Semiclassical Treatment of High-Lying Electronic States of H$_2^+$

TL;DR

This paper compares quantum mechanical and semiclassical WKB calculations for high-lying electronic states of H$_2^+$, exploring an uncharted regime with states corresponding to very high principal quantum numbers and large internuclear distances.

Contribution

It introduces a semiclassical approach to high-lying H$_2^+$ states and validates it against quantum calculations, extending understanding into previously unexplored energy regimes.

Findings

Good agreement between quantum and semiclassical energies.

Semiclassical method accurately predicts spectroscopic constants and dipole matrix elements.

High-lying states reach principal quantum numbers up to 145 with large internuclear distances.

Abstract

This work reports quantum mechanical and semiclassical WKB calculations for energies and wave functions of high-lying $^2\Sigma$ states of H$_2^+$ in atomic units. The high-lying states presented lie in an unexplored regime, corresponding asymptotically to H $(n\leq 145)$ plus a proton, with $R\leq 120,000$ $a_0$. We compare quantum mechanical energies, spectroscopic constants, dipole matrix elements, and phases with semiclassical results and demonstrate a good level of agreement. The quantum mechanical phases are determined by using Milne's phase-amplitude procedure. Our semiclassical energies for low-lying states are compared with those published previously in the literature.