Interatomic surface virial

Before proceeding with the development of relating the pressure to the surface virial, it is useful to introduce a related concept, the interatomic surface virial. An atomic surface S( 2 r) consists in general, of a number of interatomic surfaces S( 2 2 r), one with each of its bonded neighbours, that is, atoms linked to it by bond paths. Correspondingly, the surface virial for the atom will consist of a sum of contributions, one for each bonded atom, and each bonded interaction will contribute to the pressure exerted on the atom. One has... 

The interatomic surface virial vs(A B) vanishes for the separated atoms and hence vs(A B) is the contribution to the energy of formation of the molecule arising from the creation of an interatomic surface S(A fi) between atoms A and B, a most useful result. The Ehrenfest force F(A B) is attractive when the force exerted on the density of atom A is directed at B, the situation found for all bonded interactions. The interatomic surface virial vs (A B) is negative in such a case and the formation of the surface S(A Z ) contributes to the stability of the system. 

The bond path is always found to be accompanied by a shadow graph, the virial path, first discovered by Keith, Bader, and Aray [17]. The virial path is a line of maximally-negative potential energy density in three-dimensional space that links the same pair of atoms that share a bond path and an interatomic surface of zero-flux. No theoretical basis has ever been provided that requires the presence of a virial path as a doppelganger of every bond path that links two chemically bonded atoms, however, there is no known computational violation of this observation to date known to the authors. The presence of the virial path links the concept of chemical bonding directly with the concept of energetic stability as amply discussed in literature on QTAIM. 

Figure 12 Contour maps of the electron density p(r) in a, of the kinetic energy density G(r) in b, and of the virial field V(r) in c for a methylene group adjacent to a methyl group in n-butane and w-pentane. The plane shown contains the carbon and hydrogen nuclei. The C-H bond cps and the associated bond paths and intersections of the interatomic surfaces are shown in a. The corresponding cps and virial paths are shown in c. This group and its properties are transferable between these two molecules to within the experimental accuracy of heats of formation. All three fields are locally proportional to the total energy density, G(r) and V(r) by theory, p(r) by observation...

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