Data-Driven Learning of Optimal Position-Dependent Exact-Exchange Energy Density Mixing for Improved Density Functionals

A transparent, data-driven route to improve approximate density functionals by learning only the position dependence of an exact-exchange admixture in local hybrid functionals is discussed. Motivated by the scarcity of exact constraints for valence regions and by gauge ambiguity issues of exchange-energy densities, we replace hand-crafted inhomogeneity measures by neural-network local mixing functions (n-LMFs) evaluated on rung-3 or rung-4 descriptors while keeping the overall structure of the functional transparent and explainable. This limited use of machine learning (ML) has already provided a number of practical outcomes. The LH24n-B95 and LH24n functionals achieve broad main-group accuracy and, strikingly, suppress gauge artifacts without use of so-called calibration functions. The reasons for the latter observation can be visualized and analyzed directly in real space. Extending the idea to rung-5 functionals leads to the first local double hybrids in which a position-dependent exact-exchange admixture is paired with an SCS-PT2 correlation, delivering consistent gains over constant-mixing analogues. To escape the zero-sum game between delocalization errors and static-correlation errors, an n-LMF has subsequently been trained in the presence of an explicit strong-correlation factor. The resulting LH25nP functional combines the so far best rung-4 performance for main-group energetics with improved spin-restricted bond-dissociation curves, fractional-spin behavior, and reduction of spin-contamination artifacts, while remaining numerically robust. The limited ML approach preserves explainability and facilitates the transfer of insights back to rational designs.