Quantal two-center Coulomb problem treated by means of the phase-integral method. I. General theory

Abstract

The present paper concerns the derivation of phase-integral quantization conditions for the two-center Coulomb problem under the assumption that the two Coulomb centers are fixed. With this restriction we treat the general two-center Coulomb problem according to the phase-integral method, in which one uses an a priori unspecified base function. We consider base functions containing three unspecified parameters C, C^-, and Λ. When the absolute value of the magnetic quantum number m is not too small, it is most appropriate to choose Λ = |m| ≠ 0. When, on the other hand, |m| is sufficiently small, it is most appropriate to choose Λ = 0. Arbitrary-order phase-integral quantization conditions are obtained for these choices of Λ. The parameters C and C^- are determined from the requirement that the results of the first and the third order of the phase-integral approximation coincide, which makes the first-order approximation as good as possible. In order to make the paper to some extent self-contained, a short review of the phase-integral method is given in the Appendix.