Variational derivation of the atomic virial theorem

The general time-dependent virial theorem for an atom in a molecule is derived from the atomic variational principle. We shall find a close connection between the expressions so obtained for the virial and those derived in the previous section for the force. In particular, the differential force law leads directly to a corresponding local expression for the virial theorem. 

Writing the expression for fl] explicitly in terms of s, differentiating with respect to e, and noting that 

Equation (8.187) can be rearranged using eqn (8.168), Schrodinger s equation, and the identities. 

As the atom is to be bounded by a zero-flux surface at all stages of the variation, the variational constraint of eqn (8.117) must be imposed. This constraint in the present case demands that the variation 

The quantity TtlH) is the average kinetic energy of the subsystem defined specifically in terms of the density K(r) (eqns (5.48) and (5.49)) and the virial of the forces exerted on an electron in the subsystem resulting from its instantaneous interactions with the nuclei and other electrons in the system (see eqn (6.19)) 

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