Numerical methods for the inverse problem of density functional theory

Abstract

The inverse problem of Kohn-Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to onedimensional finite and periodic model systems. K E Y W O R D Sdensity functional theory, inverse problems, PDE-constrained optimization | I N TR ODU C TI ONDensity functional theory (DFT) is a very popular and formally exact quantum many-body theory that has successfully been used to model both finite and periodic systems. [1,2] Much of this success stems from the availability of approximate exchange-correlation functionals used as input in the direct problem of DFT to produce the ground-state electronic density of a given system. Although far less common, this process can be reversed and the true exchange-correlation potential can be recovered from a given electronic density by solving the inverse problem of Kohn-Sham (KS) DFT. (Other flavors of DFT exist but we focus on the standard ground-state version of KS DFT and drop the KS prefix for brevity throughout this tutorial.) Although both the forward and inverse problems use the same equations, the difference in input and output variables leads to major differences in solution methods for the two problems. We refer to the inverse problem of DFT as a density-to-potential inversion and the numerical methods required for solving this problem are the main focus of this tutorial. In this section, we give a brief introduction to inverse problems and explain how density-to-potential inversions can be used within DFT.Inverse problems are common in science and have been central in quantum mechanics since its inception. Much of what we know about the structure of matter has come from scattering experiments, [3] which can be described mathematically as inverse problems. A variety of mathematical and numerical techniques exist for solving scattering problems as well as other inverse problems.[4] These methods are often very different from the methods used for solving the direct problems due to the differences in mathematical structure and input data.According to Hadamard, [5] a problem is well-posed if a solution exists, it is unique, and it depends continuously on the data. If any of the three properties listed above is violated then the problem is ill-posed. The forward or direct problem of DFT is widely considered to be well-posed thanks to several proofs including the Hohenberg-Kohn Theorems and the constrained-search formulation of Levy and Lieb. [1] The inverse problem of DFT can also be well-posed for dis...

First step in the nuclear inverse Kohn-Sham problem: From densities to potentials

“…The CV method has been implemented following some important modifications suggested in Ref. [19]. First, a new set of variables, viz.…”

Inverting the external‐potential‐to‐electron density map in systems of electrons by minimization of a least‐squares fit functional for analytic potentials

“…Jensen and Wasserman have applied the methods of inverse problem theory to DFT. Their approach is based of the minimization of the cost functional , a least‐squares functional defined as where ρ v is the density corresponding to the external potential v , and w is a positive definite weighting function.…”

“…Conceptually, one of the simplest approaches would be the minimization of the least‐squares functional by gradient methods, where ρ t is the target density and ρ v is the density corresponding to a trial potential v . Least‐squares minimization via gradient methods is often used for inverse problems, but only recently it has been applied to the inversion of the potential‐to‐density map. In their work, Jensen and Wasserman apply existing methods of inverse problem theory to DFT, and describe a grid‐based implementation where densities and potentials are represented as discrete points on a grid.…”

“…Least‐squares minimization via gradient methods is often used for inverse problems, but only recently it has been applied to the inversion of the potential‐to‐density map. In their work, Jensen and Wasserman apply existing methods of inverse problem theory to DFT, and describe a grid‐based implementation where densities and potentials are represented as discrete points on a grid. The main goal of this article is to extend Jensen and Wasserman's work to the case where the densities and potentials have an analytical form that allows the exact (within machine accuracy) calculation of all the required integrals.…”

Constructing a non-additive non-interacting kinetic energy functional approximation for covalent bonds from exact conditions

“…This approach typically involves computationally expensive inversion methods. [7][8][9][10][11][12] The NAKE can also be obtained much more efficiently via explicit functional approximations. There are two categories of NAKE approximations: decomposable and nondecomposable approximations 13 .…”