<i>Ab initio</i>theory of superconductivity. I. Density functional formalism and approximate functionals

Abstract

An approach to the description of superconductors in thermal equilibrium is developed within a formally exact density functional framework. The theory is formulated in terms of three ``densities:'' the ordinary electron density, the superconducting order parameter, and the diagonal of the nuclear $N$-body density matrix. The electron density and the order parameter are determined by Kohn-Sham equations that resemble the Bogoliubov--de Gennes equations. The nuclear density matrix follows from a Schr\"odinger equation with an effective $N$-body interaction. These equations are coupled to each other via exchange-correlation potentials which are universal functionals of the three densities. Approximations of these exchange-correlation functionals are derived using the diagrammatic techniques of many-body perturbation theory. The bare Coulomb repulsion between the electrons and the electron-phonon interaction enter this perturbative treatment on the same footing. In this way, a truly ab initio description is achieved which does not contain any empirical parameters.