# Introduction and Overview of the Reduced Density Matrix Functional Theory

Reduced density matrix functional theory (RDMFT) is a theoretical framework for approximating the many-electron problem. In RDMFT, the fundamental quantity is the one-body, reduced, density matrix (1RDM) which plays the same role as the electronic density in density functional theory. Gilberts theorem stands in the foundations of RMDFT, and guarantees that every observable for the ground state is a functional of the 1RDM. This allows for approximating the total energy in terms of the 1RDM and minimizing it under certain conditions for the N-representability of the 1RDM. So far, in almost all practical applications Coleman’s ensemble N-representability conditions are employed which are very simple for fermionic systems. They concern the eigenvalues of the 1RDM, known as natural occupations, restricting then in the range between zero and one and their sum which is fixed to be the total number of electrons.

A certain advantage of tackling the many electron problem in this way is that the kinetic energy of the system is a simple expression in terms of the 1RDM, i.e. there is no need for a fictitious non interacting system like the Kohn-Sham system in DFT. Thus, fractional occupations enter the theory in a natural way allowing to construct simple approximations that describe accurately electronic correlations. A central and simple functional in RDMFT is the Mler functional, a relatively simple modification of the expression of the total energy in Hartree-Fock theory. This functional was shown to reproduce the correct physical picture of the dissociation of the Hydrogen molecule, although it is known to overestimate substantially the correlation energy. Several approximations were introduced in the last couple of decades, many of which are corrections to the Mller functional, and were proven to describe accurately such diverse effects and quantities like static correlations and the band gaps of materials. Unfortunately, due to the non-existence of a non-interacting systems, RDMFT calculations are demanding compared to DFT and, at present, are restricted to small molecules or simple periodic systems.