Pseudospectral Calculation of the Wavefunction of Helium and the Negative Hydrogen Ion

TL;DR

This paper demonstrates that pseudospectral methods can efficiently and accurately solve the Schrödinger equation for two-electron atoms, achieving exponential convergence without complex cusp treatments.

Contribution

It introduces a pseudospectral approach with multiple subdomains for handling wavefunction cusps, showing exponential convergence and simplicity over traditional methods.

Findings

Achieves exponential convergence in energy and wavefunction accuracy.

Uses three subdomains to effectively handle cusp behavior.

Simplifies cusp treatment without needing special logarithmic handling.

Abstract

We study the numerical solution of the non-relativistic Schr\"{o}dinger equation for two-electron atoms in ground and excited S-states using pseudospectral (PS) methods of calculation. The calculation achieves convergence rates for the energy, Cauchy error in the wavefunction, and variance in local energy that are exponentially fast for all practical purposes. The method requires three separate subdomains to handle the wavefunction's cusp-like behavior near the two-particle coalescences. The use of three subdomains is essential to maintaining exponential convergence. A comparison of several different treatments of the cusps and the semi-infinite domain suggest that the simplest prescription is sufficient. For many purposes it proves unnecessary to handle the logarithmic behavior near the three-particle coalescence in a special way. The PS method has many virtues: no explicit assumptions need be made about the asymptotic behavior of the wavefunction near cusps or at large distances, the local energy is exactly equal to the calculated global energy at all collocation points, local errors go down everywhere with increasing resolution, the effective basis using Chebyshev polynomials is complete and simple, and the method is easily extensible to other bound states. This study serves as a proof-of-principle of the method for more general two- and possibly three-electron applications.