Polyacetylene Peierls distortion

The one-dimensional chain of hydrogen atoms is merely a model. Flowever, compounds do exist to which the same kind of considerations are applicable and have been confirmed experimentally. These include polyene chains such as poly acetylene. The p orbitals of the C atoms take the place of the lx functions of the H atoms they form one bonding and one antibonding n band. Due to the Peierls distortion the polyacetylene chain is only stable with alternate short and long C-C bonds, that is, in the sense of the valence bond formula with alternate single and double bonds ... 

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Exercise 6.4. Consider a crude, but useful, model of polyacetylene (CH),, consisting of the ideal zig-zag (all-trans) chain shown below in which all the C-C bond distances are equal. Draw its TT-type band structure and show that it is Peierls unstable. Show that, unlike the H-chain case, the Peierls distortion does not correspond to a doubling of the unit cell. 

H3NBH3 is isoelectronic with ethane, H2NBH2 is isoelectronic with ethylene, andHNBH is isoelectronic with acetylene. Derive the band structure and the DOS for planar poly- -BHNH- (isoelectronic to polyacetylene) with a single B-N distance and predict its conductivity and stability with respect to a Peierls distortion. Only consider the tt electronic structure. 

Infinite linear polyenes show a bond alternation between successive long and short C-C bonds [1], a consequence of the Peierls theorem on the nonexistence of one-dimensional metals [2], This Peierls distortion (or instability) is very important both from a theoretical and a practical point of view, being a typical example of a metal-insulator transition [3]. Consider an infinite chain of equally spaced sites -(CH)-, each of them bearing one electron in a single valence orbital. In this case we have a half-filled band and the system has metallic character. If we distort the chain into an alternating sequence of short and long bonds -(CH=CH)-, the half-filled band splits into a lower one completely filled and an upper empty band, separated by a gap. This dimerized polyacetylene is an insulator. 

For chemists, it is probably not a big surprise that a one-dimensional chain of hydrogen atoms does not exist and that it will immediately decompose into isolated H2 molecules. The Peierls distortion has important consequences for one-dimensional systems, such as polyacetylene with C-C bond-length alterations (instead of equal C-C distances) [74], infinite molecules with platinum-platinum bonding such as Krogmann s salts K2[Pt(CN)4]Xo,3 3H2O with X = Cl or Br [75], or other one-dimensional systems [76], and it also affects three-dimensional systems, in particular elemental structures (see Section 3.4). From a group-theoretical point of view, Peierls distortions are characterized by a loss of translational symmetry in the above example, the nonequidistant chain of H atoms is less symmetrical (in terms of translational symmetry) than the equidistant one. 

Geometry optunizatiou by semi-empirical methods gives the bonding pattern seen in Figure 16.5. Each N atom corresponds to one CH unit in polyacetylene (PA). In PA, each CH unit contributes one % orbital and one electron. The valence band is half filled and the systan PA is therefore subject to Peierls distortiou. In (SNj, on the other hand, the sp hybridized S atom contributes two electrons to the jt-system. The jt-system of (SN)x is therefore three-quarters filled and not subject to auy Peierls distortion. However, three-quarter filling leads to other peculiarities, as we will see next. 

FIGURE 1.20. (a) Half-filled band of metallic polyacetylene, (b) Filled Peierls-distorted band of semiconducting polyacetylene. 

The Peierls distortion in polyacetylene. A. The band structure after the Peierls distortion. Note that now a finite band gap exists, and Ep lies below a. B. The two degenerate forms of polyacetylene along with the hypothetical, bond-equal form (in brackets). 

Sketch the band structure (HMO) of polyacetylene using butadiene as the unit cell. Also, sketch the consequences of a Peierls distortion. 

Polyacetylene (PA) is the simplest conjugated polymer. It can exist in cis- and transforms (cis- and trans-PA isomers). The latter is thermodynamically more stable. The transition between C-C and C = C bonds in tmns-PA does not require energy change, so the Peierls distortion opens up a substantial gap in the Fermi level. This twofold degeneration leads to the formation of mobile solitons with a length of == 15 C-C units and spin 5 = on trans-PA chains. These correspond to a break in the pattern of bond alternation," and thus determine the fundamental properties of the polymer. 

Section 15-9 Polyacetylene with Alternating Bond Lengths—Peierls Distortion... 

We have seen that Peierls distortion produces a band gap at the Fermi level in polyacetylene. Another type of change we can consider is replacement of one kind of atom in the unit cell with another. While this is not a modification that the polymer can make spontaneously, it is a way for us to predict what sort of changes in band sttucture would result if substitutional relatives of polyacetylene were synthesized. 

Figure 5.64. Simplified band diagrams for a conductive polymer. Shown are (a) polyacetylene if all C—C bond distances were equivalent (metallic), (b) actual band diagram for PA, resulting from Peierls distortion, (c) polaron mid-gap states and bands formed upon doping, and (d) soliton mid-gap state and band formed upon doping. For (c) and (d), note that at higher levels of doping, the mid-gap bands grow at the expense of the valence and conduction bands, as midgap states are taken from band edges.

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