The Closed Polyene Chain

we want to find the general solution for the system of homogeneous linear equations for the closed polyene chain with N atoms yielding the Nth degree secular equation 

The general equation for the coefficients is the same as that for the linear chain  

The general solution is now the progressive wave in complex form  

Therefore, the general solution for the N-atom ring will be  

If Ak is a normalization factor, the form of the general MO in complex form will be  

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THE CLOSED POLYENE CHAIN For the real coefficients, we have ... 

In Chapter 3, the Hiickel model of linear and closed polyene chains is used to explain the origin of band structure in the one-dimensional crystal, outlining the importance of the nature of the electronic bands in determining the different properties of insulators, conductors, semiconductors and superconductors. 

One of the interesting predictions of the Su-Schrieffer-Heeger model [158, 159] is that the electronic excitations are closely related to the topological distortions of the conjugated polyene chain. It is predicted that solitons are the elementary excitations both for optical... 

FIGURE 40. Plot of the differences in chemical shift observed between rhodopsin and the ll-ct s-retinal PSB chloride salt (open squares), and between isorhodopsin and the 9-ds-retinal PSB chloride salt (closed squares), for retinal carbons along the polyene chain. Reprinted with permission from Reference 55. Copyright (1990) American Chemical Society... 

Fig. 20. Model for the primary event in vision. Isomerization of the 11.12 bond leads to charge separation at Ihe Schiff base site. This process, as shown, can possibly be followed by proton transfer, the latter resulting from the charge separation. In rhodopsin, the second negative charge responsible for wavelength regulation is shown close to the 11,12 bond of the polyene chain. This model assumes that hypsorhodopsin is the unprotonated form of the Schiff base, and that it is formed possibly by proton transfer from the Schiff base nitrogen in some pigments. From Honig ct al. [207].

Other authors [214,222,225], agreeing that a negative charge on the protein can control the observed spectral shift, proposed that this charge is located close to the ionone ring rather than the polyene chain of the chromophore. 

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