Action integral atomic

It is this property that is common to the corresponding Lagrangian and action integrals for a quantum subsystem, the chemical atom. 

Since the zero-flux boundary condition (eqn (8.109)) is also satisfied by an atom, that is, by a quantum subsystem, the atomic action and Lagrangian integrals vanish as they do for a total closed system. Indeed, one may view the vanishing of the action integral over some total system as being the result of the action integral vanishing separately over the space of each atom in the system. 

As a system in a given quantum state changes and evolves with time under the action of the generator — HSt, the atomic surfaces evolve in a continuous manner and the property of exhibiting a zero flux in Vp(r) is continuously maintained. Thus, the atomic action integral will always vanish... 

The condition, started in eqn (8.113), that the atomic action integral vanish for all time intervals may be taken as the quantum definition of an atom. It is a direct consequence of the topological definition of an atom as the union of... 

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... 

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. 

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. 

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... 

Hamiltonian energy integral atomic action integral, eqn (8.111) atomic Lagrange integral, eqn (8.110) action integral for external fields, eqn (8.210)... 

Like all atomic properties, summation of E (Q T(Q) or 1T(Q) over all the atoms in a molecule yields the corresponding molecular average. This is a direct consequence of the mode of definition of the action integral for an atom in the extension of ScWinger s... 

The beauty of the above topological definition of the atom in a molecule lies in the fact that it coincides with the rigorous quantum mechanical definition of an open subsystem [27, 33, 34]. In particular, the atomic action integral, which is defined through the atomic one-particle Lagrangian density, is zero within the atomic volume ... 

For electronic transitions in electron-atom and heavy-particle collisions at high unpact energies, the major contribution to inelastic cross sections arises from scattering in the forward direction. The trajectories implicit in the action phases and set of coupled equations can be taken as rectilinear. The integral representation... 

Carver MB, Hanlet DV, Chapin KR. MAKSIMA-CHEMIST, A program for mass action kinetic simulated manipulation and integration using stiff techniques, Chalk River Nuclear Laboratories Report, Atomic Energy Canada Ltd. 6413, 1979 1-28. 

---