The principle of stationary action

The development given here follows closely that given by Schwinger (1951). A base vector system a t is specified by the eigenvalues x, of some complete set of commuting operators a, at the time t. One can consider having two similarly constructed operator sets at times f, and tj, a,i and a,2, respectively, which possess the same eigenvalue spectrum a, and are, therefore, connected by a unitary transformation as outlined in eqns (8.28) and (8.29). Thus, 

The transformation function connecting a base state at time (2 with that of another representation derived from the commuting set of observables p, at time f 1 may be viewed as the matrix of U12 in the original eigenvector system a,i. That is, 

As previously discussed, a description of the temporal evolution of a system is accomplished by stating the relationship between eigenvectors associated with different times or, in other words, by exhibiting the transformation function in eqn (8.71). One may expect that the quantum dynamical laws will find their proper expression in terms of the transformation function and we now present Schwinger s development (1951) of a differential formulation of this type. 

Schwinger s reasoning is, with minor modification, as follows. The operator UI2 describes the development of the system from time to time and involves not only the detailed dynamical characteristics of the system in this space-time region, but also the choice of commuting operators di,i and 0. 2 at the times and 2- Any infinitesimal change in the quantities on which the transformation function depends induces a corresponding alteration in 0 2  

The quantity iUi2 Ui2 Hermitian as a result of the unitary property U = U or UU = 1. Because of this property one has 

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The division of the molecular volume into atomic basins follows from a deeper analysis based on the principle of stationary action. The shapes of the atomic basins, and the associated electron densities, in a functional group are very similar in different molecules. The local properties of the wave function are therefore transferable to a very good approximation, which rationalizes the basis for organic chemistry, that functional groups react similarly in different molecules. It may be shown that any observable... 

Note that for general parameterizations this metric matrix is neither skew diagonal nor constant-, see below. The equations of motion expressed in Eq. (2.6) are obtained by using the Principle of Stationary Action, 5A = 0, with Lagrangian... 

With the Lagrangian in hand, the principle of stationary action... 

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 atomic statements of the Ehrenfest force law and of the virial theorem establish the mechanics of an atom in a molecule. As was stressed in the derivations of these statements, the mode of integration used to obtain an atomic average of an observable is determined by the definition of the subsystem energy functional i2]. It is important to demonstrate that the definition of this functional is not arbitrary, but is determined by the requirement that the definition of an open system, as obtained from the principle of stationary action, be stated in terms of a physical property of the total system. This requirement imposes a single-particle basis on the definition of an atom, as expressed in the boundary condition of zero flux in the gradient vector field of the charge density, and on the definition of its average properties. 

It has been shown that the principle of stationary action for a stationary state applies to a system bounded at infinity and to one bounded by a surface of zero flux in Vp(r). It is demonstrated in Chapter 8, through a variation of the action integral, that the same boundary conditions are obtained in the general time-dependent case. One may seek the most general solution to the problem of defining an open system by asking for the set of all possible subsystems to which the principle of stationary action is applicable. Thus, one must consider the variation of the energy functional f2 3 defined as... 

Bader 1975) that the condition for the satisfaction of the principle of stationary action is that each subsystem lj be bounded by a surface Si satisfying a zero-flux boundary condition of the form... 

The first step in applying the principle of stationary action is to generalize the variation of the action integral to include a variation of the time end-points and to retain the variations at the end-points in order to define the generator F t). We shall express the Lagrangian operator in terms of the complete set of... 

A form of the principle of stationary action that will prove of great use in later applications is obtained by expressing the principle of stationary action (eqn (8.79)) for an infinitesimal time interval, that is, as a variation of the Lagrange function operator. Equation (8.79) is re-expressed as... 

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... 

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. 

Now it is the essence of the principle of stationary action that the total change in action is equal to the difference in the values of the generator /( ) evaluated at the two time end-points (eqn (8.79)). In the Schrodinger representation this corresponds to equalling the difference in the... 

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... 

The derivation of the principle of statioiiary action for an atom in a molecule (eqn (8.143)) yields Schrodinger s equation of motion for the total system, identifies the observables of quantum mechanics with the variations of the state function, defines their average values, and gives their equations of motion. We have demonstrated in Chapter 6 how one can use the atomic statement of the principle of stationary action given in eqn (8.148) to derive the theorems of subsystem quantum mechanics and thereby obtain the mechanics of an atom in a molecule. The statement of the atomic action... 

This chapter concludes with the demonstration that the atomic statement of the principle of stationary action obtains in the presence of an electromagnetic field. Thus, atoms continue to exist in the presence of applied electric and/or magnetic fields and the response of each atom to such fields and its contribution to the magnetic and electric properties of the total system can be determined. 

The division of the molecular volume into atomic basins follows from a deeper analysis based on the principle of stationary action. The shapes of tlie atomic basins, and... 

We now come to the central issue of Hamilton s Principle of Variable Action generalized from one to several dimensions. Henceforth, it is assumed that the functional is stationary for fixed domain and boundary values, which is equivalent to asserting the validity of (10) as given priori, rather than obtained posteriori via the Principle of Stationary Action above. Then (9) reduces at once to... 

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