a Generalized Approach to the Roothaan Numerical Orbital Procedure: Nonmatching Integration and its Application to a Helium Isoelectronic Series

The difficulty in obtaining numerical solutions of the Hartree-Fock equations for large atomic systems has led to efforts at simplifying their treatment. Recently a technique was devised which effectively transforms the equations into a homogeneous linear system. To fully benefit from such an approach, a new integration scheme has been developed which eliminates the need for matching, providing greater flexibility when dealing with larger atoms. By integrating in a single direction, the new process offers a significant reduction in processor time with no loss in accuracy of the calculated results. It is applied to the ground state of the helium isoelectric series as a test of its feasibility. The Hartree-Fock equations are converted into a set of "difference equations" via a five-point generalization of the Numerov formula. To achieve stability, a Gaussian elimination technique is implemented which transforms the system into one whose solution depends solely on the boundary values, avoiding spurious "oscillations" in the wave functions associated with extrapolation roundoff errors. The results of this investigation were compared to those obtained from calculations performed with the original matching routine and were found to conform quite closely; agreement in many cases approached the limit of computer precision. The new procedure reduced Central Processor time up to 45% while preserving the accuracy of the earlier data.