Partition density-functional theory is a formally exact method for solving the ground-state electronic structure problem. It operates by dividing the system into noninteracting fragments along with a global effective potential, which, when solved to self-consistency, yields the exact energy and density (as the sum of the fragment densities) of the interacting system. A recent paper [Phys. Rev. A 86, 012504 (2012)] questions the uniqueness of the fragment occupations and fragment densities. We demonstrate that their analysis is incorrect. We describe some of the technical details of partition density-functional calculations and discuss the relationship to other embedding schemes.
Partition density-functional theory is a formally exact method for solving the ground-state electronic structure problem. It operates by dividing the system into noninteracting fragments along with a global effective potential, which, when solved to self-consistency, yields the exact energy and density (as the sum of the fragment densities) of the interacting system. A recent paper [Phys. Rev. A 86, 012504 (2012)] questions the uniqueness of the fragment occupations and fragment densities. We demonstrate that their analysis is incorrect. We describe some of the technical details of partition density-functional calculations and discuss the relationship to other embedding schemes.
Recently we proposed an information entropy based method for electronic structure calculations within the density-matrix functional theory(DMFT) (Phys. Rev. Lett. 128, 013001), dubbed as i-DMFT. Comments have been raised regarding the accuracy of the one-particle density-matrix compared to that from the wave function, and the universality of the functional. We address these questions and the problematic case of HeH+.
Quantum embedding is a divide and conquer strategy that aims at solving the electronic Schrödinger equation of sizeable molecules or extended systems. We establish in the present work a clearer and in-principle-exact connection between density matrix embedding theory (DMET) and density-functional theory (DFT) within the simple but nontrivial one-dimensional Hubbard model. For that purpose, we use our recent reformulation of single-impurity DMET as a Householder transformed density-matrix functional embedding theory (Ht-DMFET). On the basis of well-identified density-functional approximations, a self-consistent local potential functional embedding theory (LPFET) is formulated and implemented. Combining both LPFET and DMET numerical results with our formally exact density-functional embedding theory reveals that a single statically embedded impurity can in principle describe the density-driven Mott–Hubbard transition, provided that a complementary density-functional correlation potential (which is neglected in both DMET and LPFET) exhibits a derivative discontinuity (DD) at half filling. The extension of LPFET to multiple impurities (which would enable to circumvent the modeling of DDs) and its generalization to quantum chemical Hamiltonians are left for future work.
In [Phys. Rev. Lett. 128, 013001 (2022)] a novel ground state method was proposed. It has been suggested that this i-DMFT would be a method within one-particle reduced density matrix functional theory (DMFT), capable of describing accurately molecules at various geometries with an information-theoretical nature. We reassess this work and its suggestions from a conceptual and practical point of view, leading to the following conclusions: i) A method which assigns to each molecule M its own functional FM is not a functional theory (striking violation of "universality") ii) even for the simplest systems i-DMFT yields incorrect one-particle reduced density matrices and iii) the use of an information-theoretical concept to describe molecular dissociation limits was not essential. The latter insight may help to fix the deficiency of i-DMFT to not reproduce correctly the smaller occupation numbers and thus to not recover the important dynamic correlations.
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.
The formalism of Kohn and Sham uses a specific (model) hamiltonian which highly simplifies the many-electron problem to that of noninteracting fermions. The theorem of Hohenberg and Kohn tells us that, for a given ground state density, this hamiltonian is unique. In principle, this density can be chosen as that of the real, interacting system. To obtain the energy, or other properties of the real system, approximations are needed. Working with non interacting fermions is an important simplification, but it may be easier to produce approximations with different choices of the model hamiltonian. The feature that the exact density is (ideally) reproduced can be kept in the newly defined fictitious systems. Using model hamiltonians having the same form as the physical one, that is, being built of one- and two-body operators, allows to approach the physical hamiltonian arbitrarily close, and thus a systematic reduction of the approximations.
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.