A finite volume approach for solving the ground state helium

Abstract

A three-dimensional finite volume discretization of the time-independent Schrödinger equation is formulated for the ground state helium atom. A set of algebraic equations arising from the second-order discretization of the PDE are solved for the wavefunction using the Gauss–Seidel algebraic multigrid solver coupled with a modified Stodola–Vianello method in extracting the ground state energy. A thorough mesh refinement study results in the ground state energy of the helium atom to be −2.903 230 au, which is only 0.017% above the Frankowski–Pekeris value. Mesh convergence of the wavefunction solution suggests that it correctly approximates to the unknown exact solution.