Atoms in molecules in real space: a fertile field for chemical bonding

In this perspective, we review some recent advances in the concept of atoms-in-molecules from a real space perspective. We first introduce the general formalism of atomic weight factors that allows unifying the treatment of fuzzy and non-fuzzy decompositions under a common algebraic umbrella. We then show how the use of reduced density matrices and their cumulants allows partitioning any quantum mechanical observable into atomic or group contributions. This circumstance provides access to electron counting as well as energy partitioning, on the same footing. We focus on how the fluctuations of atomic populations, as measured by the statistical cumulants of the electron distribution functions, are related to general multi-center bonding descriptors. Then we turn our attention to the interacting quantum atom energy partitioning, which is briefly reviewed since several general accounts on it have already appeared in the literature. More attention is paid to recent applications to large systems. Finally, we consider how a common formalism to extract electron counts and energies can be used to establish an algebraic justification for the extensively used bond order-bond energy relationships. We also briefly review a path to recover one-electron functions from real space partitions. Although most of the applications considered will be restricted to real space atoms taken from the quantum theory of atoms in molecules, arguably the most successful of all the atomic partitions devised so far, all the take-home messages from this perspective are generalizable to any real space decompositions.