Uniting Nonempirical and 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): nonempirical (NE) constraint satisfaction and empirical (E)
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 traditional polynomial expansions by enabling explicit enforcement
of linear and nonlinear constraints as well as ICF smoothness using
Tikhonov and penalized B-splines (P-splines) regularization. As proof-of-concept,
we use the so-called CASE (constrained and smoothed empirical) framework
to construct a constraint-satisfying and data-driven global hybrid
that exhibits enhanced performance across a diverse set of chemical
properties. We argue that the CASE approach can be used to generate
DFAs that maintain the physical rigor and transferability of NE-DFAs
while leveraging high-quality quantum-mechanical data to remove the
arbitrariness of ansatz selection and improve performance.