Pseudospectral Calculation of Helium Wave Functions, Expectation Values, and Oscillator Strength

TL;DR

This paper demonstrates the extension of pseudospectral methods to accurately compute helium wave functions, expectation values, and oscillator strengths, achieving near state-of-the-art precision without specialized numerics.

Contribution

It introduces a method for applying pseudospectral techniques to wave functions with non-zero angular momentum and provides a systematic approach for exponential convergence in helium calculations.

Findings

Achieved highly precise oscillator strength for helium transition.

Reproduced leading order corrections with near state-of-the-art accuracy.

Demonstrated exponential convergence of energies and matrix elements.

Abstract

The pseudospectral method is a powerful tool for finding highly precise solutions of Schr\"{o}dinger's equation for few-electron problems. We extend the method's scope to wave functions with non-zero angular momentum and test it on several challenging problems. One group of tests involves the determination of the nonrelativistic electric dipole oscillator strength for the helium $1^1$S $\to 2^1$P transition. The result achieved, $0.27616499(27)$, is comparable to the best in the literature. Another group of test applications is comprised of well-studied leading order finite nuclear mass and relativistic corrections for the helium ground state. A straightforward computation reaches near state-of-the-art accuracy without requiring the implementation of any special-purpose numerics. All the relevant quantities tested in this paper -- energy eigenvalues, S-state expectation values and bound-bound dipole transitions for S and P states -- converge exponentially with increasing resolution and do so at roughly the same rate. Each individual calculation samples and weights the configuration space wave function uniquely but all behave in a qualitatively similar manner. Quantum mechanical matrix elements are directly and reliably calculable with pseudospectral methods. The technical discussion includes a prescription for choosing coordinates and subdomains to achieve exponential convergence when two-particle Coulomb singularities are present. The prescription does not account for the wave function's non-analytic behavior near the three-particle coalescence which should eventually hinder the rate of the convergence. Nonetheless the effect is small in the sense that ignoring the higher-order coalescence does not appear to affect adversely the accuracy of any of the quantities reported nor the rate at which errors diminish.