Inverse Kohn–Sham Density Functional Theory: Progress and Challenges

Abstract

Inverse Kohn–Sham (iKS) methods are needed to fully understand the one-to-one mapping between densities and potentials on which density functional theory is based. They can contribute to the construction of empirical exchange–correlation functionals and to the development of techniques for density-based embedding. Unlike the forward Kohn–Sham problems, numerical iKS problems are ill-posed and can be unstable. We discuss some of the fundamental and practical difficulties of iKS problems with constrained-optimization methods on finite basis sets. Various factors that affect the performance are systematically compared and discussed, both analytically and numerically, with a focus on two of the most practical methods: the Wu–Yang method (WY) and the partial differential equation constrained optimization (PDE-CO). Our analysis of the WY and PDE-CO highlights the limitation of finite basis sets. We introduce new ideas to make iKS problems more tractable, provide an overall strategy for performing numerical density-to-potential inversions, and discuss challenges and future directions.

A Comparison of Exact and Model Exchange–Correlation Potentials for Molecules

“…Thus, development of XC functionals that provide satisfactory accuracy for weakly and strongly correlated systems still remains a serious challenge in DFT. To that end, the inverse DFT approach − of finding the XC potential ( v xc ( r )) corresponding to a given density (ρ data ( r )) provides an instructive tool in investigating the deficiencies of existing model XC functionals. To elaborate, a comparison of the XC potentials corresponding to ground-state densities from meta-GGA or hybrid XC-based GKS calculation against the exact XC potential for the ground-state density obtained from accurate many-body calculation (e.g., configuration interaction (CI)) can inform the features missed by the approximate XC functionals and, in turn, assist in designing next-generation functionals.…”

“…Given the importance of inverse DFT in assessing and developing XC functionals, over the last three decades, several approaches have been developed to solve the inverse DFT problem. 30 While analytical evaluation of the v xc from the density is feasible for one-or two-electron systems, 33,34 systems with more than two electrons warrant numerical evaluation. Broadly speaking, the numerical approaches to inverse DFT can be categorized as an iterative procedure 25,26,35−38 or a constrained optimization.…”

Massively Parallel Fragment-Based Quantum Chemistry for Large Molecular Systems: The Serestipy Software

“…As a result, the main area of application of subsystem DFT using approximate NAKE functionals are weakly-interacting subsystems. For more strongly-interacting subsystem connected via for example covalent bonds, other ways of describing the subsystem interaction such as potential-reconstruction [60][61][62][63][64][65][66][67][68] or projection techniques [69][70][71][72] are usually more suitable.…”

Accurate Kohn-Sham auxiliary system from the ground state density of solids

“…The inversion of the KS system is not an easy task, though, and in particular a finite basis set may lead to drastically modified results [49][50][51][52][53] . Moreover, small changes in the density can yield large differences in the potential 54,55 . Altogether, a reliable inversion of the KS equations remains a difficult task even for finite systems, and while various methods have been proposed to overcome the problems, research in this direction is still ongoing [54][55][56][57][58][59][60][61][62][63][64][65] .…”