Hartree-Fock level of accuracy, to obtain the stagnation graph of D nh ðC nh Þ compounds. The third-order linear autonomous system for the flow was integrated using
Runge-Kutta procedures [108].
Stagnation graphs of unsaturated C n H n hydrocarbons in a magnetic field
orthogonal to the r h molecular plane are displayed in Fig. 7.20. It can be observed
that the SGs of the J
B
field, characterized by the symmetry elements of the magnetic groups D nh ðC nh Þ, show common features. In the outer reaches of the
molecular domain the flow is diamagnetic. It is represented in Fig. 7.20 by a green
open vortical line, coinciding with the highest symmetry axis and extending to the
boundaries of configuration space. A pair of (0, 0) transition points, at the same
distance above and below the plane of the carbon nuclei, is found for all systems.
The distance of the (0, 0) SPs increases from ≈1.3 bohr for n = 3 to ≈2.0 bohr for
n = 4, and to ≈2.5 bohr for n = 5. Approximately the same values were found for
higher n, i.e., ≈2.5 bohr for n = 6, ≈2.4 bohr for n = 7, and ≈2.13 bohr for n = 8.
The 2n + 1 SLs originating at the north (0, 0) point merge at the other; n of the
southbound lines are saddle lines and n are diamagnetic vortical lines. In Fig. 7.20,
the former are represented in blue, the latter in green. Each diamagnetic vortex
crosses the molecular plane in the region of a C–C bond, close to its midpoint, each
saddle line passes through a point of r h in the proximity of the carbon nucleus
[104]. The conservation condition (Gomes theorem) [55–57, 93], Eq. (7.59), is
satisfied. All the SLs lie on a topological surface with the shape of an oval ball,
referred to as separatrix [56], encasing domains of localized flow and a central
paramagnetic vortex represented by a red line in Fig. 7.20. All over this surface, the
element of electric current, Eq. (7.44), dI
B
¼ J
B
Á ds vanishes, since no current
density streamline can cross the separatrix.
Quite remarkably, the topological index of each of the 2n SLs constituting the
skeleton of the separatrix does not stay the same on moving south (north) from the
north (south) branching point, which can be observed in Fig. 7.20 as a colour
change corresponding to a change of local regime—from saddle to vortex and
viceversa. The change takes place at points lying on a same plane for every SL. One
can ask if such a change implies a violation of the index conservation theorem,
Eq. (7.59). The question is answered in the negative if it is assumed that, for each
SL, the scenario is modelled by the pitchfork bifurcations of Eqs. (7.69) and (7.70),
visualized in Fig. 7.6.
The corresponding maps for the streamlines of the current density flowing on the
r h plane displayed in Fig. 7.21 show patterns of vortex and saddle regime which are
elucidated by the SGs of Fig. 7.20. The existence of a central paramagnetic vortex,
documented for the first time in 1982 by Lazzeretti and Zanasi [109], is the distinctive feature of all diatropic systems with D nh ðC nh Þ magnetic symmetry. It must
exist as required by the index conservation constraint proven by Gomes, Eq. (7.59)
[55–57, 93]. The analysis of proton magnetic shielding in cyclic compounds [110]
is consistent with these results.
7 Topology of Quantum Mechanical Current Density …
193
Runge-Kutta procedures [108].
Stagnation graphs of unsaturated C n H n hydrocarbons in a magnetic field
orthogonal to the r h molecular plane are displayed in Fig. 7.20. It can be observed
that the SGs of the J
B
field, characterized by the symmetry elements of the magnetic groups D nh ðC nh Þ, show common features. In the outer reaches of the
molecular domain the flow is diamagnetic. It is represented in Fig. 7.20 by a green
open vortical line, coinciding with the highest symmetry axis and extending to the
boundaries of configuration space. A pair of (0, 0) transition points, at the same
distance above and below the plane of the carbon nuclei, is found for all systems.
The distance of the (0, 0) SPs increases from ≈1.3 bohr for n = 3 to ≈2.0 bohr for
n = 4, and to ≈2.5 bohr for n = 5. Approximately the same values were found for
higher n, i.e., ≈2.5 bohr for n = 6, ≈2.4 bohr for n = 7, and ≈2.13 bohr for n = 8.
The 2n + 1 SLs originating at the north (0, 0) point merge at the other; n of the
southbound lines are saddle lines and n are diamagnetic vortical lines. In Fig. 7.20,
the former are represented in blue, the latter in green. Each diamagnetic vortex
crosses the molecular plane in the region of a C–C bond, close to its midpoint, each
saddle line passes through a point of r h in the proximity of the carbon nucleus
[104]. The conservation condition (Gomes theorem) [55–57, 93], Eq. (7.59), is
satisfied. All the SLs lie on a topological surface with the shape of an oval ball,
referred to as separatrix [56], encasing domains of localized flow and a central
paramagnetic vortex represented by a red line in Fig. 7.20. All over this surface, the
element of electric current, Eq. (7.44), dI
B
¼ J
B
Á ds vanishes, since no current
density streamline can cross the separatrix.
Quite remarkably, the topological index of each of the 2n SLs constituting the
skeleton of the separatrix does not stay the same on moving south (north) from the
north (south) branching point, which can be observed in Fig. 7.20 as a colour
change corresponding to a change of local regime—from saddle to vortex and
viceversa. The change takes place at points lying on a same plane for every SL. One
can ask if such a change implies a violation of the index conservation theorem,
Eq. (7.59). The question is answered in the negative if it is assumed that, for each
SL, the scenario is modelled by the pitchfork bifurcations of Eqs. (7.69) and (7.70),
visualized in Fig. 7.6.
The corresponding maps for the streamlines of the current density flowing on the
r h plane displayed in Fig. 7.21 show patterns of vortex and saddle regime which are
elucidated by the SGs of Fig. 7.20. The existence of a central paramagnetic vortex,
documented for the first time in 1982 by Lazzeretti and Zanasi [109], is the distinctive feature of all diatropic systems with D nh ðC nh Þ magnetic symmetry. It must
exist as required by the index conservation constraint proven by Gomes, Eq. (7.59)
[55–57, 93]. The analysis of proton magnetic shielding in cyclic compounds [110]
is consistent with these results.
7 Topology of Quantum Mechanical Current Density …
193