Nonperturbative approach to non-condon effects: Must a nonadiabatic transition always occur at the potential surface crossing?

We present a generating function method for the computation of non-Condon effects in the context of nonadiabatic multiphonon transition theory. The method is nonperturbative in the electron-phonon coupling and nonperturbative in the dependence of the transition matrix element on nuclear coordinates, but it does assume the overall transition rate is small enough to be treated perturbatively. We do not assume the transition operator is given by the Born-Oppenheimer breakdown operator but rather consider more general forms of the transition matrix element V(x), where x represents nuclear coordinates. In particular, for narrow forms of V(x) such as a delta function, the optimum position (that position of V(x) which produces the maximal transition rate) is temperature dependent and may not coincide with the transition state. This calls into question the classical notion that a transition occurs via a thermal fluctuation to the potential surface crossing. The classical significance of the transition state does appear valid, however, at low temperature and large electron-phonon coupling. Furthermore, when V(x) is broad and the non-Condon effects are weak, the transition rate can be expressed in the Condon approximation with suitably scaled parameters. Since this scaling is temperature dependent, there may be an apparent change in the reorganization energy and in the enthalpy and entropy of activation. When many vibrational modes couple to the transition, there are circumstances under which a single promoting mode can dominate the vibrational effects. In the context of optical line shapes, this means there are circumstances under which a weakly coupled vibrational mode can cause the wings of a line to assume a strictly exponential, rather than Gaussian, shape.