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##### Density Functional Theory of Superconductivity

**Authors**: M.A.L. Marques

**Ref.**: Université Pierre et Marie Curie, Paris VI, France, 29/10/2004 (2004)

**Abstract**: A novel 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 Schrödinger 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.

The theory is then applied to the calculation of superconducting properties
of several elemental metals, MgB_{2}, and Li and Al under pressure. In
particular, we present results for the transition temperature, for the gap
at zero temperature, and for thermodynamic properties like the specific
heat. We obtain an unprecedented agreement with experimental results.
Superconductors both with strong and weak electron-phonon coupling are
equally well described.

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