with a direction and magnitude given by the vector (−x, 1−x
2
−y
2 ). The resulting
trajectories form the phase flow in Fig. 2.6, which is reminiscent of the gradient
vector field of the electron density of a diatomic molecule with one nucleus shown
(at the attractor critical point (0, 1)). The second critical point shown lies at point (0,
−1) and clearly has a qualitatively different flow pattern locally. This critical point
is both a maximum and a minimum, depending on the direction of approach to it,
and is reminiscent of a bond critical point. The two dashed trajectories lines that
terminate at this critical point form a separatrix, which is reminiscent of an interatomic surface.
A non-exhaustive list of quantum mechanical functions that have hitherto been
partitioned includes the electron density ρ(r) (the analysis of which started with
Ref. [19]), its Laplacian ∇
2 ρ(r) (started off with Refs. [20, 21] and studied for the
first time in terms of the full topology in Refs. [13, 22, 23]), the nuclear potential
V nuc (r) (studied [24] already in 1980 but the first elaborate and self-contained study
[10] appeared only 30 years later), the electron localization function (ELF) [25]
(started with Ref. [26] and reviewed in Ref. [27]), the electrostatic potential [28]
(started with thorough but stubbornly named “topographic” instead of topological
studies [29, 30] and continued with more modern work [31–34]), the virial field
[35], the magnetically induced molecular current distributions (started with [36]),
the intracule density (started with Ref. [37]), the Ehrenfest force field (topology first
investigated [38] in 2012 and then improved [39] in 2015), and finally the topological energy partitioning (Coulomb potential energy partitioning started with [40]
and culminated into the theory of Interacting Quantum Atoms (IQA) [41] (see
below), leading to energetic underpinning for the topological expression of
chemical bonding [42]) By bundling all these QCT studies under the umbrella of
the topology, the combined method is strengthened and can start competing with
the more traditional interpretative method of quantum chemistry [43–49]. This
competition should be seen in the light of falsification.
At the end of Sect. 2.2 it is useful to pause and muse about the character of the
topology as an instrument to study Nature. The language of dynamical systems,
which is rooted in topology, is at the heart of QCT. A hallmark of QCT’s partitioning is its binary character: a point in space belongs to a QCT subspace or not.
Whereas non-QCT approaches allow for more gradual transitions from one subspace (e.g. an atom) to another, QCT works with step functions. The 3D step
function defines a finite-volume subspace that remains well defined under (possibly
large) geometrical deformations. Topology does allow for large deformations in the
geometry of the objects it defines while still characterising them by the same
invariant measures. However, once beyond a certain degree of deformation, the
topological object changes. The suddenness of this change makes some researchers
uncomfortable. The comfort of a gradual change may look appealing but then one
can ask if this is a false comfort. Can a world view with only gradual change make
any clear decision on what is A and what is B? Or should one not care about being
able to make this decision? Or can one make the decision at the price of introducing
a parameter? But is then the problem of state allocation (i.e. making the aforementioned decision) not simply deferred to fixing a parameter value?
2 On Quantum Chemical Topology
33
2
−y
2 ). The resulting
trajectories form the phase flow in Fig. 2.6, which is reminiscent of the gradient
vector field of the electron density of a diatomic molecule with one nucleus shown
(at the attractor critical point (0, 1)). The second critical point shown lies at point (0,
−1) and clearly has a qualitatively different flow pattern locally. This critical point
is both a maximum and a minimum, depending on the direction of approach to it,
and is reminiscent of a bond critical point. The two dashed trajectories lines that
terminate at this critical point form a separatrix, which is reminiscent of an interatomic surface.
A non-exhaustive list of quantum mechanical functions that have hitherto been
partitioned includes the electron density ρ(r) (the analysis of which started with
Ref. [19]), its Laplacian ∇
2 ρ(r) (started off with Refs. [20, 21] and studied for the
first time in terms of the full topology in Refs. [13, 22, 23]), the nuclear potential
V nuc (r) (studied [24] already in 1980 but the first elaborate and self-contained study
[10] appeared only 30 years later), the electron localization function (ELF) [25]
(started with Ref. [26] and reviewed in Ref. [27]), the electrostatic potential [28]
(started with thorough but stubbornly named “topographic” instead of topological
studies [29, 30] and continued with more modern work [31–34]), the virial field
[35], the magnetically induced molecular current distributions (started with [36]),
the intracule density (started with Ref. [37]), the Ehrenfest force field (topology first
investigated [38] in 2012 and then improved [39] in 2015), and finally the topological energy partitioning (Coulomb potential energy partitioning started with [40]
and culminated into the theory of Interacting Quantum Atoms (IQA) [41] (see
below), leading to energetic underpinning for the topological expression of
chemical bonding [42]) By bundling all these QCT studies under the umbrella of
the topology, the combined method is strengthened and can start competing with
the more traditional interpretative method of quantum chemistry [43–49]. This
competition should be seen in the light of falsification.
At the end of Sect. 2.2 it is useful to pause and muse about the character of the
topology as an instrument to study Nature. The language of dynamical systems,
which is rooted in topology, is at the heart of QCT. A hallmark of QCT’s partitioning is its binary character: a point in space belongs to a QCT subspace or not.
Whereas non-QCT approaches allow for more gradual transitions from one subspace (e.g. an atom) to another, QCT works with step functions. The 3D step
function defines a finite-volume subspace that remains well defined under (possibly
large) geometrical deformations. Topology does allow for large deformations in the
geometry of the objects it defines while still characterising them by the same
invariant measures. However, once beyond a certain degree of deformation, the
topological object changes. The suddenness of this change makes some researchers
uncomfortable. The comfort of a gradual change may look appealing but then one
can ask if this is a false comfort. Can a world view with only gradual change make
any clear decision on what is A and what is B? Or should one not care about being
able to make this decision? Or can one make the decision at the price of introducing
a parameter? But is then the problem of state allocation (i.e. making the aforementioned decision) not simply deferred to fixing a parameter value?
2 On Quantum Chemical Topology
33