Page 301 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

284 P. LAZZERETTI ET AL.
Left-multiplying (32) by occupied in a non degenerate case, using (19),
(21), and (33) one obtains

This equation shows that second- and first-order Lagrangian multipliers are not in-
dependent, so that a specific selection of will bias Thus the choice (25) for
the first-order Lagrangian multipliers makes the sum over j in eq. (35) vanish in the
case of real perturbations, but, in general, there is no choice of in eq. (35) which
annihilates the projection of on occupied and
However, using the McWeeny approach [7], it is sufficient to calculate only the pro-
jection on the subspace of virtual zero-order orbitals in order to get the
second hyperpolarizability tensor. This projection is evaluated via a procedure similar
to the one used in solving the first-order equation (21). Taking in (32) the Hermitian
product with the unoccupied and using (19), one finds

Multiplying on the left by the ket and summing over k, one finds

where

is available from the solutions (29) to the first-order eq. (21).

4. Computational scheme

We will now discuss an iterative scheme based on the CHF approach outlined in
Sections II and III, using the McWeeny procedure [7] for resolving matrices into
components, by introducing projection operators and with respect to the
subspaces spanned by occupied and virtual molecular orbitals.
Expanding the occupied over an orthonormal atomic basis set of order m (which
is assumed independent of the perturbation), one has