where gðr; r 0 Þ and Gðr; r 1 Þ are one-electron and two-electron density matrices, respectively, Z a is the charge the nucleus a, e and m are the electronic charge and mass, respectively, and V n is the nuclear repulsion. These functions are related by the local form of the virial theorem [20]:Finally, the density of the total electronic energy is defined as:Exploration of energy distributions in molecules by use of wavefunction-based calculations [21][22][23][24][25][26] has revealed that analysis of the local electronic energy is a direct approach to characterization of bonding in molecules and crystals. In particular, it facilitates recognition of the type of atomic interactions from the properties of bond critical points at r b À h e ðr b Þ < 0 and gðr b Þ=rðr b Þ < 1, and ' 2 rðr b Þ < 0, are observed for shared-type atomic interactions whereas h e ðr b Þ q 0, gðr b Þ=rðr b Þ > 1, and ' 2 rðr b Þ > 0 are typical for intermediate and closed-shell interactions [16,18,23,27].Bader has also stressed [26, 28] that the potential energy density represents the field of the virial of the Ehrenfest force [29] acting on an electron at r, the virial field vðrÞ. Irrespective of the type of atomic interaction, each bond path in the electron density at equilibrium geometry is homeomorphically mirrored by a virial path, a line of maximum negative potential energy density linking the same nuclei [24]. The presence of the bond paths and virial path provides, according to Ref. [26], an indicator of bonding atomic interaction. A network of the virial and bond paths defines a molecular graph, which is independent of the nuclear vibrations in a stable system.It was, of course, a challenge to perform real-space energy analysis of bonding on the basis of the experimental electron density (ED). There is, fortunately, a theory which establishes the interconnection between the electron density and energy densities of different kinds. This is the density-functional theory (DFT) [30][31][32][33][34][35][36][37][38][39][40] which exploits r as a main variable and determines all the properties of atoms, molecules, and crystals in the ground electronic state [41]. Thus, DFT is a basis for quantitative characterization of bonding in terms of energy densities and other functions related to electron density. It is, therefore, attractive to combine the formalism of the DFT with experimental electron density to analyze the nature of atomic and molecular interactions in molecules and solids in terms of the local energies. This might, in principle, be done in two different ways. The exact functionals connecting r and the energy densities of electrons -the kinetic, potential, exchange and correlation densities -are, in general, unknown [41]. DFT methods therefore use either the Kohn and Sham orbital scheme [42] or approximate functionals with explicit (but nonunique) dependence of these functions on r and its derivatives [31]. The former approach might be achieved by use of an idempotent one-electron density matrix iteratively reconstructed from ...