Density functional theory provides a powerful tool for computations of the quantum state of atoms, molecules and solids, and of ab-initio molecular dynamics. It was conceived in its initial naïve and approximative version by Thomas and Fermi immediately after the foundation of quantum mechanics, in 1927. Just thirty years ago, in the middle of the sixties, Hohenberg, Kohn and Sham on the one hand established a logically rigorous density functional theory of the quantum groundstate on the basis of quantum mechanics, and on the other hand, guided by this construction, introduced an approximative explicit theory called the local-density approximation, which for computations of the quantum groundstate of many-particle systems proved to be superior to both Thomas-Fermi and Hartree-Fock theories. From that time on, density functional theory has grown vastly in popularity, and a flood of computational work in molecular and solid state physics has been the result. Motivated by its success, there has been always a tendency to widen the fields of application of density functional theory, and in these developments, some points which were left somewhat obscure in the basic theory, were brought into focus from time to time. This led in the early eighties to a deepening of the logical basis, essentially by Levy and Lieb, and finally Lieb gave the basic theory a form of final mathematical rigour. Since that treatment, however, is based on the tools of modern convex functional analysis, its implications only gradually became known to the many people who apply density functional theory. A thorough treatment of the dependence on particle number on the basis of Lieb’s theory is given for the first time in the present text.
Given a vector space of microscopic quantum observables, density functional theory is formulated on its dual space. A generalized Hohenberg-Kohn theorem and the existence of the universal energy functional in the dual space are proven. In this context ordinary density functional theory corresponds to the space of one-body multiplication operators. When the operators close under commutation to form a Lie algebra, the energy functional defines a Hamiltonian dynamical system on the coadjoint orbits in the algebra's dual space. The enhanced density functional theory provides a new method for deriving the group theoretic Hamiltonian on the coadjoint orbits from the exact microscopic Hamiltonian.
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function---the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics---extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space---putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of ``parity'' and ``displacement'' in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.
Exact density-functional theory is recast here as two parallel exact ensemble variational hierarchies: an interacting hierarchy rooted in Lieb's ensemble formulation and a noninteracting hierarchy rooted in exact noninteracting ensemble theory. In optimization terms, N-representability is primal feasibility, Legendre-Fenchel duality equates the primal and dual values, v-representability is dual attainment, and the Hohenberg-Kohn theorem gives uniqueness, modulo constants, of an attained local potential. The Kohn-Sham construction couples the interacting density-space optimality condition to a compatible noninteracting dual realization on their common N-representable density domain. State-class restrictions yield the Levy-Lieb and single-determinant branches, while fractional particle number and fractional occupations lead naturally to piecewise linearity, one-sided chemical potentials, Janak-type relations, and the derivative discontinuity. This organization locates the exactness of Kohn-Sham theory in the preservation of the interacting density-space optimization together with its compatible noninteracting realization, without implying a general many-body spectral interpretation of Kohn-Sham eigenvalues.