In 1959, at the Colorado conference on Molecular Quantum Mechanics, Charles Coulson pointed out that the description of atomic and molecular ground states involves the two-electron reduced density matrix (2RDM), only [1,2]. Indeed, since electrons interact only by pairwise interactions, the energies and other electronic properties of atoms and molecules can be computed directly from the 2RDM. This crucial insight has defined the starting point for the development of new theoretical approaches to the ground state problem, avoiding the use of the exponentially complex N-electron wave function. Since each subfields of the Quantum Sciences typically restricts to systems all characterized by a fixed pair interaction V (for instance Coulomb interaction in quantum chemistry, contact interaction in quantum optics and Hubbard interaction in solid state physics) the ground state problem should de facto involve only the one-particle reduced density matrix (1RDM). Indeed, for Hamiltonians of the form H(h) = h + V, where h represents the one-particle terms and V the fixed pair interaction, the conjugate variable to H(h) and h, respectively, is the 1RDM. The corresponding exact one-particle theory is known as Reduced Density Matrix Functional Theory (RDMFT) which is based on the existence of an exact energy functional of the 1RDM [3]. This interaction functional is universal in the sense that it depends only on the fixed interaction V but not on the one-particle terms h [3,4].
Most notably, compared to the widely used Density Functional Theory (DFT), RDMFT has some significant conceptual advantage and is therefore expected to overcome at some point the fundamental limitations of DFT. On the one hand, the kinetic energy is described in an exact way due to the access of the full 1RDM and all the effort can be spent to derive good approximations to the interaction energy. On the other hand, RDMFT allows explicitly for fractional occupation numbers and has therefore great prospects of describing systems with strong correlations, particularly static correlations [5]. For instance, benchmark calculations revealed that the common functionals in RDMFT yield correlation energies for closed shell atoms and molecules which are by one order of magnitude more accurate than B3LYP (one of the most popular density functionals in quantum chemistry) and a precision comparable to Møller-Plesset second-order perturbation theory [5-8]. RDMFT has also succeeded in predicting more accurate gaps of conventional semiconductors than semilocal DFT and demonstrated the insulating behavior of Mott-type insulators [9-11]. At the same time, involving the full 1RDM lies, however, also at the heart of possible disadvantages of RDMFT relative to DFT. The 1RDM involves quadratically more degrees of freedom than the spatial density and enforcing the orthonormality of the natural orbitals is computationally highly demanding [5]. Furthermore, for open-shell systems, the domain of the universal functional seems to be constrained by the more involved generalized Pauli constraints. All these aspects are still hampering RDMFT from reaching its full potential and effectiveness.
Significant recent progress on reduced density matrices and the theory of fermion correlation provides new ways for overcoming the problems of the recent realization of RDMFT. For instance, it has been observed that for many solid materials, one can exploit either the spatial locality or the translational symmetry in an efficient way, leading to more accurate functionals and remarkable additional insights [12-14]. To be more specific, the latter de facto simplifies RDMFT to a natural occupation number functional theory and the natural orbitals are fixed from the very beginning. Moreover, it has been shown that the one- and two-body N-representability constraints strongly shape the exact functional [15,16]. The generalized Pauli constraints manifest themselves within RDMFT in the form of an exchange force which repulsively diverges on the boundary of the allowed region of possible 1RDMs [16]. Moreover, promising ideas have been suggested for improving the accuracy of approximate functionals by taking the most significant two-body N-representability constraints into account [15]. It will be one of the crucial challenging for the next few years to combine all those new ideas to improve the foundation of RDMFT, overcome its recent limitations and to extent its scope to including also non-singlet, finite-temperature, and time-evolving systems [17,18].