Phase topology, overlap

Figure 8.20. Generation of a ID correlation function, fl (x), by autocorrelation of the ID electron density, Ap (y) for a two-phase topology. Each value of ft (x) is proportional to the overlap integral (total shaded area) of the density and its displaced ghost...

The three possible topological interactions in [ 2 + 2] cyclo-addition reactions are shown in Fig. 5.2 again the basis molecular orbitals of the ethylene components are considered. In the supra-supra and i v antara-antara combinations there are no out of phase orbital overlaps (or two if the signs are reversed on one ethylene component). In the supra intara mode there is one out of phase overlap. Since there are four electrons involved, the Mobius type interaction (i.e. supra-antard) should be preferred the other combinations should therefore be possible under photochemical control. These results accord with the previous findings of orbital symmetry theory. 

It is the present writer s opinion that the topology of this situation is fundamental and dictates its explanation. Therefore we must clearly note the topology of the physical layout of the design of the situation that exhibits the effect. The physical situation is that of an interferometer. That is, there are two paths around a central location—occupied by the solenoid—and a measurement is taken at a location, III, in the Fig. 10, where there is overlap of the wavefunctions of the test waves that have traversed, separately, the two different paths. (The test waves or test particles are complex wavefunctions with phase.) It is important to note that the overlap area, at III, is the only place where a... 

In order to eliminate the problems with the invariance, we proposed some time ago a topological approximation based on the so-called overlap determinant method [43]. This approximation is based on the transformation matrix T that describes the mutual phase relations of atomic orbitals centred on molecules R and P, and thus plays in this approach the same role as the so-called assigning tables in the overlap determinant method (Eq. 4)... 

Dewar, although none of such systems has been obtained so far, it is of great significance to develop a conception to substantiate it. The system in which the AO phase shift is inevitable differs topologically from the system in which such a shift can be avoided.... This difference depends only on the overlap topology of AOs involved in the formation of a delocalized system and is absolutely independent of the MOs that can be composed from those AOs . 

It should be emphasized that there is no essential difference between the two types of bonds other than their geometry. For AOs to fuse to MOs, it is only necessary that they should overlap. The geometry of overlap is not relevant. Bonding is a matter of the topology of orbital overlap, not geometry the only important factors are the extent of overlap and whether the phases of the AOs are the same or different in the region where they overlap. 

No such Mobius strip compound has as yet been synthesized, but the concept underlying it is very important. A system in which there is an inevitable phase dislocation is topologically distinct from one in which all phase dislocations can be avoided. Note that this distinction depends on the topology of overlap of the AOs involved in forming a delocalized system it is quite independent of the MOs that can be constructed from them. 

While neither of these theoretical possibilities has been realized for stable 7i-electron systems, it must be remembered that delocalized systems in chemistry are not all of n type. There is no reason why delocalized systems should be two dimensional or involve exclusively Ti-type overlap of p or d AOs. In a three-dimensional delocalized system involving a- as well as retype overlap, we have much more flexibility and, as we shall see presently, examples of this alternative type of topology become important. Since the MO treatment of n systems was first introduced by Huckel, the normal class of delocalized systems with no required phase dislocations may be described as HUckel systems. The second topological type with one inevitable phase dislocation may then be called anti-Huckel systems. ... 

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