Stagnation graph

A stagnation graph is composed of two types of lines, vortical lines (later called center stagnation lines by Keith and Bader ) and saddle lines, interconversion being allowed at critical points where an index theorem is obeyed. This states that on going... 

In molecular systems, at large distances from the nuclei the behavior of the current density is frequently similar to that pictured in Figure 12, so that the stagnation graph has associated vortical ends. This vortical line may branch out at critical points but the index theorem will be satisfied. ... 

Lazzeretti supplemented this analysis by that of the magnetic point-group symmetry which helps in locating in space the components of the stagnation graph. This is of the utmost importance as stagnation graphs are frequently too complicated for easy three-dimensional visualization. 

Figure 12. Stagnation graph (a) and the charge circulation (b) for a closed-shell atom under an external magnetic-field. The graph is a straight vortical line through the nucleus, the axis of Larmor rotation of the charge under the magnetic field.

Figure 13. Stagnation graph of a molecular system. At long distance from the nuclei, there is a single vortical line (a) parallel with the external field that extends outward indefinitely. This vortical line may branch into new vortical and saddle lines, (b), forming a more-complicated stagnation graph in some cases and a less-complicated one in others. On the right-hand side of part b, a crude representation is shown of the current circulation on the perpendicular plane.

Figure 17. Stagnation graph of CO2. The stagnation graph shown is compatible with the induced current density shown in Figure 14. The primary vortex (0), as well as vortices (2) and (20, are diamagnetic the inner vortex (1) corresponds to the paramagnetic circulation near the center of symmetry. Vortices (3) and (30 are the other paramagnetic circulations near the oxygen atoms.

Singularities, Stagnation Lines and Stagnation Graph of a Current Density Field... 

Fig. 7.3 Perspective view of the stagnation graph of the Gomes flow, Eq. (7.60), for z <2.5. The central stagnation line coincides with the z axis. Green (red) lines correspond to clockwise (anticlockwise) vortices in Eigs. 7.1 and 7.2, saddle lines are blue...

Fig. 7.9 Perspective view of the stagnation graph of the current density vector field induced by a magnetic field B =Bei normal to the yz plane of the nuclei of ethylene. Green (red) lines denote diamagnetic (paramagnetic) vortices, saddle lines are blue. Isolated blue points mark (3, 1) saddle-nodes, isolated green (red) points denote (3, 1) diamagnetic (paramagnetic) foci...

Magnetic Symmetry, Stagnation Graph and Induced Current Density of Mono-cyclic Neutral and Charged C H Systems in a Magnetic Field... 

Fig. 7.18 Ptaspective view of the stagnation graph of the current density vectra- field in methane CEl,. The unifinm external magnetic field B is parallel to a C3 symmetry axis along a CH bond...

Hartree-Fock level of accuracy, to obtain the stagnation graph of Dnh Cnh) compounds. The third-order hnear autonomous system for the flow was integrated using Runge-Kutta procedures [108]. 

Fig. 7.20 The stagnation graph of the current density in cyclic systems C H for n = 3,4,. Green (red) SLs denote diamagnetic (paramagnetic) vortices, saddle SLs are blue...

The stagnation graphs of 1,3-cyclopentadiene, furan, pyrrole, and thiophene, in the presence of a magnetic field perpendicular to the molecular plane [ 111], are displayed in Fig. 7.22. They show that the electron flow induced by a perpendicular magnetic field in pentatomic cyclic molecules with C2v(Cj) = TC2 TCy different from that of D h C h) compounds discussed in Sect. 7.5.2. An analogous consideration was made for... 

Fig. 7.22 The stagnation graph of five-membered cyclic compounds. Clockwise from top left 1,3-cyclopentadiene, p5urole, furan, and thiophene in the presence of a magnetic field perpendicular to the molecular plane. Diamagnetic (paramagnetic) vortices are represented by green (red) lines, and saddle lines are blue. All the stagnation lines lie on the Tcr plane of magnetic synunetry...

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