System-Specific Reparameterization of Density Functionals with Machine Learning: Application to Spin-Splitting Energies of Transition Metal Complexes

Accurate estimates of the spin-splitting energy (SSE) are essential for proper modeling of transition metal complex (TMC) catalysts and functional materials but are notoriously challenging to achieve. Over a large set of over 450 TMCs, we demonstrate that adding Hartree-Fock exchange (HFX) to semilocal density functionals (e.g., PBE or SCAN) in a system-specific fashion provides the flexibility to significantly reduce errors relative to reference wave function theory (i.e., DLPNO-CCSD(T)) values. Nevertheless, to do so requires that one knows the correct amount of HFX to add to the functional, which is system-dependent. We show that global reparameterizations, optimal ionization potential tuning, and machine learning (ML) models trained on atomic properties and molecular connectivity are all insufficient to reliably improve results relative to standard hybrid functionals. Instead, we utilize a Behler-Parrinello neural network trained on the electron density generated by a hybrid functional to predict the optimal amount of HFX to include, achieving errors on the unseen test set of <5% HFX, corresponding to 3-4 kcal/mol energetic errors. These results are competitive with the best single functionals on these data sets, as well as previous approaches that utilize system-specific functional recommendation, providing a path to a ML-derived self-tuning functional based on the electron density. This approach provides a practical and interpretable way to increase the accuracy of density functionals, which will enable more reliable and efficient screening of chemical space.