The relation between various coupling schemes for calculating nonradiative multiphonon transition rates

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Abstract
In an earlier paper (see ibid., vol.15, p.L381, 1982) the author gave a simple proof that the static, adiabatic and intermediate coupling schemes give the same nonradiative multiphonon transition rates to leading order in the off-diagonal part of the electron-phonon interaction. In that proof, the time dependence in the interaction picture of the matrices describing the transformation from one representation to another was neglected using a heuristic argument. In this paper it is shown explicitly that this time dependence can be neglected, hence firmly establishing the earlier result. The equivalence between the various coupling schemes is also established for the time rate of change of a general observable; for example the charge density. The present work is discussed in relation to the recent papers by Peuker et al. (1982) and by Wagner (1982). In particular it is shown that there is no contradiction between the present work and that due to Wagner. It is also shown that Wagner's alternative adiabatic base is none other than the base defined by Condon electronic wavefunctions and displaced lattice oscillators and that this result holds independently of the strength of the off-diagonal part of the electron-phonon interaction.