Peierls distortion models

The model of the chain of hydrogen atoms with a completely delocalized (metallic) type of bonding is outlined in the preceding section. Intuitively, a chemist will find this model rather unreal, as he or she expects the atoms to combine in pairs to give H2 molecules. In other words, the chain of equidistant H atoms is expected to be unstable, so it undergoes a distortion in such a way that the atoms approach each other in pairs. This process is called Peierls distortion (or strong electron-phonon coupling) in solid-state physics ... 

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 ... 

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... 

From more recent optical data it is proved that Qn(TCNQ)2 is a semiconductor up to 300 K, with an energy gap due mainly to a Peierls distortion on the conducting TCNQ chains [57]. However, this gap Ec = 2A is not constant, as is simply assumed in the model of Epstein et al. In fact, A appears to decrease significantly from = 1200 K at T = 0 K to = 300 K at T = 300 K, somewhat like in the salt TEA(TCNQ)2 (see above). This requires a modified approach in which the existence of a conductivity maximum cjm still implies a T-dependent mobility p, but not so steep as would require a constant gap. 

The previous treatment deals with a one-component order parameter (such as for a commensurate Peierls distortion) but does not apply to situations where the order parameter is complex with an amplitude and a phase (superconductivity, incommensurate Peierls, or spin density wave transitions). The latter situation is analogous to classical moments which can rotate freely in an XY plane. The coherence length of the XY model is less strongly divergent at low temperature than for the Ising model,... 

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. 

Why the difference Apparently one is subject to a Peierls distortion but the other is not. If true, there must be a difference in the electronic structures. The first hint comes from applying the Zintl-Klemm model. With Y and Th, it is pretty clear, we have Y3+, [BC]3- and Th4+, [BC]4-, respectively. That is, although the... 

In (2) the originally discrete ions act effectively as a continuum and there is no driving mechanism for a possible Peierls distortion towards phase locking or commensurability with respect to the lattice ("Incommensurate FrShlich Model") contrary to Ref. (5). The relevance of the model to KCP can be estimated from the approximations it implies ... 

Rather amusingly it turns out even at a very low level of description that there is a degree of concordance in general predictions concerning a class of conductive states at least for the class of "benzenoid" polymers. In particular within the framework of either the simplest Hiickel model or of the simplest resonance-theoretic rationale it seems that the same stmctural conditions arise for the occurrence of Peierls-distortion and the sometimes associated solitonic excitations. For the simple Hiickel model, starting with uniform P-parameters, such a structural condition is well-known [54-57] to be coimected with a 0-band gap for which the feimi energy Cp occurs at a rational multiple of the Brillouin-zone size, say at wave-vector k=Tip/q - then a distortion cutting the... 

Thus, in so far as energetics of low-lying excited states of odd and even carbon systems and doped systems are concerned, once electron correlations are introduced, the notion of solitons as elementary electronic excitations becomes somewhat blurred [128]. Moreover, it is important to evaluate whether the associated topological features of the low-lying excitations survive when we introduce realistic electron-electron correlations. The purpose of this chapter is to study the equilibrium geometries of excitations in the Peierls-PPP model and discover if there is a relation between the relaxed geometries and the topological lattice distortions associated with solitons in the presence of realistic electron correlations. 

What is the mechanism for conductivity in one-dimensional organic stacks or polymers This depends on the electronic structure and how it is coupled to the nuclear vibrations. We want to treat these rather different systems within the same electronic model. In a stacked n-system, the principle is the same as for ET in general. There is always some vibration that cooperates with the electronic motion to carry electrons back and forth between equivalent locations. Complications arise in onedimensional systems particularly due to the Peierls distortion. 

The Peierls distortion is confirmed in accurate models. If Pol is given the value of 3 eV in a bond-length-dependent coupling model, the band gap has the value Ipl = 1.57 eV. 

Abstract We present a theoretical study of Peierls distortion in carbon rings. We demonstrate using the Longuet-Higgins-Salem model that the appearance of bond alternation in conjugated carbon polymers is independent of the boundary conditions and does in fact appear in carbon rings just as in carbon chains. We use the Hartree-Fock approximation and density functional theory to show that this behaviour is retained at the first principles level. 

Keywords Peierls distortion Conjugated polymers Annulenes Longuet-Higgins-Salem model Density functional theory... 

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