Effects of a supplementary quadrature in the collocation method for solving the Hartree Fock equations in Ab-initio calculations

The collocation method for solving the Hartree-Fock equations of the self-consistent field in large atomic and molecular systems is analyzed and a method for improving its performances by supplementary analytical and numerical quadrature is proposed. We used monomial and Chbyshev type trial functions and the collocation points were equidistant or Chebyshev polynomial roots. The singularities have been avoided by a function change and analytical expressions have been obtained for the most part of the integrated terms in the matrix elements, except the Hartree-Fock potential which is treated separately in a similar way. This apriori analytical treatment ensures a greater speed and a lower condition number of the matrix necessary for the expansion's coefficient calculus, with an important effect on the overall precision and speed. Some numerical results are presented and compared with well-known types of orbitals, demonstrating the performance increasing in terms of precision and computing effort.