Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite Dimensions

The Structure of Density-Potential Mapping. Part I: Standard Density-Functional Theory

The Hohenberg-Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just the one-body particle density. Part I of this review aims at clarifying the status of the Hohenberg-Kohn theorem within DFT and Part II at different extensions of the theory that include magnetic fields. We collect evidence that the Hohenberg-Kohn theorem does not so much form the basis of DFT, but is rather the consequence of a more comprehensive mathematical framework. Such results are especially useful when it comes to the construction of generalized DFTs.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Density-functional theory on graphs

The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void, in general, while many insights into the topological structure of the density-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional.

Ensemble Density Functional Theory of Neutral and Charged Excitations : Exact Formulations, Standard Approximations, and Open Questions

Recent progress in the field of (time-independent) ensemble density-functional theory (DFT) for excited states are reviewed. Both Gross-Oliveira-Kohn (GOK) and N-centered ensemble formalisms, which are mathematically very similar and allow for an in-principle-exact description of neutral and charged electronic excitations, respectively, are discussed. Key exact results, for example, the equivalence between the infamous derivative discontinuity problem and the description of weight dependencies in the ensemble exchange-correlation density functional, are highlighted. The variational evaluation of orbital-dependent ensemble Hartree-exchange (Hx) energies is discussed in detail. We show in passing that state-averaging individual exact Hx energies can lead to severe (although solvable) v-representability issues. Finally, we explore the possibility of using the concept of density-driven correlation, which has been introduced recently and does not exist in regular ground-state DFT, for improving state-of-the-art correlation density-functional approximations for ensembles. The present review reflects the efforts of a growing community to turn ensemble DFT into a rigorous and reliable low-cost computational method for excited states. We hope that, in the near future, this contribution will stimulate new formal and practical developments in the field.

Predictive Mixing for Density Functional Theory (and Other Fixed-Point Problems)

Density functional theory calculations use a significant fraction of current supercomputing time. The resources required scale with the problem size, the internal workings of the code, and the number of iterations to convergence, with the latter being controlled by what is called "mixing". This paper describes a new approach to handling trust regions within these and other fixed-point problems. Rather than adjusting the trust region based upon improvement, the prior steps are used to estimate what the parameters and trust regions should be, effectively estimating the optimal Polyak step from the prior history. Detailed results are shown for eight structures using both the "good" and "bad" multisecant versions as well as the Anderson method and a hybrid approach, all with the same predictive method. Additional comparisons are made for 36 cases with a fixed algorithm greed. The predictive method works well independent of which method is used for the candidate step, and it is capable of adapting to different problem types particularly when coupled with the hybrid approach.

Erratum: Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite Dimensions [Phys. Rev. Lett. 123, 037401 (2019)]

This corrects the article DOI: 10.1103/PhysRevLett.123.037401.

Unique continuation for the magnetic Schrödinger equation

The unique-continuation property from sets of positive measure is here proven for the many-body magnetic Schrödinger equation. This property guarantees that if a solution of the Schrödinger equation vanishes on a set of positive measure, then it is identically zero. We explicitly consider potentials written as sums of either one-body or two-body functions, typical for Hamiltonians in many-body quantum mechanics. As a special case, we are able to treat atomic and molecular Hamiltonians. The unique-continuation property plays an important role in density-functional theories, which underpins its relevance in quantum chemistry.

Nonadiabatic Electron Dynamics in Tunneling Junctions: Lattice Exchange-Correlation Potential

The search for exchange-correlation functionals going beyond the adiabatic approximation has always been a challenging task for time-dependent density-functional theory. Starting from known results and using symmetry properties, we put forward a nonadiabatic exchange-correlation functional for lattice models describing a generic transport setup. We show that this functional reduces to known results for a single quantum dot connected to one or two reservoirs and furthermore yields the adiabatic local-density approximation in the static limit. Finally, we analyze the features of the exchange-correlation potential and the physics it describes in a linear chain connected to two reservoirs where the transport is induced by a bias voltage applied to the reservoirs. We find that the Coulomb blockade is correctly described for a half-filled chain, while additional effects arise as the doping of the chain changes.

Self-Consistent Density-Functional Embedding: A Novel Approach for Density-Functional Approximations

In the present work, we introduce a self-consistent density-functional embedding technique, which leaves the realm of standard energy-functional approaches in density functional theory and targets directly the density-to-potential mapping that lies at its heart. Inspired by the density matrix embedding theory, we project the full system onto a set of small interacting fragments that can be solved accurately. Based on the rigorous relation of density and potential in density functional theory, we then invert the fragment densities to local potentials. Combining these results in a continuous manner provides an update for the Kohn-Sham potential of the full system, which is then used to update the projection. We benchmark our approach for molecular bond stretching in one and two dimensions and show that, in these cases, the scheme converges to accurate approximations for densities and Kohn-Sham potentials. We demonstrate that the known steps and peaks of the exact exchange-correlation potential are reproduced by our method with remarkable accuracy.