Surface integrals proportional to the energy

According to eqn (6.103), the surface virial measures the disparity in the changes in the values of the electronic kinetic and potential energies over the basin of the atom required for the satisfaction of the virial theorem, as occasioned by the formation of chemical bonds. 

The analogous result is obtained using quantum mechanics with the pV operator being defined as (PV) = (2/3) f — (Hirschfelder et al. 1967  

The separation of the virial into basin and surface contributions requires a choice of origin and a natural and physically useful choice is to place the origin of the position vector for atom A, the vector r, at the nucleus of the atom in question. If the vector from the nucleus of atom A to that of atom B is denoted by R b, then Rab = — rb nd, recalling that the surface normals are 

For a diatomic molecule, the integral on the right-hand side of eqn (6.106) is a measure of the contribution of the interatomic surface energy to the total energy change AE. 

For real G, the statement of the atomic hypervirial theorem yielding the atomic average of the commutator is (see eqn (5.43)) 

---