Inverse Kohn-Sham Density Functional Theory: Progress and Challenges

Inverse Kohn-Sham (iKS) methods are needed to fully understand the one-toone 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.

Automated summary

Inverse Kohn-Sham (iKS) methods are necessary for understanding the mapping between densities and potentials in density functional theory. They can aid in building exchange-correlation functionals and density-based embedding techniques. Numerical iKS problems are challenging, with the Wu-Yang method (WY) and partial differential equation constrained optimization (PDE-CO) being practical approaches, but limited by finite basis sets.