The variation of Hamiltonian-based functionals

In analogy with and its many-electron subsystem counterpart, the 

The variation of 2] with the imposition of the variational constants given in eqn (5.88) at every stage of the variation and including a variation of the surface yields 

The variation of i [i/, 2] involves variation with regard to 5il/ and It is necessary to perform an integration by parts twice in succession to rid the expression of the variations of gradients of This procedure is illustrated below for a single term in 2] 

Using this result, the variation of S [i/ , 2] including a variation of the surface of 2 is found to be 

This result, which is obtained without any restrictions on 2 is identical to that obtained from the variation of [i/, 2] by constraining the subsystem to be bounded by a zero-flux surface. It is clear from eqn (E5.9) that, unlike the variation of [i, Q] or [i, 2], one obtains the same result for the variation of S [ij/, 2] whether or not the surface is varied. 

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