transition occurs from diamagnetic vortical flow in the outer reaches of the molecule to the internal region of the current density field, enclosed within the grey
surface observable in Fig. 7.31. This surface is a two-dimensional manifold of
points r at which the zz component of the shielding density, Eq. (7.56),
X I
zz
ðrÞ ¼ À
l 0
4p
zbc
r b À R I b
jr À R I j
3
J
B z
c ðrÞ
ð 7:83Þ
vanishes for any dummy atom I all along the C 5 symmetry axis, in a magnetic field
B parallel to it. It surrounds the central paratropic vortex, indicated by a red line,
and five diatropic vortices indicated by green lines, separated by the five saddle blue
lines, but it does not coincide with the separatrix. The line branching of Fig. 7.28 is
consistent with the Gomes theorem, Eq. (7.59). Planar visualisations of the density
function, Eq. (7.83), for I ¼ H, are displayed in Fig. 7.30.
Fig. 7.29 Perspective view of the J
B field in pentaprismane in a uniform external magnetic field
B ¼ B z e 3 , of magnitude 1 au, perpendicular to the pentagonal faces. Only current densities with
jJ
B j between 0.05 (blue arrows) and 0.1 au (red arrows) are plotted. The figure shows the
peripheral regions of delocalized diatropic flow, the central paratropic vortex and the local
diatropic vortices about the midpoint of the C–C bonds
7 Topology of Quantum Mechanical Current Density …
205
surface observable in Fig. 7.31. This surface is a two-dimensional manifold of
points r at which the zz component of the shielding density, Eq. (7.56),
X I
zz
ðrÞ ¼ À
l 0
4p
zbc
r b À R I b
jr À R I j
3
J
B z
c ðrÞ
ð 7:83Þ
vanishes for any dummy atom I all along the C 5 symmetry axis, in a magnetic field
B parallel to it. It surrounds the central paratropic vortex, indicated by a red line,
and five diatropic vortices indicated by green lines, separated by the five saddle blue
lines, but it does not coincide with the separatrix. The line branching of Fig. 7.28 is
consistent with the Gomes theorem, Eq. (7.59). Planar visualisations of the density
function, Eq. (7.83), for I ¼ H, are displayed in Fig. 7.30.
Fig. 7.29 Perspective view of the J
B field in pentaprismane in a uniform external magnetic field
B ¼ B z e 3 , of magnitude 1 au, perpendicular to the pentagonal faces. Only current densities with
jJ
B j between 0.05 (blue arrows) and 0.1 au (red arrows) are plotted. The figure shows the
peripheral regions of delocalized diatropic flow, the central paratropic vortex and the local
diatropic vortices about the midpoint of the C–C bonds
7 Topology of Quantum Mechanical Current Density …
205