Title: Enhancing Exchange-Correlation Functionals Through Derivative-Informed Learning

Abstract

While machine-learned (ML) exchange-correlation (XC) functionals are designed to supplant human-crafted density functional approximations by deriving directly from reference data, they have yet to consistently surpass the performance of traditional hybrid functionals, which scale as $\mathcal{O}(N^4)$. This study investigates a hybrid-distillation framework where $\mathcal{O}(N^3)$-scaling ML-XC functionals are trained to replicate B3LYP/def2-SVP targets. To achieve this, we propose the Derivative Informed XC-Loss (DI-Loss), a novel loss function that leverages additional information from the reference hybrid functional by enforcing supervision over the first and second derivatives of the energy on the Grassmannian of permissible density matrices. Instead of merely aligning the self-consistent fixed point, DI-Loss ensures that the local first- and second-order responses of the learned functional mirror those of the target functional.

Evaluation across four distinct architectures demonstrates that DI-Loss consistently enhances primary energy metrics. When averaged uniformly across these architectures, the mean absolute error (MAE) for total energy drops by 66% compared to methods relying solely on energy and density supervision. Specifically, the density-sensitive mean-field energy metric, $E_\rho$, shows an average improvement from 1.2 to 0.8 mEh. However, improvements in dipole and $\mathcal{L}_2$ density errors are not uniform. Furthermore, we demonstrate that utilizing densities derived from these distilled functionals can reduce the number of self-consistent field (SCF) iterations required for hybrid functionals by up to 50%. In downstream time-dependent density functional theory (TDDFT) applications, Hessian supervision leads to better excited-state predictions; notably, the XCdiff method reduces the mean excitation-energy MAE by 19% to 35%.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC