Derivative Informed Learning of Exchange-Correlation Functionals

Machine-learned (ML) XC functionals aim to replace human-designed density functional approximations by learning directly from reference data, but they still do not consistently outperform traditional $\mathcal{O}(N^4)$-scaling hybrid functionals. We therefore study a hybrid-distillation setting, where $\mathcal{O}(N^3)$-scaling semilocal ML-XC functionals are trained to reproduce B3LYP/def2-SVP targets. We introduce Derivative Informed XC-Loss (DI-Loss), a loss that incorporates additional information from the reference hybrid functional by supervising first and second derivatives of the energy on the Grassmannian of admissible density matrices. Rather than only matching the self-consistent fixed point, DI-Loss aligns the local first- and second-order response of the learned functional with that of the target functional. Across four evaluated architectures, DI-Loss consistently improves the main energy metrics. Averaged uniformly across architectures, the total-energy MAE decreases by 66% relative to energy and density supervision alone. The density-sensitive mean-field energy metric $E_\rho$ improves from 1.2 to 0.8 mEh on average, while dipole and $\mathcal{L}_2$ density errors do not improve uniformly. We further show that densities from the distilled functionals reduce hybrid-functional SCF iterations by up to 55%. In downstream TDDFT calculations, Hessian supervision improves excited-state predictions, with XCdiff reducing the mean excitation-energy MAE by 24-35% across molecule sizes on QM40.

Accurate computer simulations are an important tool for designing new molecules and materials, with applications ranging from drug discovery to batteries, catalysts, and sustainable chemicals. These simulations can predict properties before a compound is synthesized in the lab, but there is a persistent trade-off: the most accurate quantum-chemical methods are often too expensive to use at large scale, while cheaper methods can miss important effects. This work uses machine learning to make high-quality chemistry calculations cheaper. We focus on density functional theory, one of the most widely used approaches in computational chemistry. In density functional theory, the main source of error is an approximation called the exchange-correlation functional, which determines how electrons interact beyond simple electrostatics. More accurate functionals, such as hybrid functionals, are useful but computationally expensive. Cheaper functionals are faster, but usually less accurate. Our idea is to train a machine-learned functional to imitate a more expensive hybrid functional, so that it can provide similar predictions at substantially lower computational cost. Instead of only teaching the model to match final energies and electron densities, we also teach it how the reference calculation locally responds when the electron density is changed. In other words, the model learns not only the correct answer, but also the local shape of the quantum-mechanical energy landscape around that answer. This additional derivative information makes the learned functional more reliable. Across several neural-network architectures, our derivative-informed training objective improves the main energy metrics compared with standard energy and density supervision. On average, it reduces total-energy errors by 66%. The learned functionals also produce electron densities that can be used to initialize more expensive hybrid-functional calculations, reducing the number of self-consistent solver iterations by up to 55%. This means that even when the expensive calculation is still needed, the machine-learned model can help reach the answer faster. We also test whether the learned functionals capture information relevant beyond ground-state energies. In time-dependent density functional theory, which is used to predict optical and excited-state properties, the curvature of the energy functional is crucial. By supervising this curvature during training, our method improves excited-state predictions, reducing excitation-energy errors by 24–35% in our experiments. Overall, this work shows how machine learning can distill expensive quantum-chemical models into cheaper ones while preserving important physical behavior. This could make accurate molecular and materials modelling more practical for large-scale screening tasks, such as searching for better drug candidates, catalysts, or functional materials.