A Mathematical Theory for Vibrational Levels Associated with Hydrogen Bonds II: The Non--Symmetric Case

We propose an alternative to the usual time--independent Born--Oppenheimer approximation that is specifically designed to describe molecules with non--symmetrical hydrogen bonds. In our approach, the masses of the hydrogen nuclei are scaled differently from those of the heavier nuclei, and we employ a specialized form for the electron energy level surface. As a result, the different vibrational modes appear at different orders of approximation. Although we develop a general theory, our analysis is motivated by an examination of the F H Cl- ion. We describe our results for it in detail. We prove the existence of quasimodes and quasienergies for the nuclear vibrational and rotational motion to arbitrary order in the Born--Oppenheimer parameter epsilon. When the electronic motion is also included, we provide simple formulas for the quasienergies up to order epsilon cubed that compare well with experiment and numerical results.