Ring critical points

There are two other types of critical points, having either one or zero negative eigenvalues in the density Hessian. The former is usually found in the centre of a ring (e.g. benzene), and consequently denoted a ring critical point, the latter is typically found at the centre of a cage (e.g. cubane), and denoted a cage critical point. They corresponds to local minima in the electron density in two or three directions. 

The nature of a critical point is denoted as (rank, signature). For example, a minimum in p is designated by (3, + 3) and a maximum by (3,-3). The two remaining types of critical points, (3,-1) and (3, +1) are saddle points, called bond critical point and ring critical point, respectively. 

Figure 3.10 Schematic representation of all the critical points in the dimeric structure (BH3-NH3)2. The hydrogen atoms are represented by the small gray spheres, and the bond critical points are represented by the small black spheres. The ring critical points are noted as two large spheres. A dotted line shows the dihydrogen bond s bond path. (Reproduced with permission from ref. 16.)...

Figure 5.2 Molecular graph of [4-(( )-but-l-enyl)-2.6-dimthoxyphenyl]pyridine-3-carboxylate with bond critical and ring critical points. (Reproduced with permission from ref. 5.)...

Figure 5.8 Molecular graph of compound 2A obtained at the B3LYP/6-31G + (d,p) level with the bond critical points noted as the smallest spheres and ring critical points noted as squares. (Reproduced with permission from ref. 12.)...

Fig. 3.21 Molecular graph with the atoms, bond critical points (red), ring critical points (yellow), and bond paths of two conformers of the CF3 and t-Bu derivatives...

Fig. 3.2. Profiles of the charge density along the C2 symmetry axis for the dissociation of the water molecule. In (a) p decreases monotonically from a maximum value at the oxygen nucleus. As Ro is increased, an abrupt change occurs in the form of the curve when a point of inflexion appears (b). This unstable critical point exists only for this single configuration X, of the nuclei and a further increase in Rq results in Its bifurcation to yield a maximum, corresponding to the formation of a bond critical point and a minimum, corresponding to the formation of a ring critical point a.s typified by (c). The system has entered the region of the ring structure.

In addition to the fundamental difference in the form of the bridgehead charge density between the two sets of molecules, there are also substantial quantitative differences in the values of p. Thus, its value at the bond critical point in [l.l.l]propellane is 0.203 au, four-fifths of the value for a normal C-C bond, while the value of p at the corresponding cage critical point in the bicyclic molecule is 0.098 au. It is also clear from Fig. 3.8(d) that the values of p at the bond critical point and the three adjacent ring critical points are almost equal, giving rise to a very broad bonded maximum in p in the bridgehead interatomic surface in [l.l.ljpropellane. 

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