Variation of the atomic action integral

We are now in a position to perform a generalized variation of the action integral for an open system to demonstrate that Schwinger s principle of stationary action can be extended in such a manner as to provide a quantum definition of an atom in a molecule. We shall be considering the change in the atomic action integral 2] of eqn (8.111) ensuing from variations 

The trial function O, where = T + P, and its first and second derivatives vanish whenever an Mectronic position vector is of infinite length. 

In terms of a region Q(, t) can be defined that is bounded by a zero-flux surface in Vp, where 

Moreover, it is required that, as tends to P at any time t, fl( , t) is continuously deformable into the region 0( ) = fX , t) associated with the atom. The region 2( , t) thus represents the atom in the varied total system, which is described by the trial function 4 (r, t, t) just as D(t) represents the atom when the total system is in the state described by 

Requiring the fulfilment of condition (2) amounts to imposing the variational constraint that the divergence of Vp, integrate to zero at all stages of the variation, i.e. that 

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Before pursuing the variation of the atomic action integral, it is helpful to first recover the statement of the principle of stationary action in the Schrodinger representation for the total system. If one sets the boundary of the region Cl at infinity in eqn (8.118) to obtain the variation of the total system action integral 2 [ ]> and restricts the variation so that ST vanishes at the time end-points and the end-points themselves are not varied, then only the terms multiplied by the variations in the first integral on the right-hand side remain. The Euler equation obtained by the requirement that this restricted... 

Proceeding as before in the field-free case, the variations in the state function are replaced by operators which act as generators of infinitesimal unitary transformations. That is, 5 P = ( — lh)F where F is an infinitesimal Hermitian operator (F = eG). Introducing the notion of generators into the result for the variation of the atomic action integral yields... 

One recognizes the first term in eqn (8.136) as the variation in the quantum mechanical current density (eqns (5.94) and (5.95)). It is obtained by combining the surface term arising from the variation with respect to VT with the surface term arising from the imposition of the variational constraint, eqn (8.135). Thus the variation of the surface of the subsystem together with the restriction that the subsystem be an atom bounded by a zero-flux surface causes the quantum mechanical current density to appear in the variation of the action integral, a term whose presence is a necessary requirement for the proper description of the properties of an open system. It is now demonstrated that eqn (8.136) is the atomic equivalent of the principle of stationary action. 

Equation (8.144) is an alternative form of the expression given in eqn (8.125) for the total system. The principle of stationary action for a subsystem can be expressed for an infinitesimal time interval in terms of a variation of the Lagrangian integral, similar to that given in eqn (8.127) for the total system. For the atomic Lagrangian, assuming F to have no explicit time dependence, this statement is... 

Thus, the atomic Lagrangian and action integrals in the presence of an electromagnetic field, like their field-free counterparts, vanish as a consequence of the zero-flux surface condition (eqn (8.109)). These properties are common to the corresponding integrals for the total system and it is a consequence of this equivalence in properties that the action integrals for the total system and each of the atoms which comprise it have similar variational properties. 

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... 

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