An interesting comparison is offered by the prismane molecule, a polycyclic
hydrocarbon with formula C 6 H 6 , whose carbon and hydrogen atoms are arranged in
the shape of a six-atom triangular prism. The magnetic symmetry is the same as
cyclopropane’s D 3h ðC 3h Þ. It is a valence isomer of benzene, far less stable than the
archetypal aromatic molecule. A spatial ring current model for this molecule has
been reported [115].
7.5.5 Cubane and Pentaprismane
Cubane is a Platonic hydrocarbon with chemical formula C 8 H 8 , possessing octahedral O h symmetry. Its special magnetic properties have been investigated by
some authors, whose work was reviewed in a recent paper [116]. A compact spatial
model for the electronic current density vector field induced in the cubane molecule
by a magnetic field perpendicular to a face has been proposed to interpret its
magnetic response.
A perspective view of the SG of the cubane molecule is shown in Fig. 7.25. It
conveys the essential information needed to understand the maps the J
B
field,
Figs. 7.26 and 7.27. The central vortical SL coincides with the C 4 ðzÞ symmetry
axis. Branching of this SL takes place at a pair of far-off critical points, where
transition occurs from diamagnetic flow in the tail regions to paratropic regime.
The central paramagnetic axial vortex is represented in Fig. 7.25 by a red line
extending to great lengths in the direction of the B field. Four green diamagnetic
vortical lines lie on the Tr d planes intersecting along the z axis and containing
HCCH bonds, and four blue saddle SLs are situated on the Tr v planes bisecting CC
bonds. The Gomes theorem, Eq. (7.59), is obeyed: by counting the i indices of the
SLs one has +5 − 4 = +1, i.e., the overall circulation is vortical.
On crossing the faces of the carbon cage parallel to the xy plane, short green
segments, representing diamagnetic vortical flow embedded in the blue saddle SLs,
are observed. This feature is similar to that observed in aromatic compounds
examined in Sect. 7.5.2 and it can tentatively be interpreted in terms of pitchfork
bifurcations taking place on a same plane parallel to r h .
Maps of J
B
field, Figs. 7.26 and 7.27, on planes orthogonal to B constitute a
basic model useful to describe the magnetic properties of cubane. They show the
diamagnetic circulation in the tail regions of the molecule, typical of all diamagnetic
systems, and the set of vortex and saddle SPs at the intersection of corresponding
SLs with the plot plane.
To sum up, the magnetic response of the C 8 H 8 molecule to a magnetic field
applied at right angles to a face of the cube is characterized by a strong paratropic
AV flowing inside the lateral surface of the cube. The intensity of the paratropic
ring currents sustained by r-electrons in cubane is higher than that induced in the pelectrons of benzene. Therefore, even if overall diamagnetism prevails, as proven
200
P. Lazzeretti
hydrocarbon with formula C 6 H 6 , whose carbon and hydrogen atoms are arranged in
the shape of a six-atom triangular prism. The magnetic symmetry is the same as
cyclopropane’s D 3h ðC 3h Þ. It is a valence isomer of benzene, far less stable than the
archetypal aromatic molecule. A spatial ring current model for this molecule has
been reported [115].
7.5.5 Cubane and Pentaprismane
Cubane is a Platonic hydrocarbon with chemical formula C 8 H 8 , possessing octahedral O h symmetry. Its special magnetic properties have been investigated by
some authors, whose work was reviewed in a recent paper [116]. A compact spatial
model for the electronic current density vector field induced in the cubane molecule
by a magnetic field perpendicular to a face has been proposed to interpret its
magnetic response.
A perspective view of the SG of the cubane molecule is shown in Fig. 7.25. It
conveys the essential information needed to understand the maps the J
B
field,
Figs. 7.26 and 7.27. The central vortical SL coincides with the C 4 ðzÞ symmetry
axis. Branching of this SL takes place at a pair of far-off critical points, where
transition occurs from diamagnetic flow in the tail regions to paratropic regime.
The central paramagnetic axial vortex is represented in Fig. 7.25 by a red line
extending to great lengths in the direction of the B field. Four green diamagnetic
vortical lines lie on the Tr d planes intersecting along the z axis and containing
HCCH bonds, and four blue saddle SLs are situated on the Tr v planes bisecting CC
bonds. The Gomes theorem, Eq. (7.59), is obeyed: by counting the i indices of the
SLs one has +5 − 4 = +1, i.e., the overall circulation is vortical.
On crossing the faces of the carbon cage parallel to the xy plane, short green
segments, representing diamagnetic vortical flow embedded in the blue saddle SLs,
are observed. This feature is similar to that observed in aromatic compounds
examined in Sect. 7.5.2 and it can tentatively be interpreted in terms of pitchfork
bifurcations taking place on a same plane parallel to r h .
Maps of J
B
field, Figs. 7.26 and 7.27, on planes orthogonal to B constitute a
basic model useful to describe the magnetic properties of cubane. They show the
diamagnetic circulation in the tail regions of the molecule, typical of all diamagnetic
systems, and the set of vortex and saddle SPs at the intersection of corresponding
SLs with the plot plane.
To sum up, the magnetic response of the C 8 H 8 molecule to a magnetic field
applied at right angles to a face of the cube is characterized by a strong paratropic
AV flowing inside the lateral surface of the cube. The intensity of the paratropic
ring currents sustained by r-electrons in cubane is higher than that induced in the pelectrons of benzene. Therefore, even if overall diamagnetism prevails, as proven
200
P. Lazzeretti