Page 162 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

QUANTUM CHEMISTRY COMPUTATIONS IN MOMENTUM SPACE 147

Furthermore their incidence are very dependent upon the nature of the properties [52,53].
Due to computer limitations, basis sets cannot be extended indefinitely and direct
numerical evaluations seem the ultimate solution for molecules [54]. In position space
this is a viable alternative for diatomic molecules [55,56], but it cannot be extended easily
to polyatomic systems. Formulated in momentum space, the HF equations have not
explicit solutions and the difficulties to express them in terms of basis functions are
analogous to those encountered in the r-space. However the momentum space HF
equations give way to numerical approaches in which Coulombic interactions become
tractable even for polyatomic molecules [7] ; among other advantages, these equations,
Eqs. 13 and 20, do not require coordinate systems adapted to the geometry of the
molecules to remove Coulombic singularities. In both equations the only singular
contribution comes from the factor.

3.1. VARIATION-ITERATION PROCEDURE

In both position and momentum spaces, iterative procedures are necessary to solve the HF
equations. Starting from a trial orbital an approximate orbital, is
obtained after k+1 iterations from Eq. 13 rewritten as:

The procedure is repeated until convergence is reached. Since we are interested in bound
states where no problem of divergence or cusps conditions is raised. But the
method can be adapted to more general situations by introducing a translation of the
energy origin in Eq. 21.

Numerical and computational problems associated with the implementation of the
approach for routine use fall in two main categories : (a) numerical integration and (b)
enforcement of the orthogonality and renormalization of the numerical orbitals during the
iteration steps. Many different integration schemes have been considered in the past,
some of which will be detailed in the section 3.2. As concerns orthonormalization, at