A finite difference Newton-Raphson solution of the atomic Hartree-Fock problem

Abstract

The SCF iteration is coupled with a finite difference Newton-Raphson algorithm to solve the set of coupled second-order integrodifferential equations with split boundary conditions which constitutes the atomic HF problem. In the new method the two-point boundary conditions at r = 0 and r = ∞ as well as the Lagrange multipliers are incorporated into a large system of nonlinear algebraic equations which are solved by means of a generalized Newton-Raphson iteration which converges rapidly and efficiently. The need to estimate initial slopes of the radial functions and values of Lagrange multipliers has been completely eliminated. As an example a calculation of the 1 S 22 S open-shell configuration of Li is presented. Through the use of Richardson extrapolation an accuracy of nine significant figures has been achieved. The new method is easier to apply and more versatile than the conventional methods. Although only Li and Be have been attempted so far (each with complete success) the method can certainly handle very large systems.