Radiationless decay of vibronically coupled electronic states

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Abstract
Radiationless transitions from an optically prepared state to the ground state are studied on a model consisting of three electronic states and two harmonic modes of vibration. The effect of the upper excited state on the nonradiative decay properties of the lower excited state is investigated for systems in which these states are coupled through the same non-totally-symmetric mode that couples the lower excited state to the ground state. If only this mode is considered, the model is exactly solvable and allows one to test the assumption that the initially prepared state is an adiabatic Born–Oppenheimer state. This assumption is found to be accurate unless the zeroth-order adiabatic vibrancy state from which the transition originates is very close to, e.g., within one vibrational quantum of, a zeroth-order state of the upper excited state manifold. Strong nonadiabatic mixing occurs when a vibrationally excited level of the lower excited state is in resonance with a level of the upper state. In general, the proximity of the two excited states increases the ability of the coupling mode to act as an energy accepting mode for radiationless decay to the ground state. This is shown by comparison with a totally symmetric, displaced oscillator in the adiabatic approximation. As a result vibronic coupling between two excited states may affect the energy gap law in that the roughly exponential decrease of the nonradiative rate constant with increasing energy gap is reduced or even reversed when the excited state approaches a second excited state to which it is vibronically coupled by the inducing mode. The model is also used to test, both analytically and numerically, the validity of approximate formulas for the calculation of matrix elements of the nuclear kinetic-energy operator. It is shown that the corresponding integrals are not normally separable into inducing and accepting mode integrals and that, when separation is possible, the accepting mode integrals are not simply overlap integrals. Treatments based on the Herzberg–Teller expansion and either Rayleigh–Schrödinger or Tanaka–Fukuda perturbation theory are shown to give rise to very large errors. These differences are traced back to differences in the diabatic basis sets used to expand adiabatic wavefunctions.