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 2. When the electronic motion is also included, we provide simple formulas for the quasienergies up to order 23 that compare well with experiment and numerical results. ∗Partially Supported by National Science Foundation Grant DMS–0600944.