Cage 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 analysis of bond, ring, and cage critical points, and of the behavior of the electron density in their vicinity, is a subject of considerable interest for the analysis of chemical... 

Cage critical point, electron density analysis,... 

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. 

Table 3. Number of hydrogen bonds (nHB), hydrogen bond critical points (nHBCP), ring critical points (nRCPs), cage critical points (nCCPs), stabilization energies (SEs, kcal/mol), range of electron density (p(rc)), and Laplacian of electron density (V p(rc)) for the most stable water clusters...

Figure 9-9. The calculated molecular graph of exo, exo-tetracyclo[6.2.1.P . 0 ] dodecane which consists of two fused norbornanes rings. The H-H bond path links the nuclei of the two bridgehead hydrogen atoms. This results in the closure of two rings concomitant with the appearance of two ring critical points (yellow) and a cage critical point between them (green).

Fig. 7 Schematic representation of the location of a bond (red), a ring (blue) and the cage critical point (black) originated from the interaction of hexafluorobenzene with fluoride...

It was discussed in Section III that the structure of a system, as characterized by its molecular graph, is determined by the number of bond, ring and cage critical points. These are stable critical points in the sense that they retain their properties and hence the charge density retains its corresponding form as the nuclei are displaced. A structure which contains only stable critical points will thus persist over some range of all possible... 

Fig. 17.7 The molecular graph of the cluster containing four acetylene molecules big black and grey circles correspond to C and H-attractors, respectively, small red, yellow and green circles designate, bond, ring and cage critical points, respectively...

Intracule-Extracule. - The properties of the electron intracule and one-electron densities of the ground state of harmonium were determined and expressed in terms of asymptotic expansions and Pade approximants. Topologies of both densities were investigated in detail. An attractor composed of a cage critical point and a (1,-1) critical sphere. 

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