Spline Hartree-Fock Program (SPHF)

In this program, the non-relativistic radial functions are expanded in a B-spline basis. As a result, the usual differential equations can be replaced by systems of non-linear, generalized-eigenvalue equations of the form (Hᵃ − εₐₐB)Pₐ = 0, where a designates the orbital. In this form the off-diagonal Lagrange multipliers must first be eliminated through projection operators. When two orthogonal orbitals of the same symmetry are both varied, the orbitals need to be rotated for a stationary solution before projection operators are applied. For multiply occupied shells, and when multiple shells are improved at the same time, Newton-Raphson method for updating all orbitals has better numerical properties and a more rapid rate of convergence than the eigenvalue process. Both ground and excited states may be computed using a default universal grid.

Spline expansions facilitate the mapping of a solution from one grid to another. The concept of two phases was introduced. In the first phase, a low-order spline with a small number of grid-points was used, so that the program could “learn” the values of parameters, like the range of an orbital, and then switches to higher-order splines with more grid points to improve accuracy. Finally, the Virial Theorem is applied to display the ratio V/T which should be -2 for an exact calculation.