• Band structure;
  • zero-point motion;
  • electron-phonon coupling;
  • first-principle calculations.


The modifications of the electronic band structure of solids due to electron-phonon interactions (temperature and zero-point motion effects) have been explored by Manuel Cardona from both the experimental and theoretical sides. In the present contribution, we focus on the theoretical approaches to such effects. Although the situation has improved since the seventies, the wish for a fully developed theory (and associated efficient implementations) is not yet fulfilled. We review noticeable semi-empirical and first-principle studies, with a special emphasis on the Allen-Heine-Cardona (AHC) approach. We then focus on the non-diagonal Debye-Waller contribution, appearing beyond the rigid-ion approximation, in a Density-Functional Theory (DFT) approach. A numerical study shows that they can be sizeable (10%–50%) for diatomic molecules. We also present the basic idea of a new formalism, based on Density-Functional Perturbation Theory, that allows one to avoid the sums over a large number of empty states, and speed up the calculation by one order of magnitude, compared to the straightforward implementation of the AHC approach within DFT.