Extending the Topological Analysis and Seeking the Real-Space Subsystems in Non-Coulombic Systems with Homogeneous Potential Energy Functions

Abstract

It is customary to conceive the interactions of all the constituents of a molecular system, i.e. electrons and nuclei, as Coulombic. However, in a more detailed analysis one may always find small but non-negligible non-Coulombic interactions in molecular systems originating from the finite size of nuclei, magnetic interactions, etc. While such small modifications of the Coulombic interactions do not seem to alter the nature of a molecular system in real world seriously, they are a serious obstacle for quantum chemical theories and methodologies which their formalism is strictly confined to the Coulombic interactions. Although the quantum theory of atoms in molecules (QTAIM) has been formulated originally for the Coulombic systems, some recent studies have demonstrated that most of its theoretical ingredients are not sensitive to the explicit form of the potential energy operator. However, the Coulombic interactions have been explicitly assumed in the mathematical procedure that is used to introduce the basin energy of an atom in a molecule. In this study it is demonstrated that the mathematical procedure may be extended to encompass the set of the homogeneous potential energy functions thus relegating adherence to the Coulombic interactions to introduce the energy of a real-space subsystem. On the other hand, this extension opens the door for seeking novel real-space subsystems, apart from atoms in molecules, in non-Coulombic systems. These novel real-space subsystems, quite different from the atoms in molecules, call for an extended formalism that goes beyond the orthodox QTAIM. Accordingly, based on a previous proposal the new formalism, which is not confined to the Coulombic systems nor to the atoms in molecules as the sole real-space subsystems, is termed the quantum theory of proper open subsystems (QTPOS) and its potential applications are detailed. The harmonic trap model, containing non-interacting fermions or bosons, is considered as an example for the QTPOS analysis. The QTPOS analysis of the bosonic systems is particularly quite unprecedented not attempted before.