Page 163 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

148 M. DEFRANCESCHI ET AL.
each step the new iterates even if initially orthonormal, need to be
renormalized and orthogonalized to form true canonical HF orbitals. Great care must be
exercised in selecting orthogonalization procedures, for instance the so-called Löwdin's
symmetric orthogonalisation procedure [57], often used in Quantum Chemistry, mixes all
the orbitals simultaneously, tends to contaminate all the iterates, and impairs the
convergence of the iterative steps. Schmidt orthogonalization does better (since it allows
to choose the sequence of orthogonalization) but looses track of the symmetry of these
orbitals. Finally, the canonical orthogonalization performs a maximum in mixing of all
+
states. We have shown [31] for the ground state of Be and B that the Gram-Schmidt
procedure turns out to be more appropriate in most cases, but with very good trial
functions, Löwdin's symmetric procedure yields equivalent results. In all cases reported
in Table 1, Gram-Schmidt orthonormalization has been used.

3.2. NUMERICAL TECHNIQUES

Different integration schemes have been considered. To cancel the singularity factor in
Eq. 13 by the integration volume element, Navaza and Tsoucaris have proposed the use of
spherical polar coordinates. However, because of the convolution integrals, interpolation
schemes are needed in these coordinates since arguments (p-q) do not necessarily belong
to the grid points. The computation time increases as the square, , of the number of
points of the integration grid, and for large systems, this time becomes prohibitive.
Another point of view has been to focus on these convolution integrals and treat them via
a more economical fast Fourier transform procedure. In this case, the computation time
increases only as , but at the expense of an approximate treatment of the
singular factor [58,59]. Variants [60,61] based on the Fock transformation have also
been proposed to deal with the infinite limits of integration resorting to a one-to-one
correspondence between intervals and . At the present time,
none of the approaches has been satisfactory enough to bring the fully numerical
momentum quantum chemistry calculations beyond a stage of prematurity. Furthermore,
computational tests [25] on helium atom have shown the importance of accuracy and
convergence of the integrals. It seems that straightforward numerical calculations are not
readily applicable and our work is now directed toward mixed numerical and analytical
procedures.