Quantal Two-Centre 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-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.

arXiv comment23 pages, RevTeX, 4 EPS figures, submitted to J. Math. Phys

References in corpus (2)

• Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method III. Quantization Conditions in the General Case Expressed in Terms of Complete Elliptic Integrals. Numerical Illustration
• Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method II. Quantization Conditions in the Symmetric Case Expressed in Terms of Complete Elliptic Integrals. Numerical Illustration

Cited by in corpus (3)

• Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method III. Quantization Conditions in the General Case Expressed in Terms of Complete Elliptic Integrals. Numerical Illustration
• Quantal Two-Centre Coulomb Problem treated by means of the Phase-Integral Method II. Quantization Conditions in the Symmetric Case Expressed in Terms of Complete Elliptic Integrals. Numerical Illustration
• Semiclassical Treatment of High-Lying Electronic States of H${}_2^{+}$