The TV looks like a doughnut with a very small central hole, completely encased
within an S
2 surface, that is, a separatrix with the shape of a topological sphere,
which forms a boundary marking the limits to the rest of the vector field. This
separatrix is filled by asymptotic paths, referred to as homoclinic trajectories [57],
see Fig. 7.35. In the streamline plot of Fig. 7.36, the toroidal flow is represented by
two juxtaposed vortices, one diamagnetic and one paramagnetic. Their centres are
found at the intersection of the green and red SLs of the stagnation loop of the TV
with the r h ðyzÞ plane.
Fig. 7.34 Perspective view of the toroidal flow in the basin of the Li atom within the LiH
molecule in a magnetic field perpendicular to the bond. All the streamlines flow through the centre
and down around the sides of the stagnation loop containing one green and one red segment. The
ð3; þ 1Þ saddle-node, observed in the foreground as a source, is connected by black trajectories to
its conjugated ð3; À1Þ (sink) partner lying behind the torus. A blue closed asymptotic line defines
the intersection of the separatrix containing the torus with the plane of the nuclei perpendicular to
the applied field. On this plane, the ð3; Æ1Þ points look like saddles. The other homoclinic blue
line joining the ð3; Æ1Þ points lies on a plane normal to the bond axis. A wavy asymptotic line,
which connects the ð3; Æ1Þ points passing inside the stagnation loop, is best observed in Fig. 7.35
210
P. Lazzeretti
within an S
2 surface, that is, a separatrix with the shape of a topological sphere,
which forms a boundary marking the limits to the rest of the vector field. This
separatrix is filled by asymptotic paths, referred to as homoclinic trajectories [57],
see Fig. 7.35. In the streamline plot of Fig. 7.36, the toroidal flow is represented by
two juxtaposed vortices, one diamagnetic and one paramagnetic. Their centres are
found at the intersection of the green and red SLs of the stagnation loop of the TV
with the r h ðyzÞ plane.
Fig. 7.34 Perspective view of the toroidal flow in the basin of the Li atom within the LiH
molecule in a magnetic field perpendicular to the bond. All the streamlines flow through the centre
and down around the sides of the stagnation loop containing one green and one red segment. The
ð3; þ 1Þ saddle-node, observed in the foreground as a source, is connected by black trajectories to
its conjugated ð3; À1Þ (sink) partner lying behind the torus. A blue closed asymptotic line defines
the intersection of the separatrix containing the torus with the plane of the nuclei perpendicular to
the applied field. On this plane, the ð3; Æ1Þ points look like saddles. The other homoclinic blue
line joining the ð3; Æ1Þ points lies on a plane normal to the bond axis. A wavy asymptotic line,
which connects the ð3; Æ1Þ points passing inside the stagnation loop, is best observed in Fig. 7.35
210
P. Lazzeretti