The variational definition of a subsystem and its properties

It will be demonstrated that the generalization of Schrodinger s variation of the functional [i ] to a subsystem has two important consequences (1) The variation of the subsystem energy functional 2] yields the hypervirial 

The extension of Schrodinger s energy functional to the many-electron case, including the Lagrange multiplier 1 is (in analogy with eqn (5.58)) 

The trial functions l representing variations in are given by eqn (5.69) and substitution of (r) for into n] yields fi]- At the point of variation, f = and fJ] equals fi]. The variations 5ij/ and dij/ are not given prescribed values on any of the boundaries, including the boundary of the subsystem. Instead only the natural boundary condition, that V,t/ nj and Vji/ n, together with ij/ and, vanish on all infinite boundaries, will be invoked. The functional [(, fi] is to be varied not only with respect to however, but also with respect to the surface defining the subsystem fJ. Only by having the surface itself considered to be a function of p can the definition of the subsystem be determined entirely in a non-arbitrary way by the variational procedure. 

The general expression for the variation of an integral including a variation of its surface for variations with respect to is given by 

No particular difficulties arise in passing from the one- to the many-electron case in the variation of [lA, f2]. Referring to eqns (5.72) and (5.74), one has 

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