Page 304 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer
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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.
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