Classical structure

In a final step, we follow the ideas of Ehrenfest [252], who first looked for classical structures in the equations of quantum mechanics, and look at the time... 

The classical structures of pyrrole, furan and thiophene (31) suggest that these compounds might show chemical reactions similar to those of amines, ethers and thioethers (32) respectively. On this basis, the initial attack of the electrophile would be expected to take place at the heteroatom and lead to products such as quaternary ammonium and oxonium salts, sulfoxides and sulfones. Products of this type from the heteroaromatic compounds under consideration are relatively rare. 

Fig. 5.11. Contrasting potential energy diagrams for stable and unstable bridged norbomyl cation. (A) Bridged ion is a transition state for rearrangement between classical structures. (B) Bridged ion is an intermediate in rearrangement of one classical structure to the other. (C) Bridged nonclassical ion is the only stable structure.

Non-classical structures are predicted to be unstable relative to classical structures (for example ethyl cation). 

In contrast to the results of the reaction of tertiary and secondary alkyl cations with carbon monoxide (Figs. 1-5), which were obtained under thermodynamically controlled conditions, the results of the carbonylation with the vinyl cations were obtained under kinetically controlled conditions. This presents a difficulty in explaining the occurrence of the 1,2-CH3 shift in the reaction 16->-17, because it involves a strong increase in energy. The exclusive formation of the Z-stereoisomer 18 on carbonylation of the 1,2-dimethylvinyl cation 16 is remarkable, but does not allow an unambiguous conclusion about the detailed structure— linear 19 or bent 20—of the vinyl cation. A non-classical structure 21 can be disregarded, however, because the attack... 

On the other hand the alkyl- or amino-substituted congeners should adopt a more classical structure, in which the twofold coordinated phosphorus atom is bent, 8b. A first hint of the cation was reported by mass-spectroscopic investigations [67]. The synthetic verification of this prediction starts from amino-substituted diphosphene 22 via protonation with CF3SO3H [68] (Scheme 15). 

Figure 3 The numbers at each site in the top half (above the dotted line connecting the extreme atoms to the left and right of the diagram) are the numbers of classical structures which can be constructed with hydrogen (muonium) attached to the position indicated and the unpaired electron at the indicated site. The corresponding numbers in the bottom half are the spin densities in atomic units from UHFAA calculations on the fully optimised geometry of CeoMu using an ST0-3G basis set within the ROHF method.

Resonance theory [15] contains essentially three assumptions beyond those of the valence bond method. Perhaps the most serious assumption is the contention that only unexcited canonical forms, non-polar valence bond structures or classical structures need be considered. Less serious, but no more than intuitive, is the proposition that the molecular geometry will take on that expected for the average of the classical structures. This is extended to the measurement of stability being greater the greater the number of classical structures. These concepts are still widely used in chemistry in very qualitative ways. 

Molecules of the size of the fullerenes require such approaches to help rationalise the results. Even though the number of classical structures can be very large the same qualitative reasoning can be used as with the smaller molecules, typically benzene. The enumeration of classical structures is easily accomplished using computers. It is simply a problem of determining how many ways a set of points (in our case carbon atoms) can be coimected given rules governing their connectivity. 

It is common in the interpretation of electron spin resonance spectroscopy of organic radicals to draw classical structures to rationalise the observed distribution of spin. In this spirit the number of possible classical structures of CeoMu, with the muon... 

Figure 4 Plot of spin density as a function of the number of classical structures.

In Fig. 3 the number of possible classical structures arising from the spin being localised at each carbon atom (top half) is compared to the UHFAA spin density results (lower half). Note that the number of classical structures when the unpaired electron is at sites 2, 5 and 6 is the same as for the double bonds involving atoms 1 and 2, 1 and 5 or 1 and 6 in Ceo- The correlation between the number of classical structures and the spin density is excellent. With only one exception all centres with the number of classical structures larger than 2200 show positive spin density and all those less than 2200 show negative spin density. This anticipated correlation can be further quantified. 

Figure 5 Plot of calculated bond lengths of CeoMu as a function of the number of classical structures. Some points are labelled by (i-j) which identify the bond between atom i and atom j.

Table 4 Correlation of the number of classical structures with calculated bond lengths for Ceo and Cn. The bond environment column describes the arrangements of the carbon atoms which have the bond in common. The column labelled with a f is the bond order calculated using resonance theory as described in the text.

Typical bond Calculated bond length No. of classical structures Bone order Interpretation of bond Bond Environment... 

Figure 8 Measure of delocalisation of each defect type predicted by resonance theory. The loops enclose centres which have numbers of classical structures larger than. 74 times the greatest number in the type. The cut-off point for type bi (or type 63) centres is particularly arbitrary since the delocalisation is spread around the equator. The small circles are the point of muonium attachment. The dotted circle is coincident with the equator of Cra-...

C8a (A = 0.077°) and an elongation of the 4a-8a bond (1.41 A), indicating a weakened sp bonding interaction between C4a and C8a in the aromatic system. It is clear that the structural features of 5, and other bridging cations give convincing support to Winstein s 3-center, 2-electron non-classical structures. ... 

These difficulties have led to the convention of representing molecules that cannot adequately be written as a single classical structure by a combination of two or more classical structures, the so-called canonical structures, linked by a double-headed arrow. The way in which one of these structures can be related to another often being indicated by curved arrows, the tail of the curved arrow indicating where an electron pair moves from and the head of the arrow where it moves to ... 

The polar character of the group R will also have a bearing on the formation and stabilization of the furan ring. In accordance with the theory of resonance, the classic structure... 

In contrast to the simple olefins, aryl-substituted olefins dissolve in sulfuric acid to give comparatively stable carbonium ions, as is shown by the -factors, the spectra, and the recovery of the olefin on dilution.262 In some cases it is neccessary to extrapolate the freezing point depression to zero time owing to a slow sulfonation. Because of the similarity in the spectra it is believed that these carbonium ions have the classical structures shown below.263... 

---