Infinite polyene

This latter case is realised in the elements sulphur, selenium and tellurium. On the other hand a linear infinite polyene — C = G — C = C — with one n electron per atom corresponds to the first case. 

The polyene case (today it would be called polyacetylene) is especially interesting, for some years ago it occasioned a great deal of discussion. Would an infinite polyene localize, 90 Eventually, Salem and Longuet-Higgins demonstrated that it would.69 Polyacetylenes are an exciting field of modern research.70 Pure polyacetylene is not a conductor. When it is doped, either partially filling the upper band in 89 or emptying the lower, it becomes a superb conductor. 

At the extreme of an infinite polyene, however, simple Hiickel theory reduces the HOMO—LUMO gap to zero, since the secants in diagrams like Fig. 1.33, would become infinitely small. Such a polyene would have equal bond lengths between each pair of carbon atoms, the gap between the HOMO and the LUMO would be zero. 

Figure 3 The development of a polyene band structure from the molecular orbitals (MOs) of ethylene. From left to right, the molecular orbitals progressively develop into a band structure as the length of the conjugated chain is increased. For shorter polyene chains, A represents the HOMO-LUMO gap. Forthe infinite polyene chain, vb and cb denote the valence band and the conduction band, respectively, and E is the band gap...

To give a general description of covalent solids and metals, the band theory arising from the infinite polyene chain must be extended to three dimensions. The properties of solids depend largely on the way in which electrons fill the different available bands. 

Extrapolation ofS and S, Sq Transition Energies to Long Carotenoids The Infinite Polyene Limit... 

Table 1. Extrapolations of carotenoid/Iinear polyene transition energies to the infinite polyene limit...

Tavan P and Schulten K (1987) Electronic excitations in finite and infinite polyenes. Phys Rev B 36 4337-4358... 

Tavan P and Schulten K (1987) Electronic excitations in finite and infinite polyenes. Phys Rev B 36 4337 358 Thornber JP, Morishige DT, Anandan S and Peter GF (1991) Chlorophyll-carotenoid proteins of higher plant thylakoids. In Scheer H (ed) Chlorophylls, pp 549-585. CRC Press, Boca Raton... 

The electrical properties of an infinite polyene have intrigued theoretical chemists for more than fifty years. Shortly after Hiickel introduced his 7i-electron theory for unsaturated systems in 1931, speculations began to appear about the carbon-carbon bond lengths to be expected in an infinite polyene. The question was interesting because, if the bonds were of equal length, 7i-electron theory predicted that the 7i-molecular orbitals would form a continuous band which would be half-filled, whereas alternating carbon-... 

At the infinite polyene, each of the four sets of MOs contains an infinite number of MOs, each of which is a particular linear combination of one of the MOs of the monomer. All MOs in a given set trace their origin to the same monomer MO. Such a collection of orbitals is called a band. An individual MO within a band is called a crystal orbital or a band orbital. Just like MOs, crystal orbitals can hold two electrons, the spins of which must be paired. The energies of the individual crystal orbitals within a band are not equally spaced, although the... 

This is shown in Fig. 28.10, where the calculated wavenumbers for three Ag modes (1 1, V2, and 1 3) of an ideally infinite polyene chain are plotted as functions of... 

Fig. 28.10 Plot of the calculated Ag frequencies for an infinite polyenic chain versus effective force constant (solid line). Data points are the experimental frequencies (both for primary and satellite peaks) of different samples of PA with different exciting laser lines values lower than 3 mdyn/A correspond to infrared doping induced or photoinduced bands. (Figure and data from Ref. 41.)...

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