Role of Kinetic Energy in the Franck–Condon Principle

Abstract

During an electronic transition in a molecule the nuclear positions and momenta (hence kinetic energies) both tend to remain fixed according to the Franck principle. According to the quantum mechanical Franck–Condon principle, the most probable changes in vibrational quantum numbers or nuclear energy conform to the Franck principle. As usually implemented, it is assumed that the most probable nuclear‐energy transitions are approximately at frequencies corresponding to the $R$ values $\mathrm{(R\hspace{0.17em}=\hspace{0.17em}}\mathrm{internuclear\; distance})$ of classical turning points of the nuclear motion, where the classical kinetic energy is zero. It is shown here for some model diatomic cases that, in transitions to or from a bound state with enough vibrational energy, an equally strong maximum of transition probability as a function of frequency can occur classically corresponding to $R$ values where the nuclei in the initial state have maximum kinetic energy. Quantum mechanically, this means for example that in such cases, transitions to states of the final‐state continuum with considerable kinetic energy can have a relatively high probability if the potential curves of the two electronic states differ strongly in a suitable way. Formulas for the “fluctuation intervals” between diffraction maxima in the intensity structure of the continuum are derived. Applications to the observed emission spectrum of the iodine molecule are made in an accompanying paper.