The Franck—Condon principle for radiationless transitions in molecules and a selection rule for morse oscillators

Abstract

The Franck—Condon (FC) principle for the tunnel radiationless transition (RT) is formulated. It reads that the RT occurs at constant values of the nuclear coordinates q* and of the classical momenta p*. However, unlike the optical transitions, q* and p* take non-physical values since the tunnel RT is a classically forbidden process. As a result of energy conservation, the potential surfaces of two given electronic states cross with one another at the nuclear configuration q*. It is concluded that the electronic-orbital selection rules and the numerical values of the purely electronic matrix elements are governed by the configuration q* rather than by the equilibrium nuclear configuration as was supposed previously. The configuration q* for the T 1 → S 0 intersystem crossing in aromatic hydrocarbons is described in terms of a large displacement of only one H atom from its equilibrium position along the CH bond (perhaps, there is also some out-of-plane displacement). Using the FC principle, it is found that the anharmonicity of the local CH bond vibrations results in a strong dependence of the T 1 → S 0 intersystem crossing rates upon the sign of the CH bond-length change between two given electronic states, Δ q = R T 1 - R S 0 . Namely, the following “selection rule” holds: these RTs are allowed at Δ q < 0 and they are forbidden at Δ q > 0, the prohibition factor being of the order of 10 2–10 4. Finally, an oscillatory dependence of the FC factors upon Δ q is explained, using the FC principle, in terms of quantum interference in the total transition probability, of which the amplitude is a sum of the transition amplitudes due to different crossing points q*. The interference effects are believed to be insignificant for RTs in molecules with large energy gaps and so they are eliminated in the usual manner by adding up the partial probabilities rather than the partial amplitudes. The classical FC factor thus obtained smoothly depends upon Δ q (and upon other parameters as well). This procedure also provides an analytical continuation of the FC factor to non-integral quantum numbers.