Numerical solution of the Thomas-Fermi equation for molecules

Numerical solutions of the multicentered Thomas-Fermi equation for molecules are obtained by use of an efficient multigrid scheme based on the standard finite-difference discretization of the Laplacian in a modification of this nonlinear equation. It is found that the difference between the molecular potential and the sum of atomic potentials is bounded and varies smoothly through the nuclei, and is thus a good choice for the dependent variable to be discretized. This choice also results in a general program that is applicable to any molecule whose structure is known; the input data are the nuclear charges and their locations. The electron density and the difference between the electron density and the sum of atomic densities are obtained for N₂ and compared with the results of molecular-orbital calculations. The method's applicability to larger molecules is illustrated by a calculation of the electron density of C₆{H}₆$. Its extension to crystalline solids with cubic unit cells is straightforward.