A Note on the Initial Condition of the Differential Equation Which Defines Proper Quantum Topological Subspaces

Abstract

We analyze Bader’s variational procedure (BVP) to define Proper Quantum Topological Subspaces (PQTS) in multiatomic systems (MAS). In particular, we explicitly treat the problem of the initial condition of the partial differential equation resulting from BVP and defining the boundary of PQTS. We show that in general for MAS where nuclei correspond to spherical regions of constant (uniform) distribution the proper initial condition associated with the equation defining single atoms automatically emerges from the fundamental physical and mathematical hypothesis on which BVP is based. Our results justify the uniqueness of Bader’s partition of atoms in molecules on the basis of an a priori mathematical argument implicitly contained in the theory rather than on an a posteriori chemical one as done so far.