Toroidal flow, see Fig. 7.32, is characterized by remarkable properties: vanishing
induced magnetic dipole moment, induced magnetic field with the shape of a
topological circumference confined inside the torus surface, and induced anapole
moment [119]. The presence of TVs in maps of J
B has been detected in some
molecules [103, 120–122].
The anapole moment can be expressed [118] via the current density J
B induced
by a uniform magnetic field B,
A a ¼ À
1
6
Z
ðr
2
d ab À r a r b ÞJ
B
b d
3 r:
ð7:84Þ
The anapole susceptibility is a nonsymmetric second-rank tensor [123, 124]
defined by
a ab ¼
@A a
@B b
:
ð7:85Þ
Fig. 7.31 Isoshielding
density surfaces R
I
zz ðrÞ ¼
0:0 au (represented in grey)
for any dummy atom I lying
on the C 5 symmetry axis of
pentaprismane in a magnetic
field BkC 5 . Vortex and saddle
SLs of the SG in Fig. 7.28 lie
on this surface, which
separates the inner part of the
J
B field from the peripheral
region of delocalized currents
7 Topology of Quantum Mechanical Current Density …
207