Mayer bond order (MBO) allows partitioning of total charge in a given system into overlap population components which can be interpreted as charges shared among atoms and retained by them through atomic orbitals. In this work, we formulate a spatial distribution of these partitioned components, rendering a breakup of the total charge density into individual densities of charges shared between all the available pairs of atoms, as well as charges exclusively retained by each of the atoms themselves. The spatial density of the interatomic MBOs in particular facilitates an unbiased physical description of electrons shared between two atoms, thus essentially constituting a plottable representation of a covalent bond, obtained without inducing any explicit localization of electrons between atoms, which otherwise is an inherent source of bias. We demonstrate the proposed formulation in the basis of Wannierized atomic orbitals constructed from first principles, in a few representative varieties of systems with varying degrees of interatomic hybridization, including scenarios of multicentered bonds in molecules, to metavalent bonding in periodic systems introduced and debated in the past few years. Pertinently, in GeTe, we find two electrons (2e) contributed by collinear p orbitals in each of the three Ge–Te–Ge(Te–Ge–Te) segments passing through Te(Ge), constituting a compact distribution of 2e over the 3 atom segments (3c), along with the relatively inert s electrons maintaining a spherical shape, to facilitate near completion of subshell filling of both the atoms, thus supporting the prevalence of 3c-2e metavalent bonding in the class of narrow band gap rock-salt structures.
Subrahmanian et al. (2026) studied this question.