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

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