The H$_2^+$ molecular ion: a solution

TL;DR

This paper develops a highly accurate uniform approximation for the eigenfunctions and energies of the H$_2^+$ molecular ion across various internuclear distances, combining WKB and perturbation methods.

Contribution

It introduces a compact uniform approximation for eigenfunctions and energies of H$_2^+$, with high accuracy, by combining WKB expansion and perturbation theory.

Findings

Relative accuracy of eigenfunctions $ extless 10^{-5}$

Total energy $E(R)$ accurate to 10-11 s.d. for $R ext{ in } [0,50]$

Calculated separation constants and oscillator strength consistent with existing data

Abstract

Combining the WKB expansion at large distances and Perturbation Theory at small distances it is constructed a compact uniform approximation for eigenfunctions. For lowest states $1s\si_{g}$ and $2p\si_{u}$ this approximation provides the relative accuracy $\lesssim 10^{-5}$ (5 s.d.) for any real $x$ in eigenfunctions and for total energy $E(R)$ it gives 10-11 s.d. for internuclear distances $R \in [0,50]$. Corrections to proposed approximations are evaluated. Separation constants and the oscillator strength for the transition $1s\si_{g} \rar 2p\si_{u}$ are calculated and compared with existing data.