A Quantitative Explanation of the Dynamics Underlying the Franck-Condon Principle: A Mostly Classical Viewpoint

摘要

The Franck-Condon Principle (FCP), the key concept for describing vibronic spectra, results from the large disparity between the time scales of electrons and nuclei in molecules. Yet in an undergraduate lecture, the FCP is nearly always rationalized from a time independent quantum mechanical viewpoint, due to students' lack of knowledge of time dependent quantum theory. Here we present a quantitative way to describe the essence of the FCP from a mostly classical, time dependent approach. On the short time scales relevant to the FCP, the phase space centers of time evolving wavepackets follow their classical counterparts; thus, it is not unreasonable to describe the FCP using classical/semiclassical mechanics. In this work, we show how the dynamical interplay between the FCP and the Born-Oppenheimer approximation (BOA) leads to the observed vibronic spectrum of a molecule. A student who has learned about quantum operators and harmonic oscillators, and the BOA, can understand nearly all the theory presented herein. We provide, as a supplement, an Excel Workbook and set of Suggested Exercises. These Exercises, along with the Workbook, can be used to lead a student through nearly all the equations presented in this paper, and can thus augment the typical Quantum Chemistry undergraduate vibronic spectroscopy laboratory experiment or be used as an assignment in an undergraduate course in Quantum Chemistry.