Quantum topology of molecular charge distributions. III. The mechanics of an atom in a molecule

The variation of the atomic action integral yields Schrödinger’s equations-of-motion and an atomic statement of the variational principle. The atom and its average properties are defined by the variational principle together with the requirement that they satisfy the Heisenberg equation-of-motion. Schrödinger’s equations define the components of the energy-momentum tensor. The divergence equations satisfied by the spatial components of this tensor, the stress tensor, yield an expression for the resultant force exerted on a single electron at a given point in space and at a given time by the average motion of the other particles in the system. The integration of this force density over the space of an atom yields an equation-of-motion for an atom in a molecule identical to that determined by the generalized variational principle. Thus the action principle yields a local description of the mechanical properties of an atom and a definition of their average values. The force density is expressible in terms of a stress tensor which is a functional of the first-order density matrix. The virial of this force yields the potential energy density of a single electron moving in the average field generated by the motion of the other particles in the system. This enables one to define the total energy density as a functional of the first-order density matrix. The atomic average of this energy density is identical to the average energy of an atom in a molecule as defined through the atomic statement of the variational principle. The energy density defined in this manner has the essential property of predicting that if the distribution of charge for an atom is identical in two different systems or at different sites within a given system (e.g., a solid) then the atom will contribute identical amounts to the total energy in both instances. Matter is manifest in real space through its distribution of charge. Those properties of matter which are extensive are so as a consequence of the periodicity of the charge distribution at the atomic level. The equivalence of the quantum mechanical definition of an atom with that determined by the topological properties of the charge distribution is demonstrated. It is shown that the special properties possessed by the atomic action integral are a direct consequence of the topological definition of an atom as the union of an attractor and its basin. The relationship between the topological and mechanical properties of the charge density is pursued through a study of the topological properties of the first-order density matrix. As demonstrated, the properties of the first-order matrix determine both the topology and the mechanics of the charge density.