Finite-element solution of the Schrödinger equation for the helium ground state

Abstract

The finite-element method has been used to obtain numerical solutions to the Schr\"odinger equation for the ground state of the helium atom. In contrast to the globally defined trial functions of the standard variational approach, the finite-element algorithm employs locally defined interpolation functions to approximate the unknown wave function. The calculation reported herein used a three-dimensional grid containing nine nodal points along the radial coordinates of the two electrons and four nodal points along the direction corresponding to the cosine of the interelectronic angle. This produced an energy of -2.9032 a.u., which lies 0.017% above the Frankowski-Pekeris value. The values of 〈${r}^{n}$〉, for n=-2,-1, 1, and 2, are closer to those of Frankowski and Pekeris than from all of the variational calculations with the exception of the calculation performed by Weiss, whose energy and 〈${r}^{n}$〉 values are comparable to those of the finite-element computation.