Page 197 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

180 Y. G. SMEYERS ET AL.

this orthogonality requirement is automatically achieved, when both wavefunctions
exhibit different multiplicities or different spatial symmetries. In the second case, the
promoved and excited spinorbitals, and possess also different symmetries.
When both wavefunctions exhibit the same multiplicity and the same spatial symme-
try, it is convenient that the excited function should be orthogonal to the fundamental
one [15]. One way to achieve partially this requirement is orthogonalized the excited
orbital to its companion at each step of the iterative procedure. Remember
that and possess the same symmetry.
In any cases, the orthogonality requirement applied to the orbitals:

implies some modifications in the formulae of the previous paragraph in order to avoid
some singularities [7]. In particular, new cross Fock operators have to be redefined:

and

in which the sumations are restricted to the nonorthogonal orbitals.
In addition, partial cross Fock operators are also to be defined for evaluating the
matrix elements in which the orthogonal orbitals are involved:

and

For the same reason, new density projection operators are redefined:

as well as limited projection operators to the k or u orbital space: