Single-particle basis for atomic properties

The atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

It has been shown that the principle of stationary action for a stationary state applies to a system bounded at infinity and to one bounded by a surface of zero flux in Vp(r). It is demonstrated in Chapter 8, through a variation of the action integral, that the same boundary conditions are obtained in the general time-dependent case. One may seek the most general solution to the problem of defining an open system by asking for the set of all possible subsystems to which the principle of stationary action is applicable. Thus, one must consider the variation of the energy functional f2 3 defined as 

Bader 1975) that the condition for the satisfaction of the principle of stationary action is that each subsystem lj be bounded by a surface Si satisfying a zero-flux boundary condition of the form 

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