Learning on-top: regressing the on-top pair density for real-space visualization of electron correlation

The on-top pair density [Π(r)] is a local quantum chemical property, which reflects the probability of two electrons of any spin to occupy the same position in space. Simplest quantity related to the two-particles density matrix, the on-top pair density is a powerful indicator of electron correlation effects and, as such, it has been extensively used to combine density functional theory and multireference wavefunction theory. The widespread application of Π(r) is currently hindered by the need for post-Hartree-Fock or multireference computations for its accurate evaluation. In this work, we propose the construction of a machine learning model capable of predicting the CASSCF-quality on-top pair density of a molecule only from its structure and composition. Our model, trained on the GDB11-AD-3165 database, is able to predict with minimal error the on-top pair density of organic molecules bypassing completely the need for ab-initio computations. The accuracy of the regression is demonstrated using the on-top ratio as a visual metric of electron correlation effects and bond-breaking in real-space. In addition, we report the construction of a specialized basis set, built to fit the on-top pair density in a single, atom-centered expansion. This basis, cornerstone of the regression, could be potentially used also in the same spirit of the resolution-of-the-identity approximation for the electron density.