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(<b>r</b>) and ∇ρ(<b>r</b>))─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(<b>r</b>) 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.