calculation

In order to determine the HPHF wave-functions, the HPHF Fock operators (24) and
(37) for the ground and excited states, respectively, have to be expressed in matrix
form, in which the orbitals are developed on a basis function set. So, we have for the
ground state: