Introduction of arbitrary powers of the elliptical coordinate μ is shown to give a better approximate wave function for the hydrogen molecule-ion than any previous function of comparable complexity. The three-parameter two-configuration functionψ=Nμne−αμ(1+bν2)is found to give an energy of —0.602566 a.u. and an equilibrium internuclear separation of 1.9956 a0 compared with the results for n=0 of —0.602386 a.u. and 1.9966 a0, respectively, and with the exact values of —0.60263 a.u. and 2.0 a0, respectively. This agreement is maintained over a range of internuclear distances. In the Appendix a discussion is given of various methods for computing kinetic energy integrals, and a special difficulty is pointed out that arises when functions in prolate spheroidal coordinates are employed.
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Geller et al. (1962) studied this question.