Guaranteed convergence of the Kohn-Sham equations

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.

Kohn-Sham Equations as Regularizer: Building Prior Knowledge into Machine-Learned Physics

Including prior knowledge is important for effective machine learning models in physics and is usually achieved by explicitly adding loss terms or constraints on model architectures. Prior knowledge embedded in the physics computation itself rarely draws attention. We show that solving the Kohn-Sham equations when training neural networks for the exchange-correlation functional provides an implicit regularization that greatly improves generalization. Two separations suffice for learning the entire one-dimensional H_{2} dissociation curve within chemical accuracy, including the strongly correlated region. Our models also generalize to unseen types of molecules and overcome self-interaction error.

Confirmation of the PPLB Derivative Discontinuity: Exact Chemical Potential at Finite Temperatures of a Model System

The landmark 1982 work of Perdew, Parr, Levy, and Balduz (often called PPLB) laid the foundation for our modern understanding of the role of the derivative discontinuity in density functional theory, which drives much development to account for its effects. A simple model for the chemical potential at vanishing temperature played a crucial role in their argument. We investigate the validity of this model in the simplest nontrivial system to which it can be applied and which can be easily solved exactly, the Hubbard dimer. We find exact agreement in the crucial zero-temperature limit and show the model remains accurate for a significant range of temperatures. We identify how this range depends on the strength of correlations. We extend the model to approximate free energies accounting for the derivative discontinuity, a feature missing in standard semilocal approximations. We provide a correction to this approximation to yield even more accurate free energies. We discuss the relevance of these results for warm dense matter.

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

The exact Kohn-Sham iteration of generalized density-functional theory in finite dimensions with a Moreau-Yosida regularized universal Lieb functional and an adaptive damping step is shown to converge to the correct ground-state density.

Quantifying Density Errors in DFT

We argue that any general mathematical measure of density error, no matter how reasonable, is too arbitrary to be of universal use. However, the energy functional itself provides a universal relevant measure of density errors. For the self-consistent density of any Kohn-Sham calculation with an approximate functional, the theory of density-corrected density functional theory (DC-DFT) provides an accurate, practical estimate of this ideal measure. We show how to estimate the significance of the density-driven error even when exact densities are unavailable. In cases with large density errors, the amount of exchange-mixing is often adjusted, but we show that this is unnecessary. Many chemically relevant examples are given.

Exact Density Functional Obtained via the Levy Constrained Search

A stochastic minimization method for a real-space wave function, Ψ(r₁, r₂...r _n), constrained to a chosen density, ρ(r), is developed. It enables the explicit calculation of the Levy constrained search, F[ρ] = min_Ψ→ρ⟨Ψ| T̂ + V̂ _ee|Ψ⟩, which gives the exact functional of density functional theory. This general method is illustrated in the evaluation of F[ρ] for densities in one dimension with a soft-Coulomb interaction. Additionally, procedures are given to determine the first and second functional derivatives, δ F/δρ(r) and δ² F/[δρ(r)δρ(r')]. For a chosen external potential, v(r), the functional and its derivatives are used in minimizations over densities to give the exact energy, E _v, without needing to solve the Schrödinger equation.

Spin-Charge Separation in Finite Length Metallic Carbon Nanotubes

Using time-dependent density functional theory, we study the optical excitations in finite length carbon nanotubes. Evidence of spin-charge separation is given in the spacetime domain. We demonstrate that the charge density wave is due to collective excitations of electron singlets, while the accompanying spin density wave is due to those of electron triplets. The Tomonaga-Luttinger liquid parameter and density-density interaction are extrapolated from the first-principles excitation energies. We show that the density-density interaction increases with the length of the nanotube. The singlet and triplet excitation energies, on the other hand, decrease for increasing length of the nanotube. Their ratio is used to establish a first-principles approach for deriving the Tomonaga-Luttinger parameter (in excellent agreement with experimental data). Time evolution analysis of the charge and spin line densities evidences that the charge and spin density waves are elementary excitations of metallic carbon nanotubes. Their dynamics show no dependence on each other.

The Hubbard dimer: a density functional case study of a many-body problem

This review explains the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site Hubbard model. The relationship to traditional quantum chemistry is included. Even in this elementary example, where the exact ground-state energy and site occupations can be found analytically, there is much to be explained in terms of the underlying logic and aims of density functional theory. Although the usual solution is analytic, the density functional is given only implicitly. We overcome this difficulty using the Levy-Lieb construction to create a parametrization of the exact function with negligible errors. The symmetric case is most commonly studied, but we find a rich variation in behavior by including asymmetry, as strong correlation physics vies with charge-transfer effects. We explore the behavior of the gap and the many-body Green's function, demonstrating the 'failure' of the Kohn-Sham (KS) method to reproduce the fundamental gap. We perform benchmark calculations of the occupation and components of the KS potentials, the correlation kinetic energies, and the adiabatic connection. We test several approximate functionals (restricted and unrestricted Hartree-Fock and Bethe ansatz local density approximation) to show their successes and limitations. We also discuss and illustrate the concept of the derivative discontinuity. Useful appendices include analytic expressions for density functional energy components, several limits of the exact functional (weak- and strong-coupling, symmetric and asymmetric), various adiabatic connection results, proofs of exact conditions for this model, and the origin of the Hubbard model from a minimal basis model for stretched H2.

Ground-state energy as a simple sum of orbital energies in Kohn-Sham theory: a shift in perspective through a shift in potential

It is observed that the exact interacting ground-state electronic energy of interest may be obtained directly, in principle, as a simple sum of orbital energies when a universal density-dependent term is added to w([ρ];r), the familiar Hartree plus exchange-correlation component in the Kohn-Sham effective potential. The resultant shifted potential, w[over ¯]([ρ];r), actually changes less on average than w([ρ];r) when the density changes, including the fact that w[over ¯]([ρ];r) does not undergo a discontinuity when the number of electrons increases through an integer. Thus, the approximation of w[over ¯]([ρ];r) represents an alternative direct approach for the approximation of the ground-state energy and density.