H₂⁺: A Problem in Perturbation Theory

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 H 2 + and HeH 2+ . 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 H 2 + at large values of the internuclear separation R but convergent for HeH 2+ ; this explains why the H 2 + series is inaccurate and the HeH 2+ 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 H 2 + series, giving an accurate 1sσ energy for all R and an improved 2pσ energy.