Page 299 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

282 P. LAZZERETTI ET AL.
All the quantities appearing in (15) are expanded in powers of a formal perturbation
parameter which is finally put equal to unity, so that, for instance,

The matrix of Lagrange multipliers is usually chosen diagonal to zero order, so that

and

with

To first order in

with

Taking the Hermitian product with in eq. (21) one has

Taking the product in eq. (21) with where k labels another occupied orbital in
a non degenerate problem, using (19) and (20),

Owing to the arbitrary nature of the Lagrange multipliers, one can choose

so that the projection of the first-order i-th orbital on the subspace spanned by occ–1
occupied MO’s vanishes. From the orthogonality condition (22) one has,
for the i-th orbital,

For real perturbations, e.g., in the presence of a static electric field, which is the case
studied in the present paper, the zero- and first-order orbitals can always be chosen
real, so that also

At any rate, the projection of the first-order orbitals on the subspace of occupied
is not needed within the McWeeny approach [7], where choice (25) is implicitly
assumed; it is sufficient to calculate the projection on the subspace of virtual