From the very beginning quantum statistical mechanics can be formulated in terms of a thermodynamic potential, which has the electron density as the basic variable. By employing a well-known procedure in quantum statistical mechanics for the construction of stationarity principles [T. D. Lee and C. N. Yang, Phys. Rev. 113, 1165 (1959); 117, 22 (1960)], the variational principles of Hohenberg, Kohn [P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964)], and Mermin [N. D. Mermin, Phys. Rev. A 137, 1441 (1965)] are obtained as are generalizations thereof.
From the very beginning quantum statistical mechanics can be formulated in terms of a thermodynamic potential, which has the electron density as the basic variable. By employing a well-known procedure in quantum statistical mechanics for the construction of stationarity principles [T. D. Lee and C. N. Yang, Phys. Rev. 113, 1165 (1959); 117, 22 (1960)], the variational principles of Hohenberg, Kohn [P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964)], and Mermin [N. D. Mermin, Phys. Rev. A 137, 1441 (1965)] are obtained as are generalizations thereof.
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.
Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg, Kohn, and Sham [4,5] in the mid-1960s. According to this theory, the fundamental physical information about a many-body system is provided by single-particle densities in a three-dimensional space, which are obtained variationally within a time-independent framework. When compared with other previous formalisms, DFT presents two clear advantages: (1) it is able to treat many-body problems in a sufficiently accurate way and (2) it is computationally simple. This explains why it is one of the most widely used theories to deal with electronic structure-the electronic ground-state energy as a function of the position of the atomic nuclei determines molecular structures and solids, providing at the same time the forces acting on the atomic nuclei when they are not at their equilibrium positions. At present, DFT is used routinely to solve many problems in gas phase and condensed matter. Furthermore, it has made possible the development of accurate molecular dynamics schemes in which the forces are evaluated quantum mechanically ‘‘on the fly.’’ Nonetheless, DFT is a fundamental tool provided the systems studied are relatively large; for small systems, standard methods based on the use of the wave function render quite accurate results [6]. Moreover, it is also worth stressing that all practical applications of DFT rely on essentially uncontrolled approximations [7] (e.g., the local density approximation [4,5], the local spin-density approximation, or generalized gradient approximations [8]), and therefore the validity of DFT is conditioned to its capability to provide fairly good values of the experimental data.
For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density j(r) are sufficent to determine the one-body potential V(r) and vector potential A(r) is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional ${\mathcal{E}}_{{V}_{0},{\mathbf{A}}_{0}}(\ensuremath{\rho},\mathbf{j})=\ensuremath{\langle}\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j}),H({V}_{0},{\mathbf{A}}_{0})\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j})\ensuremath{\rangle}$ can be minimal for densities that are not the ground-state densities of the fixed potentials V0 and A0. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and j, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.
The study of quantum statistical mechanics begins with a review of the basic principles of quantum mechanics. Schrödinger’s equation is introduced and Eigenstates (or stationary states) are defined. Model probability for quantum statistics is assumed to have a uniform distribution in phases. Wave functions for many-body systems are defined. The density matrix is introduced. The Planck entropy and the microcanonical ensemble are defined. The differences between classical and quantum statistical mechanics are all based on the differing concepts of a microscopic ‘state’. While the classical microscopic state (specified by a point in phase space) determines the exact position and momentum of every particle, the quantum mechanical state determines neither; quantum states can only provide probability distributions for observable quantities.
Abstract An “apparatus” operator approach is presented for the extension of the density functional theory of Hohenberg, Kohn, and Mermin. Using the Mermin entropy principle, a one‐to‐one correspondence is established between the density matrix for the system and the electron charge density for a finite‐temperature system in the presence of an apparatus. In the zero‐temperature limit in the absence of an apparatus, the Hohenberg–Kohn theory is recovered. The central aspect of this new density functional theory is that the principle of maximum entropy is applied to the system plus its surroundings under the additional constraint that the electron charge density is given. The system is treated as a subsystem of a composite system and is not necessarily in the equilibrium state as in the Mermin theory. As an example, it is shown how, in principle, excited states are encompassed by the theory.
Given a general, N-particle Hamiltonian operator, analogs of the Hohenberg-Kohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Lieb's Legendre transform ansatz to the generalized Hohenberg-Kohn functional not only solves the upsilon-representability problem for these entities, but, more importantly, also solves the N-representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional. If all the operators of interest are incorporated in the restricted maximization, however, the variational principle dictates that exact results are obtained for the systems of interest. This might have important implications for practical work not only for density matrices but also for density functionals. A follow-up paper will present a useful alternative approach to the upsilon- and N-representability problems based on the constrained search formalism.