Learning Molecular Conformational Energies Using Semilocal Density Fingerprints

We present a general theoretical framework for transforming the information contained in semilocal descriptors of the electron density (i.e., ρ(r) and ∇ρ(r))─the quantum mechanical objects at the very heart of density functional theory (DFT)─into feature vectors that can be used for machine learning (ML). As a proof of principle, we consider the task of learning molecular conformational energies and introduce the semilocal density fingerprint (SLDF) descriptor, which transforms the most energetically relevant information in ρ(r) into compact fixed-size feature vectors that are invariant to translations, rotations, and atomic permutations, unique (at least up to the semilocal DFT exchange-correlation energy), smooth with respect to changes in the atomic positions, and more transferable to systems containing elements from wider swaths of the periodic table. We demonstrate that SLDF-based ML models can predict benchmark conformational energies that are often >100 times more accurate than semilocal DFT, with some predictions approaching spectroscopic accuracy (≈1 cm^-1). Under the most stringent learning strategy─wherein the training set does not include any conformations of the molecules in the test set─SLDF-based ML models exhibit a sizable degree of transferability and generate conformational energies that are 10-fold more accurate than semilocal DFT for molecules containing first- and second-row main-group elements. As a final assessment of transferability, we demonstrate that SLDF-based ML models can rectify the qualitatively incorrect semilocal DFT description of the oxirene potential energy surface without seeing a single oxirene conformation.