Deep Neural Network Computes Electron Densities and Energies of a Large Set of Organic Molecules Faster than Density Functional Theory (DFT)

Density functional theory (DFT) is one of the main methods in Quantum Chemistry that offers an attractive trade off between the cost and accuracy of quantum chemical computations. The electron density plays a key role in DFT. In this work, we explore whether machine learning - more specifically, deep neural networks (DNNs) - can be trained to predict electron densities faster than DFT. First, we choose a practically efficient combination of a DFT functional and a basis set (PBE0/pcS-3) and use it to generate a database of DFT solutions for more than 133,000 organic molecules from a previously published database QM9. Next, we train a DNN to predict electron densities and energies of such molecules. The only input to the DNN is an approximate electron density computed with a cheap quantum chemical method in a small basis set (HF/cc-VDZ). We demonstrate that the DNN successfully learns differences in the electron densities arising both from electron correlation and small basis set artifacts in the HF computations. All qualitative features in density differences, including local minima on lone pairs, local maxima on nuclei, toroidal shapes around C-H and C-C bonds, complex shapes around aromatic and cyclopropane rings and CN group, etc. are captured by the DNN. Accuracy of energy predictions by the DNN is ~ 1 kcal/mol, on par with other models reported in the literature, while those models do not predict the electron density. Computations with the DNN, including HF computations, take much less time that DFT computations (by a factor of ~20-30 for most QM9 molecules in the current version, and it is clear how it could be further improved).