applicable to these systems. Therefore, there is a real demand to extend the formalism of the orthodox QTAIM to non-Coulombic systems.
The programme of reconsidering the QTAIM formalism in the case of
non-Columbic interactions was started sometime ago and it was demonstrated that
the subsystem variational procedure and the subsystem hypervirial theorem are both
insensitive to the nature of the potential energy operator as far as there is a bound
quantum state in the system [40, 41]. This is also true for the local zero-flux
equation of the one-particle density which is the equation of deriving the
inter-atomic surfaces for both the Coulombic and non-Coulombic systems [40, 41].
However, upon considering the Hookean molecules, i.e. model systems where
some of the Coulombic interactions have been replaced with the harmonic potential,
it emerged that the AIM structures derived from the topological analysis were not
the one expected based on “chemical intuition”, which is routed in previous
experiences with the Coulombic systems [41]. Thus, the use of topological analysis
and the local zero-flux equation do not automatically guarantee that the emerging
“real-space” subsystems are the usual AIM, also called topological atoms. In present study more examples of exotic real-space subsystems in non-Coulombic systems are presented.
In contrast to the previous studies [40, 41], the focus of this contribution is on
the part of the QTAIM formalism that is sensitive to the nature of the potential
energy operator namely, the basin energy of an atom in a molecule [1].
Accordingly, the definition of the basin energy is extended beyond the Coulombic
potential energy function demonstrating that for the subset of homogeneous
potential energy functions the regional virial theorem may be used to derive
well-defined, origin-independent, basin energies.
4.2 The Generalized Subsystem Virial
Theorem for the Homogeneous Potential
Energy Functions
The atomic/regional theorems, emerging from the subsystem hypervirial theorem
[9, 42, 43], are insensitive to details of the potential energy operator and are true as
far as a system is composed of a single type of quantum particles and there is a
bound stationary state emerging from the interaction of quantum particles with each
other and the external fields. This insensitivity is compelling since the orthodox
formalism may be employed with least modifications for non-Coulombic systems
however the regional/basin energies have been derived employing explicitly the
properties of the Coulombic potential (see particularly Sect. 6.3 in [1]). In the
present section the very definition of the basin energy is extended to include the set
of the homogeneous potential energy functions (for an elementary discussion on the
homogeneous potential energy functions see Chap. 14 in [44]).
4 Extending the Topological Analysis and Seeking the Real-Space …
91
The programme of reconsidering the QTAIM formalism in the case of
non-Columbic interactions was started sometime ago and it was demonstrated that
the subsystem variational procedure and the subsystem hypervirial theorem are both
insensitive to the nature of the potential energy operator as far as there is a bound
quantum state in the system [40, 41]. This is also true for the local zero-flux
equation of the one-particle density which is the equation of deriving the
inter-atomic surfaces for both the Coulombic and non-Coulombic systems [40, 41].
However, upon considering the Hookean molecules, i.e. model systems where
some of the Coulombic interactions have been replaced with the harmonic potential,
it emerged that the AIM structures derived from the topological analysis were not
the one expected based on “chemical intuition”, which is routed in previous
experiences with the Coulombic systems [41]. Thus, the use of topological analysis
and the local zero-flux equation do not automatically guarantee that the emerging
“real-space” subsystems are the usual AIM, also called topological atoms. In present study more examples of exotic real-space subsystems in non-Coulombic systems are presented.
In contrast to the previous studies [40, 41], the focus of this contribution is on
the part of the QTAIM formalism that is sensitive to the nature of the potential
energy operator namely, the basin energy of an atom in a molecule [1].
Accordingly, the definition of the basin energy is extended beyond the Coulombic
potential energy function demonstrating that for the subset of homogeneous
potential energy functions the regional virial theorem may be used to derive
well-defined, origin-independent, basin energies.
4.2 The Generalized Subsystem Virial
Theorem for the Homogeneous Potential
Energy Functions
The atomic/regional theorems, emerging from the subsystem hypervirial theorem
[9, 42, 43], are insensitive to details of the potential energy operator and are true as
far as a system is composed of a single type of quantum particles and there is a
bound stationary state emerging from the interaction of quantum particles with each
other and the external fields. This insensitivity is compelling since the orthodox
formalism may be employed with least modifications for non-Coulombic systems
however the regional/basin energies have been derived employing explicitly the
properties of the Coulombic potential (see particularly Sect. 6.3 in [1]). In the
present section the very definition of the basin energy is extended to include the set
of the homogeneous potential energy functions (for an elementary discussion on the
homogeneous potential energy functions see Chap. 14 in [44]).
4 Extending the Topological Analysis and Seeking the Real-Space …
91