H₂⁺: A Problem in Perturbation Theory

Abstract

A résumé is given of how accurately perturbation theory (using a ground-state hydrogen-like atom perturbed by a point charge) predicts the 1sσ and 2pσ energies of H2+ and HeH2+. Corresponding delta-function models are discussed; here it is possible to test the convergence of the perturbation expansions. The models indicate that the series for the 1sσ energy is divergent for H2+ at large values of the internuclear separation R but convergent for HeH2+; this explains why the H2+ series is inaccurate and the HeH2+ series accurate. Both series converge for small R. The 2pσ energies are represented by divergent series which are asymptotically convergent if R is sufficiently large. A `resonance' correction is justified for the H2+ series, giving an accurate 1sσ energy for all R and an improved 2pσ energy.