Atomic action principle

The statement of the atomic action principle given in eqn (8.145) is a variational principle which enables one to derive the properties of an atom in a molecule—it is an atomic variation principle. We shall use it first to derive the atomic statement of the Ehrenfest force law, the equation of motion of an atom in a molecule. This is accomplished through a variation of fl]... 

The averaging of these commutators in the manner determined by the atomic action principle (eqn (8.225)) serves to define the atomic averages of the kinetic energy and of the virials of the internal and external forces acting on the basin of the atom, i.e. 

If the topological property which defines an atom is also one of physical significance, then it should be possible to obtain from quantum mechanics an equivalent mechanical definition. As demonstrated in Chapters 5 and 8, this can be accomplished through a generalization of the quantum action principle to obtain a statement of this principle which applies equally to the total system or to an atom within the system. The result is a single variational principle which defines the observables, their equations of motion, and their average values for the total system or for an atom within the system. 

In order for the scalar product of n, the vector normal to the surface (see Fig. 2.5), with Vp to vanish, it is necessary that the atomic surface not be crossed by any trajectories of Vp and as such it is referred to as a zero-flux surface. The state function ij/ and n, where the gradient is taken with respect to the coordinates of any one of the electrons, vanish on the boundary of a bound system at infinity. Thus, p and Vp vanish there as well and a total isolated system is also bounded by a surface satisfying eqn (2.9). Since the generalized statement of the action principle applies to any region bounded by such a surface, the zero-flux surface condition places the description of the total system and the atoms which comprise it on an equal footing. 

The Lagrangian-based functional [i, 2] or 2] derives directly from the Lagrangian as employed in the quantum action principle. For a total system, both the Lagrangian- and Hamiltonian-based functionals yield identical variational results. This equivalence in variational behaviour is maintained for the corresponding subsystem functionals only if the subsystem is bounded by a zero-flux surface. Only an atomic region ensures an equivalence in both the values and the variational properties of the two types of functionals (eqns (E5.6a,b)) thereby preserving the properties obtained for a total system. 

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... 

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... 

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... 

ACTION PRINCIPLE FOR A QUANTUM SUBSYSTEM 8.4 8.4.5 Summary of the atomic variational principle... 

Actually, the entire present development stays under the valence or long-range regime of electrons in atoms and molecules in various forms and approximations of conceptual DFT. The minus sign in Eq. (4.203) agrees with the opposite phenomenological behavior in density and potential variation, as provided by Poisson equation—for instance (Putz et al., 2005), and is in accordance with alternative derivation based on ehemical action principle and virial theorem (Putz, 2009a). 

This extension of quantum mechanics is indeed possible within the framework of the generalized action principle and, as stated in Section 1, its use has resulted in the identification of the proper open systems of quantum mechanics with the topological atom. The derivation of the quantum mechanics of a proper open system cannot be given here, but one can present an outline of the essential ideas. It is based on the reformulation of physics provided by the work of Feynman and Schwinger, an approach that makes possible the asking and answering of questions not possible within the traditional Hamiltonian framework. It is important that the reader appreciate that the alternative to the use of existing... 

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... 

---