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

摘要

The contour integrals, occurring in the arbitrary-order phase-integralquantization conditions given in a previous paper, are in the first- andthird-order approximations expressed in terms of complete elliptic integrals inthe case that the charges of the Coulomb centres are equal. The evaluation ofthe integrals is facilitated by the knowledge of quasiclassical dynamics. Theresulting quantization conditions involving complete elliptic integrals aresolved numerically to obtain the energy eigenvalues and the separationconstants of the $1s\sigma$ and $2p\sigma$ states of the hydrogen molecule ionfor various values of the internuclear distance. The accuracy of the formulasobtained is illustrated by comparison with available numerically exact results.