Variational derivation of the atomic force law

The atomic statement of the principle of stationary action, eqn (6.3), yields a variational derivation of the hypervirial theorem for any observable G, a derivation which applies only to a region of space H bounded by a surface satisfying the condition of zero flux in the gradient vector field of the charge density, 

We shall use the principle of stationary action to obtain a variational definition of the force acting on an atom in a molecule. This derivation will illustrate the important point that the definition of an atomic property follows directly from the atomic statement of stationary action. To obtain Ehrenfest s second relationship as given in eqn (5.24) for the general time-dependent case, the operator G in eqn (6.3) and hence in eqn (6.2) is set equal to pi, the momentum operator of the electron whose coordinates are integrated over the basin of the subsystem 1. The Hamiltonian in the commutator is taken to be the many-electroii, fixed-nucleus Hamiltonian 

The symbol V will be used to denote the complete potential energy operator, the sum of the electron-nuclear V , electron-electron and nuclear-nuclear potential energy operators, 

The commutator of this Hamiltonian and the momentum operator of a single electron is, as in eqn (5.23), equal to ihVV. 

The method of obtaining the subsystem average of the commutator and hence of the force acting on the atom SJ, is determined by the definition of the functional H] via eqn (6.3). It is demonstrated in Section 6.2 that the mode of integration used in the definition of the subsystem functional n], (sec eqn (5.72) and discussion following) is the only one which leads to a physically realizable boundary condition. Because of eqn (6.3), this same mode of integration defines the atomic average of the commutator and thus of the atomic force, F( 2), 

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