between different canonical forms of the Jacobian [15]. Since a (0, 0) point
corresponds to a transition between pure imaginary and pure real eigenvalues,
the branching must necessarily take place at an SP characterized by three zero
eigenvalues [55, 56].
The reason for the denomination “branching point” used for (0, 0) singularities is
easily understood. Consider, for instance, a molecule of D nh symmetry, in the
singlet electronic ground state, in the presence of a magnetic field B along the
highest symmetry axis C n . According to the general analysis of Sect. 7.5.2, in the
outer reaches of the molecular domain, the induced electronic current density is
diamagnetic, that is, it flows in planes at right angles to B, like the Larmor current
that takes place in atoms. In the proximity of the North and South poles, at large
distance from the molecular plane, the diamagnetic regime is represented by the
primary (2, 0) vortical stagnation line parallel to B and C n . Transition to different
regimes, e.g., from vortex to saddle flow, or vice versa, takes place closer to the
centre of charge, i.e., in the regions of higher electron density qðrÞ, where the
primary vortex line may split up into saddle and vortical lines.
The splitting of a SL into several SLs is regulated by a fundamental topological
theorem proved by Gomes [55–57, 93] in the form of an index conservation constraint. Recall that, according to footnote 3, the index of a saddle (vortex) line is −1
(+1). When an SL of index i 0 splits into m new lines, the sum of the indices of the
SLs which emerge from the branching point must satisfy the condition
X m
k¼1
i k ¼ i 0 :
ð7:59Þ
For instance, a vortex line may bifurcate giving rise to two new vortex lines and
one saddle line. This bifurcation conserves the total index +1.
7.3.1 The Gomes Flow
A simple model of velocity vector field exhibiting branching of vortex and saddle
stagnation lines has been considered by Gomes [56] via the system of differential
equations
_
x ¼ y
3
þ yðz
2
À z À 2Þ
_
y ¼ Àx
3
À xðz
2
þ z À 2Þ
_
z ¼ 0
8
<
:
ð7:60Þ
describing trajectories everywhere parallel to the xy plane, shown in Figs. 7.1 and
7.2 for some z values. They display the change of regime occurring in different
7 Topology of Quantum Mechanical Current Density …
167
corresponds to a transition between pure imaginary and pure real eigenvalues,
the branching must necessarily take place at an SP characterized by three zero
eigenvalues [55, 56].
The reason for the denomination “branching point” used for (0, 0) singularities is
easily understood. Consider, for instance, a molecule of D nh symmetry, in the
singlet electronic ground state, in the presence of a magnetic field B along the
highest symmetry axis C n . According to the general analysis of Sect. 7.5.2, in the
outer reaches of the molecular domain, the induced electronic current density is
diamagnetic, that is, it flows in planes at right angles to B, like the Larmor current
that takes place in atoms. In the proximity of the North and South poles, at large
distance from the molecular plane, the diamagnetic regime is represented by the
primary (2, 0) vortical stagnation line parallel to B and C n . Transition to different
regimes, e.g., from vortex to saddle flow, or vice versa, takes place closer to the
centre of charge, i.e., in the regions of higher electron density qðrÞ, where the
primary vortex line may split up into saddle and vortical lines.
The splitting of a SL into several SLs is regulated by a fundamental topological
theorem proved by Gomes [55–57, 93] in the form of an index conservation constraint. Recall that, according to footnote 3, the index of a saddle (vortex) line is −1
(+1). When an SL of index i 0 splits into m new lines, the sum of the indices of the
SLs which emerge from the branching point must satisfy the condition
X m
k¼1
i k ¼ i 0 :
ð7:59Þ
For instance, a vortex line may bifurcate giving rise to two new vortex lines and
one saddle line. This bifurcation conserves the total index +1.
7.3.1 The Gomes Flow
A simple model of velocity vector field exhibiting branching of vortex and saddle
stagnation lines has been considered by Gomes [56] via the system of differential
equations
_
x ¼ y
3
þ yðz
2
À z À 2Þ
_
y ¼ Àx
3
À xðz
2
þ z À 2Þ
_
z ¼ 0
8
<
:
ð7:60Þ
describing trajectories everywhere parallel to the xy plane, shown in Figs. 7.1 and
7.2 for some z values. They display the change of regime occurring in different
7 Topology of Quantum Mechanical Current Density …
167