  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 apart from basin energy of an atom in a molecule, its
theoretical ingredients are not sensitive to the explicit form of the potential
energy operator. In this study, it is demonstrated that the basin energy may be
defined not only for coulombic systems but for all real-space subsystems of
those systems that are described by any member of the set of the homogeneous
potential energy functions. 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 call for an extended
formalism that goes beyond the orthodox QTAIM, which is not confined to the
Coulombic systems nor to the atoms in molecules as the sole real-space
subsystems. It is termed the quantum theory of real-space open subsystems
(QTROS) and its potential applications are detailed. The harmonic trap model,
containing non-interacting fermions or bosons, is considered as an example for
the QTROS analysis. The QTROS analysis of bosonic systems is particularly quite
unprecedented, not attempted before.
