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
  The contour integrals, occurring in the arbitrary-order phase-integral quantization conditions given in a previous paper, are in the first- and third-order approximations expressed in terms of complete elliptic integrals in the case that the charges of the Coulomb centres are equal. The evaluation of the integrals is facilitated by the knowledge of quasiclassical dynamics. The resulting quantization conditions involving complete elliptic integrals are solved numerically to obtain the energy eigenvalues and the separation constants of the $1s\sigma$ and $2p\sigma$ states of the hydrogen molecule ion for various values of the internuclear distance. The accuracy of the formulas obtained is illustrated by comparison with available numerically exact results. 
N. Athavan, N. Fr\"oman and M. Lakshmanan