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The adiabatic approximation and perturbation theory to second order in atomic displacement are used to construct a theory of thermal-electron energy shifts. Both intraband and interband 'self-energy' and 'Debye-Waller' corrections are included. A sum rule based on translational invariance clarified the intimate relation between these two types of correction. Various kinds of cancellation between the terms can be seen explicitly for the two limiting cases of a very narrow band and a small gap in a nearly-free-electron band structure.
Allen et al. (Mon,) studied this question.