Uniting non-empirical and empirical density functional approximation strategies with the constrained and smoothed empirical (CASE) framework

Density functional theory (DFT) has become the de facto standard when performing electronic structure calculations for large molecules and materials. While DFT is an exact theory, its accuracy depends on the density functional approximation (DFA) used to describe the non-trivial exchange-correlation interactions between electrons in molecules and matter. To date, DFAs have been primarily designed using non-empirical strategies (i.e., satisfaction of physical constraints and appropriate norms) or empirical strategies (i.e., data-driven optimization). Here we present a general framework that unites these seemingly contrasting strategies. The proposed method employs B-splines, bell-shaped spline functions with compact support, to construct the inhomogeneity correction factors (ICF) in the DFA, which offers several advantages over traditional polynomial expansions: (1) the ability to enforce linear and non-linear constraints as well as ICF smoothness using Tikhonov and penalized B-splines regularization; and (2) the flexibility to leverage quantum-mechanical and/or experimental data during the optimization. As proof-of-concept, we use the resulting Constrained And Smoothed Empirical (CASE) framework to construct a constraint-satisfying and data-driven hybrid DFA that exhibits enhanced performance across a diverse set of chemical properties. We argue that CASE can generate DFAs that maintain the physical rigor and transferability of non-empirical DFAs while leveraging high-quality data to remove the arbitrariness of ansatz selection and improve performance.