This paper examines local anharmonic vibrations in molecules using an analysis that starts with an ab initio potential energy surface, fits a model potential constructed of Gaussian basis functions, and proceeds to a quantum mechanical analysis of the anharmonic modes using Cartesian harmonic oscillator basis functions in a variational calculation. The objective of this work is to suggest methods, with origins in nuclear and molecular (electronic) cuantum mechanics, that should be useful for the accurate analysis of the local anharmonic motions of hydrogen, and perhaps other atoms or small molecular fragments, residing in molecularly complicated but otherwise harmonic environments.