Towards the development of the quantum mechanics of a subspace

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Abstract
The development of the quantum mechanics for a subspace of a total system is pursued via the derivation or introduction of the following: (1) the variational principle for a subspace; (2) the relationship between the time dependent subspace average of an operator and its associated commutator; (3) the unique set of quantum properties found for a subspace bounded by a surface of zero flux as defined by ∇ρ (r) ⋅n (r) =0, ‐r‐S (r), and denoted as a ’’virial fragment’’; (4) the special variational properties of the zero‐flux boundaries in one‐electron systems; (5) operators which cause the energy integral to be stationary over a subspace; (6) the fragment virial theorem; (7) the importance of the scaling of the coordinates of a single electron in the determination of subspace properties; (8) a reformulation of the total Hamiltonian for a many‐particle system as a sum of single‐particle Hamiltonians through the use of a complete set of virial sharing operators; (9) new interpretation of the energy of the equilibrium geometry of a fixed‐nucleus Hamiltonian and of the kinetic and potential energies in the molecular virial theorem as purely electronic; (10) a proof that the energy for a molecule in an equilibrium geometry is a sum of one‐electron contributions; (11) the crucial nature of the principle stated in (10) for the definition of the energy of a subspace; (12) introduction of the principle of equal sharing of the energy of interaction between identical particles into the definition of a subspace energy; (13) a proof of the nonpartitionability of an atom based on the principle outlined in (12); (14) discussion of the information required to define a fragment and its energy. As a consequence of the above theorems and results there emerges a unique definition of a unit cell in a solid, of an atomiclike fragment in a molecule and of a molecule in a system of interacting molecules, for these are the subsystems bounded by surfaces of zero flux.

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