Analytical Model of Electron Density and Its Machine Learning Inference

We present an analytical model representation of the electron density ρ(r) in molecules in the form of expansions of a few functions (exponentials and a Gaussian) per atom. Based on a former analytical model of ρ(r) in atoms, we devised its molecular implementation by introducing the anisotropy inherent in the electron distribution of atoms in molecules by means of proper anisotropic functions. The resulting model named A2MD (anisotropic analytical model of density) takes an analytical form highly suitable for obtaining the electron density in large biomolecules as its computational cost scales linearly with the number of atoms. To obtain the parameters of the model, we first devised a fitting procedure to reference electron densities obtained in ab initio correlated quantum calculations. Second, in order to skip costly ab initio calculations, we also developed a machine learning (ML)-based predictor that used neural networks trained on broad molecular datasets to determine the parameters of the model. The resulting ML methodology that we named A2MDnet (A2MD network-trained) was able to provide reliable electron densities as a basis to predict molecular features without requiring quantum calculations. The results presented together with the low computational scaling associated to the A2MD representation of ρ(r) suggest potential applications to obtain reliable electron densities and ρ(r)-based molecular properties in biomacromolecules.