Rearrangements CUMULATIVE

The third mechanism starts with addition of the AT-allylamine 103 to the cumulated acceptor system of an allene carbonester 108 (Acc=CHC02Me) to form an intermediate iV-allyl ammonium amide enolate 109 (allene carbonester Claisen rearrangement). The anion stabilizing group is exclusively placed... 

The calculated AH, assuming no interaction between the double bonds, is 2(-126) = -252. The more negative the observed value of AH compared to -252, the less stable the diene the less negative the observed value, the more stable the diene. Conjugated dienes are most stable and cumulated dienes are least stable under the prof>er conditions allenes tend to rearrange to conjugated dienes. 

Equation (108) shows that the cumulant expansion for In g s,td) may be rearranged to give... 

To find the cumulants of 17, this must be rearranged according to the powers of u ... 

Figure 8. Time evolution of the cumulated number of rearranging events per atom in respective shells at 600 K. The indices specifying the shells are allocated for each shell. The center atoms is denoted by core. As mentioned in the caption of Fig. 7, the initial configuration of A47B20 consists of five shells. The fifth shell is indicated by thick solid line with a caption surface. The dotted line denotes a typical increasing trend of the cumulated number of rearranging events in a completely melting cluster.

The title reactions are discussed in a series of reviews . However, the most complete and detailed description of rearrangements involving allenes was presented in Huntsman s comprehensive survey, wherein the cumulated systems were considered as either the starting materials, the intermediates or the reaction products. More recently, very detailed reviews devoted to vinyl cations and containing numerous examples of rearrangements of cumulenes and other polyenes were published . Therefore, this section will cover only relatively recent publications. 

In our own work we found the sensitivity to small intensities to be a real problem. One possible solution is to consider not P(y) but the cumulative distribution function that is the fraction of all transitions with intensity up to y. In actual applications we note that for either experimental or computational data, the distribution of intensities is generated as a histogram. Say the intensity axis is divided into 1 bins of equal size and P is the fraction of transitions with intensities in the nth bin. The histogram is now rearranged... 

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