Linear chain polyene

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... 

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. 

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...

Figure 3.2 Electronic bands in linear polyene chain (single 0)...

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°). 

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). 

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. 

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