The density functional theory for superconductors developed in the preceding article is applied to the calculation of superconducting properties of several elemental metals. 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. This demonstrates that, as far as conventional superconductivity is concerned, the first-principles prediction of superconducting properties is feasible.

Reduced density matrix functional theory for the case of solids is presented and an exchange-correlation functional based on a fractional power of the density matrix is introduced. We show that compared to other functionals, this produces more accurate behavior for total energies as a function of particle number for finite systems. Moreover, it captures the correct band-gap behavior for conventional semiconductors, as well as strongly correlated Mott insulators, where a gap is obtained in the absence of any magnetic ordering. DOI: 10.1103/PhysRevB.78.201103 PACS number͑s͒: 71.10.Ϫw, 71.20.Nr, 71.27.ϩa, 71.45.Gm One of the most dramatic failures of the usual localdensity approximation ͑LDA͒ or generalized-gradient-type approximations to the exchange-correlation ͑xc͒ functional of density-functional theory ͑DFT͒ is the incorrect prediction of a metallic ground state for the strongly correlated Mott insulators, of which transition-metal oxides ͑TMOs͒ may be considered as prototypical. For some TMOs ͑NiO and MnO͒ spin-polarized calculations do show a very small band gap ͑up to 95% smaller than experiments͒ but only as the result of antiferromagnetic (AFM) ordering; however, all TMOs are found to be metallic in a spin-unpolarized treatment. On the other hand, it is well known experimentally that these materials are insulating in nature even at elevated temperatures ͑much above the Néel temperature͒. 1 This indicates that the magnetic order is not the driving mechanism for the gap and is just a co-occurring phenomenon. A real challenge for any kind of ab initio theory then is the prediction of an insulating state for these strongly correlated materials in the absence of magnetic order. Until now the main focus of reduced density matrix functional theory ͑RDMFT͒ has been on finite systems such as atoms and molecules 2-11 with various xc functionals, [3][4][5][6][7][8][11][12][13][14][15] which are essentially modifications of the original Müller functional. 2 In the present work we extend RDMFT to the case of solid-state systems and introduce a functional which generates not only accurate gaps for conventional semiconductors, but demonstrates insulating behavior for Mott-type insulators in the nonmagnetic phase.Formally, the one-body reduced density matrix ␥ for a pure state of N electrons is defined as ͑spin degrees of freedom are omitted for simplicity͒ ␥͑r,rЈ͒ = N ͵ ⌿͑r,r 2 , ... ,r N ͒⌿ ‫ء‬ ͑rЈ,r 2 , ... ,r N ͒d 3 r 2 , ... ,d 3 r N . ͑1͒Diagonalization of this matrix produces a set of natural orbitals, 16 i , and occupation numbers, n i , leading to the spectral representationwhere the necessary and sufficient conditions for ensemble N representability 17 require 0 Յ n i Յ 1 for all i, and ͚ i n i = N. In terms of ␥, the total ground-state energy of the interacting system is 18 ͑atomic units are used throughout͒where ͑r͒ = ␥͑r , r͒, V is a given external potential, and E xc is what we call the xc energy functional. Minimizing the total energy with E xc ͓␥͔ =− 1 2 ͉͐␥͑r , rЈ͉͒ 2 / ͉r − rЈ͉d 3 rd 3 rЈ is equival...

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. In this way, a truly ab-initio description is achieved which does not contain any empirical parameters.