Variational Hirshfeld Partitioning: General Framework and the Additive Variational Hirshfeld Partitioning Method

We introduce the general mathematical framework of variational Hirshfeld partitioning, wherein the best possible approximation to a molecule's electron density is obtained by minimizing the <i>f</i>-divergence between the molecular density and a non-negative linear combination of (normalized) basis functions. This framework subsumes several existing methods that variationally optimize their pro-atoms, like (Gaussian) iterative stockholder analysis (ISA and GISA) and minimal basis iterative stockholder partitioning (MBIS), and provides a solid foundation for developing mathematically rigorous partitioning schemes. In this paper, we delve into the mathematical underpinnings of Hirshfeld-inspired partitioning schemes and show that among all the valid <i>f</i>-divergence measures only the extended Kullback-Leibler is a suitable choice. This led us to develop a novel partitioning scheme, called additive variational Hirshfeld (AVH), which constructs the pro-molecular density as a convex linear combination of the densities from selected states of isolated atoms and atomic ions. The AVH method is size-consistent with a unique solution and provides a straightforward approach for adding constraints for fragment properties. It also results in an intuitively appealing valence-bond-like decomposition of the molecular density as a weighted average of the densities of the atomic states in the molecule; that is, the AVH atomic density is a minimal deformation of the corresponding isolated atomic reference state's density. Compared to other variational Hirshfeld variants, our numerical results show that AVH yields chemically interpretable and sensible atomic charges that are not excessively large and demonstrate computational robustness.