Page 196 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

HARTREE-FOCK MODEL TO THE LOWEST SINGLET AND TRIPLET EXCITED STATES 179

which is the expression of the Generalized Brillouin Theorem for the HPHF function
written as a function of the orbitals. A similar equation can be deduced when a
orbital is replaced by a one. The next step will now be to write equations
(19) as pseudo-eigenvalue equations to be solved in an iterative way just as in the
Unrestricted Hartree-Fock method.
For this purpose, let us define the following density projection operators:

and let us introduce them in (19). After some straightforward operations, we obtain:

From this equation the following HPHF Fock operator for determining the orbitals
can be extracted:

A similar operator for determining the orbitals can be obtained in the same
way.
Let us remark that operator (21) is not symmetric. But, it can be symmetrized easily
just by adding the adjoint of the asymmetric part:

In addition, since the action of operator on a virtual orbital is zero, it is seen
that this adjoint will not affect the results. So that the complete operator may
be written as:

2.2. APPLICATION TO EXCITED STATES
The HPHF wavefunction for an excited state is constructed by substi-
tuting in the HPHF ground state wavefunction (1) an occupied spinorbital by an
virtual one. In order to avoid the possible collapsing of to so-constructed excited
wavefunction onto the ground state one during the variational process, it is conve-
nient that the excited function should be orthogonal to the former. In some cases,