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COUPLED HARTREE-FOCK APPROACH 287
The iteration starts with The second-order repulsion matrix is
defined analogously to (55). and X matrices have been computed only once
to solve the first-order CHF problem (i.e., to determine the polarizability and the
first hyperpolarizability . and are saved onto a file to be processed at
each step of the iterative calculation (63)-(66): it seems worthy of notice that the
present CHF algorithm, based on the Hartree-Fock propagator (30), is quite general,
compact and suitable for efficient sequential determination of both first- and second-
order perturbed orbitals. In addition, it can be easily extended to perturbations of
higher order.
So far we have considered an orthonormal basis set In actual calculations, em-
ploying non orthogonal sets of Gaussian functions with overlap matrix
it is customary to orthogonalize according to the Löwdin procedure, i.e.,
with similar equations for first- and second-order perturbed matrices. In the second-
order iteration eqs. (63) and (65) are replaced by
for a non orthogonal basis.
The expression for the electronic contribution to electric dipole moment,
is not affected by transformation (68)-(72), owing to the trace theorem. In addition,
it can be shown that the iterative steps (49)-(52) are formally the same for a non
orthogonal basis, as the formula for the polarizability
is also invariant under Löwdin orthogonalization. The overlap matrix appear only to
third order in the expression for the first hyperpolarizability [10],
where and are permutations of the expression in square brackets.