We describe an approach for calculations of phonon contributions to the electron spectral function, including both quasiparticle properties and satellites. The method is based on a cumulant expansion for the retarded one-electron Greens function and a many-pole model for the electron self-energy. The electron-phonon couplings are calculated from the Eliashberg functions, and the phonon density of states is obtained from a Lanczos representation of the phonon Greens function. Our calculations incorporate ab initio dynamical matrices and electron-phonon couplings from the density functional theory code ABINIT. Illustrative results are presented for several elemental metals and for Einstein and Debye models with a range of coupling constants. These are compared with experiment and other theoretical models. Estimates of corrections to Migdals theorem are obtained by comparing with leading order contributions to the self-energy, and are found to be significant only for large electron-phonon couplings at low temperatures.
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