CASE21: Uniting Non-Empirical and Semi-Empirical Density Functional Approximation Strategies using Constraint-Based Regularization

In this work, we present a general framework that unites the two primary strategies for constructing density functional approximations (DFAs): non-empirical (NE) constraint satisfaction and semi-empirical (SE) data-driven optimization. The proposed method employs B-splines -- bell-shaped spline functions with compact support -- to construct each inhomogeneity correction factor (ICF). This choice offers several distinct advantages over a polynomial basis by enabling explicit enforcement of linear and non-linear constraints as well as ICF smoothness using Tikhonov regularization and penalized B-splines (P-splines). As proof of concept, we use this approach to construct CASE21 -- a Constrained And Smoothed semi-Empirical hybrid generalized gradient approximation that completely satisfies all but one constraint (and partially satisfies the remaining one) met by the PBE0 NE-DFA and exhibits enhanced performance across a diverse set of chemical properties. As such, we argue that the paradigm presented herein maintains the physical rigor and transferability of NE-DFAs while leveraging high-quality quantum-mechanical data to improve performance.