A Mathematical Theory for Vibrational Levels Associated with Hydrogen Bonds I: The Symmetric Case

We propose an alternative to the usual time--independent Born--Oppenheimer approximation that is specifically designed to describe molecules with 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. Consequently, anharmonic effects play a role in the leading order calculations of vibrational levels. Although we develop a general theory, our analysis is motivated by an examination of symmetric bihalide ions, such as FHF- or ClHCl-. We describe our approach for the FHF- ion in detail.