Theoretical development of a virial relationship for spatially defined fragments of molecular systems

The original statement of the proposal that a molecular charge distribution can be spatially partitioned by a particular surface into fragments whose average kinetic and potential energies obey the virial relationship, involves an arbitrary choice of origin for the definition of the nuclear virial. A development of the fragment virial relationship is presented here which closely parallels Slater's derivation of the molecular virial theorem. This development provides an independent condition for the determination of the origin and demonstrates that it is determined by a property of the system. The surface which partitions the charge distribution ρ(x) of the total system into fragments (A) and (B) is defined by the gradient vector ${\nabla }_{\rho }(\mathrm{x})$ , passing through the point at which ρ(x) attains its minimum value between a pair of adjacent nuclei. The consequences of the restraints which the fragment virial relationship places on a molecular system are discussed with reference to the question of the transferability of the spatially defined fragments between different systems and with respect to the possibility of imposing the restraints directly in the calculation of the properties of a system.