Peierls alternation

It is well known that metallic electronic structure is not generally realised in low-dimensional materials on account of metal-insulator transition (or Peierls transition [14]). This transition is formally required by energetical stabilisation and often accompanied with the bond alternation, an example of which is illustrated in Fig. 4 for metallic polyacetylene [15]. This kind of metal-insulator transition should also be checked for CNT satisfying 2a + b = 3N, since CNT is considered to belong to also low-dimensional materials. Representative bond-alternation patterns are shown in Fig. 5. Expression of band structures of any isodistant tubes (a, b) is equal to those in Eq.(2). Those for bond-alternation patterned tube a, b) are given by. 

Fig. 4. Peierls transition in metallic polyacetylene and accompanied generation of bond alternation. Note that the semiconductive (or insulating) structure accompanied with the bond alternation is the more energetically stable.

Alternative evaluations of the absorption spectrum (or the optical conductivity) of Peierls systems within the FGM have been reported in Refs. (451 and [46], In those... 

The Peierls order parameter is the alternating pail of the hopping amplitudes [.see Eq. (3.11)] and now consists of two parts... 

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 ... 

The most striking implication of the electron lattice coupling in ID chains is the appearance of the semiconducting state the equal bond ID lattice (metallic state) is unstable (33) with respect to a lattice distorsion and this so called static Peierls instability is the origin of the opening of the intrinsic band gap at the edge of the B.Z. with an infinite density of states there and the presence of band alternation. 

In contrast, in the SSH model, the electrical bandgap arises because of the alternation between single and double carbon-carbon bonds, a signature of the Peierls distortion in a ID system. When a perfect ID chain of equidistant carbon atoms is considered, the electronic structure resulting from the electronic coupling between the atomic Pz-orbitals is that of a half-filled n band, implying a metallic... 

On the other hand, benzannulation in the 3,4 positions of PT, would produce the nearly degenerate structures 2a and 2b (Scheme I) because the gain in aromaticity of the benzene ring"Tn 2b is expected to outweigh the loss of thiophene aromaticity in 2a. The result would be amelioration of bond alternation ("Peierls distortion" of the backbone) with a concomitant decrease in bandgap. 

This gap may be further widened by electron-phonon interaction effects, as in the Peierls process. An accepted terminology [64] is to partition the gap A into the sum of an extrinsic one Ae coming from the nonequivalence of bonds (including the effect of the a electrons), and an intrinsic one A, due to ir-electron effects alone. The relative importance of Ae and A,-, or the magnitude of the bond alternation, are more difficult to calculate. 

The Peierls distortion to be expected for the (Ag-F) chain is an alternation of linear, symmetrical (F-Ag-F) species and Ag, interatomic distances in the chain becoming short about the Ag(III) and long about the Ag(I). But the polar field about each Ag(II) in the chain must frustrate the required centrosymmetric F ligand displacement about each Ag atom. It is perhaps for... 

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