Page 302 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

COUPLED HARTREE-FOCK APPROACH 285
The projection operators have matrix representations

and

A perturbation expansion analogous to (17) holds for any matrix, for instance, the
Fock matrix

and the density matrix

The first- and second-order coefficients and can also be resolved into projec-
tions on the subspaces of occupied and virtual molecular orbitals:

The projections with k labeling another occupied orbital, vanish according to
choice (25), and the projection vanishes for a real perturbation, see eq. (27).
Any matrix A can now be resolved into projection components [7] with respect to
the occupied and virtual subspaces, that is,

For instance, the first-order density matrix can be written

where

The iterative scheme for the first-order coefficients becomes