Page 193 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

176 Y. G. SMEYERS ET AL.
2.Theory
2.1. GENERAL APPROACH

The HPHF wavefunction for an 2n electron system, in a ground state of S quantum
number, even or odd, is written as a linear combination of only two DODS Slater
determinants, built up with spinorbitals which minimize the total energy [1-2]:

where and are two spinorbitals of opposite spin belonging to a same electron
pair, so that
This linear combination is obtained by projection of one of the determinants on the
spin eigenstates with S even or odd:

where is a permutation operator which interchanges all the and spin functions
in the initial determinant.
Since the HPHF wavefunction for singlet states does not contain any triplet contam-
ination, this model was seen to produce relatively good results for singlet ground
states, very close to those of the fully projected one [1-10].
The Brillouin’s theorem has been shown to hold in the case of the HPHF function [2].
As a result, any variations of the orbitals which minimizes the HPHF total energy,
can be expressed as:

where is the HPHF function in which an occupied orbital has been replaced
for an virtual one.
Introducing the HPHF wave-function expression (1) in (3), and taking into account
the idempotency of operator the following equation may be obtained:

where is a Slater determinant in which one i occupied orbital has been substituted
by a t virtual one.
In order to solve equation (4), the following matrix elements between Slater Deter-
minants have to be considered:

Since and are constructed with the same set of orthonormal spinorbitals, the
two first matrix elements can easily rewritten, according to the Slater’s rules [13], as: