Atomic action and Lagrangian integrals

The subsystem or atomic Lagrangian integral is defined by the standard mode of integration used in the definition of the subsystem functional [ F, iJ] for a stationary state and for the definition of subsystem properties. 

When T and T satisfy Schrodinger s equations, the atomic Lagrangian integral reduces to 

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 

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