A falsifiable definition of “ring currents” [15] as a synonim of delocalized
currents in a molecule may be attempted within the lexicon of topology, delocalized
currents are those flowing in the domain which extends beyond the separatrix
surrounding the nuclear skeleton. This formulation would seem ad hoc for cyclic,
neutral or charged, planar molecules C n H n in the presence of a magnetic field
orthogonal to the molecular plane, i.e., compounds with D nh ðC nh Þ magnetic symmetry, see Sect. 7.5.2, in which the separatrix is defined by the surface containing
n vortical and n saddle stagnation lines: delocalized currents flow on the outside for
a great enough distance to reach the tail regions of the electron cloud. It seems
appropriate also to the case of cyclopropane and prismane discussed in Sect. 7.5.4,
cubane and pentaprismane, Sect. 7.5.5.
However, the same definition would apply to the case of p-electron currents in the
same molecules, in which the separatrix coincides with the single vortical line
through the centre of the molecule. It would be also applicable to diamagnetic atoms,
in which the delocalized flow beyond the nucleus consists of concentric circular
streamlines about a vortical stagnation axis identifiable with the separatrix [60].
7.8 Current Density Induced by a Pair of Magnetic Dipole
Moments and Nuclear Spin-Spin Coupling
The Ramsey theory of indirect nuclear spin-spin coupling [12, 13] can be reformulated in terms of linear superposition of two current density fields, J
m I and J
m J ,
induced in the electrons of a molecule by nuclear magnetic dipoles m I and m J [3,
51, 52, 61]. Graphical representations of the interference pattern within the total
current density vector field, together with corresponding density maps, Eq. (7.57),
are very useful to elucidate coupling pathways and to rationalize the exchange of
spin information between coupled nuclei.
A paradigmatic application has been reported for ethane [16]. It shows that the
Fermi contact contributions to experimental nuclear spin-spin coupling constant are
easy to explain in terms of current densities (7.25), which transport spin polarization along the coupling pathway, and associated plots of property density,
Eq. (7.57). Same-spin electron correlation, the only kind of correlation recovered by
the Hartree Fock wavefunction considered in Ref. [16], determines the alignment of
the nuclear dipoles at its ends, as shown in the current-density maps reported for
ethane, Fig. 7.44.
According to experimental and theoretical results, the magnetic dipoles of the
vicinal protons are anti-parallel, in the configuration of lower energy. Therefore, the
physically acceptable magnetic symmetries are C 2v for the eclipsed, and C 2h C s
ð Þ for
the staggered ethane. The current density fields induced in the electrons by two
anti-parallel and parallel nuclear magnetic dipoles at vicinal protons of the eclipsed
ethane, are shown respectively on the left and on the right of Fig. 7.44.
The streamlines in Fig. 7.44a cannot cross the r v plane. At a vanishingly small
distance from this plane, the trajectories flow parallel. In Fig. 7.44d, the Tr v plane
7 Topology of Quantum Mechanical Current Density …
219
currents in a molecule may be attempted within the lexicon of topology, delocalized
currents are those flowing in the domain which extends beyond the separatrix
surrounding the nuclear skeleton. This formulation would seem ad hoc for cyclic,
neutral or charged, planar molecules C n H n in the presence of a magnetic field
orthogonal to the molecular plane, i.e., compounds with D nh ðC nh Þ magnetic symmetry, see Sect. 7.5.2, in which the separatrix is defined by the surface containing
n vortical and n saddle stagnation lines: delocalized currents flow on the outside for
a great enough distance to reach the tail regions of the electron cloud. It seems
appropriate also to the case of cyclopropane and prismane discussed in Sect. 7.5.4,
cubane and pentaprismane, Sect. 7.5.5.
However, the same definition would apply to the case of p-electron currents in the
same molecules, in which the separatrix coincides with the single vortical line
through the centre of the molecule. It would be also applicable to diamagnetic atoms,
in which the delocalized flow beyond the nucleus consists of concentric circular
streamlines about a vortical stagnation axis identifiable with the separatrix [60].
7.8 Current Density Induced by a Pair of Magnetic Dipole
Moments and Nuclear Spin-Spin Coupling
The Ramsey theory of indirect nuclear spin-spin coupling [12, 13] can be reformulated in terms of linear superposition of two current density fields, J
m I and J
m J ,
induced in the electrons of a molecule by nuclear magnetic dipoles m I and m J [3,
51, 52, 61]. Graphical representations of the interference pattern within the total
current density vector field, together with corresponding density maps, Eq. (7.57),
are very useful to elucidate coupling pathways and to rationalize the exchange of
spin information between coupled nuclei.
A paradigmatic application has been reported for ethane [16]. It shows that the
Fermi contact contributions to experimental nuclear spin-spin coupling constant are
easy to explain in terms of current densities (7.25), which transport spin polarization along the coupling pathway, and associated plots of property density,
Eq. (7.57). Same-spin electron correlation, the only kind of correlation recovered by
the Hartree Fock wavefunction considered in Ref. [16], determines the alignment of
the nuclear dipoles at its ends, as shown in the current-density maps reported for
ethane, Fig. 7.44.
According to experimental and theoretical results, the magnetic dipoles of the
vicinal protons are anti-parallel, in the configuration of lower energy. Therefore, the
physically acceptable magnetic symmetries are C 2v for the eclipsed, and C 2h C s
ð Þ for
the staggered ethane. The current density fields induced in the electrons by two
anti-parallel and parallel nuclear magnetic dipoles at vicinal protons of the eclipsed
ethane, are shown respectively on the left and on the right of Fig. 7.44.
The streamlines in Fig. 7.44a cannot cross the r v plane. At a vanishingly small
distance from this plane, the trajectories flow parallel. In Fig. 7.44d, the Tr v plane
7 Topology of Quantum Mechanical Current Density …
219