Machine Learning Force Fields Without Precomputed Labels: Variational Kohn-Sham Training

Molecular dynamics at the accuracy of density functional theory (DFT) is limited by the cost of a self-consistent field (SCF) calculation at every step. Machine learning force fields (MLFFs) remove that cost from the simulation by regressing energies and forces onto reference DFT calculations, but move it to the dataset: every training label is itself a converged SCF calculation, and a new system, level of theory or range of conditions requires a new set of them. We merge labelling and training into a single optimisation. Because the Kohn-Sham energy is a variational functional of the electronic density, a network predicting the electronic state of a molecule can be trained by minimising the Kohn-Sham energy of its own prediction, averaged over geometries taken from an inexpensive empirical force field trajectory; the physics supplies the training signal and no labelled dataset is precomputed. The network emits a matrix in the atomic-orbital basis whose eigendecomposition yields an admissible density, energies follow from the functional and forces from the standard analytic nuclear gradient at that density, so inference requires no SCF iteration. Across ten drug-like and small organic molecules the model reaches a mean force error of $0.98$ meV/Å against converged DFT, and on a temperature-extrapolation benchmark it reaches $5.6$ meV/Å force RMSE far outside the training distribution, against $62.0$ for the strongest supervised baseline. Against supervised training of the same network on the same geometries it is as accurate or better while using $12$–$17$% less GPU time, and it drives stable molecular dynamics at a third of the cost per step of DFT.