Quantization in molecular potentials

We present a modified Bohr-Sommerfeld quantization rule for calculating energy levels in a molecular potential; it is based on the appropriate consideration of the reflection phase at the outer turning point. When the attractive tail of the potential is given by an inverse power law with a power greater than two, the reflection phase is accurately approximated by a simple algebraic formula. In an example defined by a 12-6 Lennard-Jones potential, this simple modified quantization rule gives highly accurate energy levels in the whole spectrum; the error relative to the level spacing is uniformly near 2 × 10^-4. Very close to threshold, where the conventional WKB expansion breaks down completely, our simple formula can be supplemented by an accurate 'effective-range' expansion. This yields energies for the most weakly bound level with residual errors which are smaller than the level spacing to the second most weakly bound level by seven powers of ten.