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We extend the two leading methods for the \emph{ab initio} computationaldescrip tion of phonon-mediated superconductors, namely Eliashberg theory anddensity fu nctional theory for superconductors (SCDFT), to include plasmoniceffects. Furth ermore, we introduce a hybrid formalism in which the Eliashbergapproximation fo r the electron-phonon coupling is combined with the SCDFTtreatment of the dynam ically screened Coulomb interaction. The methods havebeen tested on a set of we ll-known conventional superconductors by studyinghow the plasmon contribution a ffects the phononic mechanism in determining thecritical temperature (\tc). Our simulations show that plasmonic SCDFT leads toa good agreement between predict ed and measured \tc's, whereas Eliashbergtheory considerably overestimates the plasmon-mediated pairing and, therefore,\tc. The hybrid approach, on the other hand, gives results close to SCDFT andoverall in excellent agreement with exper iments.

Ab initio theory of plasmonic superconductivity within the Eliashberg and density-functional formalisms