Machine learning the Hohenberg-Kohn map to electronic excited states

Time-Dependent Density-Functional Theory (TDDFT) is the workhorse for computing electronic excitations in molecules and materials; however, the approximations inherent in practical TDDFT calculations, together with their computational expense, motivate finding a cheaper, more direct map for electronic excitations. The existence of such a map is provided by the Hohenberg-Kohn theorem of density-functional theory, which proves a bijection between the ground-state electron density and the external potential of a many-body system. This guarantees a one-to-one map from the electron density to all observables of interest, including electronic excited-state energies. Here, we show that multistate density and energy functionals can be constructed via machine learning. The framework is used to perform the first excited-state molecular dynamics simulations with a machine-learned functional on malonaldehyde and correctly capture the kinetics of its excited-state intramolecular proton transfer, allowing insight into how mechanical constraints can be used to control the proton transfer reaction in this molecule.