Estimating the Accuracy of the Variational Energy: The Hydrogen Atom in a Magnetic Field as an Illustration

TL;DR

This paper introduces a numerical method to evaluate the accuracy of variational energies for a hydrogen atom in a magnetic field, using the Non-Linearization Procedure and orthogonal collocation, with applications to different trial functions.

Contribution

It presents a novel numerical approach to estimate the accuracy of variational energies and related properties for atomic systems in magnetic fields.

Findings

Validated the accuracy of variational energy at $b3=1$ a.u.

Assessed the local accuracy of trial functions.

Analyzed the cusp parameter and quadrupole moment accuracy.

Abstract

For a hydrogen atom subject to a constant magnetic field, we report a numerical realization of the two-dimensional Non-Linearization Procedure (NLP) to estimate the accuracy of the variational energy associated with a given trial function. Relevant equations of the NLP, which resemble those describing a dielectric medium with a space-dependent permittivity and charge distribution, are solved numerically with high accuracy by using an orthogonal collocation method. As an illustration, we consider three different trial functions including the one proposed in Phys. Rev. A 103, (2021) 032820. For a magnetic field strength of $\gamma = 1$ a.u., we establish the accuracy of the variational energy as well as the local accuracy of the trial function. Additionally, the accuracy of the cusp parameter and the quadrupole moment, found by means of the trial function, is investigated