Linear polyenes, numbering system

The most easily obtained information from such calculations is the relative orderings of the eneigy levels and the atomic coefficients. Solutions are readily available for a number of frequently encountered delocalized systems, which we will illustrate by referring to some typical examples. Consider, first, linear polyenes of formula C H 2 such as 1,3-butadiene, 1,3,5-hexatriene, and so forth. The energy levels for such compounds are given by the expression... 

The spectra of linear polyenes are modelled well as one-dimensional free-electron systems. The cyanine dyes are a classical example. They constitute a class of long chain conjugated systems with an even number n of 7r-electrons distributed over an odd number N = n — 1 of chain atoms. The cyanine absorption of longest wavelength corresponds to promotion of an electron from the highest occupied energy level, En/2 to the lowest unoccupied level, such that in terms of a free-electron model... 

General solutions exist for the Huckel determinants of special systems with any number n of carbon atoms, namely for the linear polyenes (Equation 4.7) and for the monocyclic systems (Equation 4.8). 

For odd-membered Craig or Mason-type (pd) systems, both formulas (8) and (9) lead to the same result all MO s occur only singly with always one non-bonding MO in a pattern similar to that of odd-membered linear polyenes. Closed-shell configurations are therefore predicted for each even number of 7r electrons for both Craig and Mason-type (pd)n systems. 

How can we develop a more quantitative model of these bands One strategy is to model the infinite linear polyene not as a linear combination of smaller polyenes, but rather as an infinite cyclic tt system, -(CH) -. If n really is infinity, there is no difference between a line and a circle (from the viewpoint of electronic structure, that is, but not from the viewpoint of topology ). Recall Eq. 14.56, here restated as Eq. 17.1. This gives the HMO energies for any cyclic polyene, where i is the orbital number and N is the number of vertices (atoms or p orbitals in the cycle [infinity in this case]). In this equation / runs from f = 0, 1, 2... up to N/2 for even Nor (N-l)/2forodd N. As such, this index directly corresponds to the number of nodal planes in the MO. 

Here, the bond lengths become approximately equal and the lowest energy electronic transition moves rapidly toward the red with Increasing number of double bonds (Table 11) whereas the shift is much smaller with the linear polyenes that have fixed single and double bonds. The location of the ions electronic transitions may be estimated using a particle in a box model,just as that for benzene may be estimated using a particle on a circle model. The jT-electron systems in these ions are closely related to those in the aromatic systems. 

Although a number of iron carbonyl complexes have been prepared with cyclic conjugated polyene systems such as cyclooctatetraene, linear conjugated polyene systems have been studied only up to 1,3,5-hexatriene,... 

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