Relativistic two-electron atomic and molecular energies using $LS$ coupling and double groups: role of the triplet contributions to singlet states

TL;DR

This paper calculates relativistic energies of two-electron systems, emphasizing triplet contributions to singlet states using double-group symmetry and high-precision variational methods, confirming theoretical predictions about fine-structure effects.

Contribution

It introduces a detailed computational approach combining double-group symmetry with variational solutions of the Dirac-Coulomb-Breit equation for two-electron systems.

Findings

Triplet contributions significantly affect singlet state energies.

Energy calculations achieve sub-parts-per-billion precision.

Results confirm the $ ext{α}^4$ dependence of relativistic corrections.

Abstract

The triplet contribution is computed to the 1 and 2 $^1S^\text{e}_0$ states of the He atom, to the $1\ ^1S^\text{e}_0$ state of the Li$^+$ and Be${^{2+}}$ ions, and to the $X\ ^1\Sigma_\text{g}^+$ ground state of the H$_2$ molecule by extensive use of double-group symmetry (equivalent to $LS$ coupling for the atomic systems) during the course of the variational solution of the no-pair Dirac-Coulomb-Breit wave equation. The no-pair Dirac-Coulomb-Breit energies are converged within a sub-parts-per-billion relative precision using an explicitly correlated Gaussian basis optimized to the non-relativistic energies. The $\alpha$ fine-structure constant dependence of the triplet sector contribution to the variational energy is $\alpha^4E_\text{h}$ at leading order, in agreement with the formal perturbation theory result available from the literature.