  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.
