Local virial relationship

The average of the atomic virial V,(12), eqn (8.192), as defined by the atomic variational principle, is the virial of the quantum mechanical force density as defined in the differential force law, averaged over the atomic volume. By taking the virial of F(r, t) in eqn (8.179), one obtains 

One proceeds as in the case of a stationary state (eqn (6.25) and following). Using the identity 

Its integral over an atomic volume yields the unique value 

Integration of eqn (8.201) over an atomic volume for which the integral of V p(r) vanishes yields, term for term, the atomic virial theorem for a time-dep endent system (eqn (8.193)) or for a stationary state (eqn (6.23)). Thus, eqn (8.201) is, in terms of its derivation and its integrated form, a local expression of the virial theorem. The atomic virial theorem provides the basis for the definition of the average energy of an atom, as discussed in Chapter 6. 

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Considering the virial theorem as applied to a harmonic oscillator, for a gas in quasi-equlIibrium we expect a close relationship between the localization and the temperature. The mean squared position spread (proportional to the potential energy) would be proportional to the mean squared momentum... 

In describing thermodynamic and equilibrium statistical-mechanical behaviors of a classical fluid, we often make use of a radial distribution function g r). The latter for a fluid of N particles in volume V expresses a local number density of particles situated at distance r from a fixed particle divided by an average number density p = NjV), when the order of IjN is negligible in comparison with 1. Various thermodynamic quantities are related to g(r). For a single-component monatomic system of particles interacting with a pairwise additive potential 0(r), the relationship connecting the pressure P to g(r) is the virial theorem, ... 

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