Evaluation of the Bethe logarithm: from atom to chemical reaction

TL;DR

This paper introduces a computational method for accurately calculating the Bethe logarithm, enabling routine quantum electrodynamics corrections for small molecular systems, with high precision results demonstrated for various molecules.

Contribution

A new computational scheme based on Schwartz' method and Hylleraas functional minimization for Bethe logarithm evaluation, applicable to small polyatomic and polyelectronic molecules.

Findings

Achieved convergence of Bethe logarithm with better than 1:10^3 relative precision

Successfully applied method to various small molecular systems including H2, H3+, He2+, and H+H2

Demonstrated applicability near potential energy surface minima

Abstract

A general computational scheme for the (non-relativistic) Bethe logarithm is developed opening the route to `routine' evaluation of the leading-order quantum electrodynamics correction (QED) relevant for spectroscopic applications for small polyatomic and polyelectronic molecular systems. The implementation relies on Schwartz' method and minimization of a Hylleraas functional. In relation with electronically excited states, a projection technique is considered, which ensures positive definiteness of the functional over the entire parameter (photon momentum) range. Using this implementation, the Bethe logarithm is converged to a relative precision better than 1:10$^3$ for selected electronic states of the two-electron H$_2$ and H$_3^+$, and the three-electron He$_2^+$ and H+H$_2$ molecular systems. The present work focuses at nuclear configurations near the local minimum of the potential energy surface, but the computations can be repeated also for other structures.