The Linear Polyene Chain

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

In conclusion, we see that the general solution for the linear chain will be  

We give below the derivation of the detailed formulae for the open linear chain with N = 4 (butadiene). 

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The first (bonding) and the last (antibonding) energy levels of the linear polyene chain are reported with respect to Nin Table 3.1 in units of (3. In the last column of the table is the difference in energy between two successive levels. The asymptotic approach of x towards 2 and of % towards -2 is apparent from the numbers given in the table, as well as is the decreasing distance between two successive levels, which tends to zero for N —> oo. 

A common example of the Peieds distortion is the linear polyene, polyacetylene. A simple molecular orbital approach would predict S hybddization at each carbon and metallic behavior as a result of a half-filled delocalized TT-orbital along the chain. Uniform bond lengths would be expected (as in benzene) as a result of the delocalization. However, a Peieds distortion leads to alternating single and double bonds (Fig. 3) and the opening up of a band gap. As a result, undoped polyacetylene is a semiconductor. 

These results can be easily established in general as follows. Using formula (3.14) for the orbital energy of the kth MO of the N-atom linear polyene chain ... 

Table 3.1 Orbital energies (units of /3) of first and last level and energy difference between two successive levels in a linear polyene chain as a function of the number of carbon atoms N...

For an even cyclopolyene (alternating hydrocarbon) the 7r-electron density is equal to unity at each atom. Therefore, from (12), an increase in /r-bond energy always follows the introduction of a heteroatom. But at the same time the cyclic delocalization energy, which is the difference in 7r-bond energy between the cyclic and the corresponding linear polyenes, will be decreased because the maximum atom self-polarizability of a linear polyene chain always has a higher value than that of the corresponding cyclopolyene 12°). 

Kohler BE and Woehl JC (1995) A simple model for conjugation lengths in long polyene chains. J Chem Phys 103 6253-6256 Kohler BE, Spiglanin TA, Hemley RJ and Karplus M (1984) Vibrational analysis of the lowest B state of 1,3,5,7-octatetraene. J Chem Phys 80 23-30 Kohler BE, Spangler C and Westerfield C (1988) The 2 Ag state in the linear polyene 2,4,6,8,10,12,14,16-octadecaoctaene. J Chem Phys 89 5422-5428... 

The symmetrized Density Matrix Renormalisation Group Procedure145 using a Hubbard-Peierls model Hamiltonian has given for a linear polyene chain quite satisfactory results for aL(static) and for yL(coa co, co2, ) (coa = co + a>2 + ) and for (yL/jV)(static). 

We carry out an analysis of the spectrum the Hiickel model of dimerized polyacetylene, both for cycUc and open chain boundary conditions, with special emphasis on the linear polyene with odd number of ir bonds. We also perform explicit perturbation expansion up to second order in the nuclear displacements. The asymptotic analysis of first- and second-order perturbation theory reveals that some behaviours assumed in the literature are not correct. 

This reaction scheme also accounts for the selective suppression of the two families of aromatic compounds formed in the pyrolysis of PVC induced by metal oxides. In fact, metal oxides (or better, the metal chlorides) are able to accelerate the rate of crosslinking of the linear polyene sequences and this process would compete with the parallel intramolecular cyclization of the polyenes. In the meantime, the amount of crosslinked polyene chains produced would be appreciably higher with respect to that formed in the pyrolysis of pine PVC. In fact, a reasonable increase of the residual char formed in the pyrolysis of PVC has also been observed. 

This formula was estimated for a linear polyene chain. We tried to apply it for the oli-gothiophenes and foimd that this formula excellently fits our experimental results. Figure 17.10 shows our fitting results for the different models. 

The convergence of the linear polyene Sq to S2 excitation energy to a finite limit as the length of the conjugated chain increases is strong evidence that there is bond length alternation in the long chains. This is supported by the limited number of direct structure determinations that have been published. 

Figure 2. Results of calculations for conjugated polyenes. (Left) Results of full calculations, where the dashed line (la) gives the normalized r-energies for linear polyenes and the solid line (Ila) gives the energies for cyclic polyenes. The line at 2.14 eV represents the limit for an infinite chain. (Right) Results of calculations in which the antisymmetry principle was not included. The energies for the linear polyenes and the 4n + 2 cyclic polyenes are unchanged, but the An cyclic polyenes are calculated to be too stable. (Reprinted with permission from ref 124a. Copyright 1996 Elsevier Science.)...

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