Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method I. General Theory

The present paper concerns the derivation of phase-integral quantization conditions for the two-centre Coulomb problem under the assumption that the two Coulomb centres are fixed. With this restriction we treat the general two-centre Coulomb problem according to the phase-integral method, in which one uses an {\it a priori} unspecified {\it base function}. We consider base functions containing three unspecified parameters $C, \tilde C$ and $\Lambda$. When the absolute value of the magnetic quantum number $m$ is not too small, it is most appropriate to choose $\Lambda=|m|\ne 0$. When, on the other hand, $|m|$ is sufficiently small, it is most appropriate to choose $\Lambda = 0$. Arbitrary-order phase-integral quantization conditions are obtained for these choices of $\Lambda$. The parameters $C$ and $\tilde C$ 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.