The Su-Schrieffer-Heeger model

In a TT-electron theory the ion represents the CH group, so there are three ionic degrees of freedom per unit cell. These ionic degrees of freedom may be formally represented as collective, normal modes. Su, Schrieffer, and Heeger, in their treatment of frans-polyacetylene, introduced a simplification to this problem (Su et al. 1970). This simplification was to consider only the normal mode that predominantly couples to the tr-electrons. For polyacetylene this is the carbon bond stretching vibration. Thus, projecting the ionic coordinates onto the chain axis, denoted by the x-axis, we have the Su-Schrieffer-Heeger (SSH) Hamiltonian, defined as, 

The Su-Schrieffer-Heeger model in the limit of static nuclei is known as the Peierls model. This is defined and discussed in Section 4.2. 

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Wan et al.236 have used the Su-Schrieffer-Heeger model to calculate a for five different C78 isomers and predict a great variation of optical properties with symmetry and shape. 

One of the interesting predictions of the Su-Schrieffer-Heeger model [158, 159] is that the electronic excitations are closely related to the topological distortions of the conjugated polyene chain. It is predicted that solitons are the elementary excitations both for optical... 

Fig. 2.7. The bond stretching mode of trana-CB.x projected onto the a -axis. 2.8.2 The Su-Schrieffer-Heeger model...

In the limit that the coherence length is much larger than the lattice spacing, and provided that we are only interested in the low energy physics near to the Fermi surface, a continuum version of the Su-Schrieffer-Heeger model can be derived. This model, derived by Takayama, Lin-Liu, and Maki, is known as the TML model (Takayama et al. 1980). It provides useful analytical results that agree with the Su-Schrieffer-Heeger model in the continuum limit. 

So far in this chapter we have described the static geometrical distortions associated with the electronic states, without paying much regard as to how these distortions arise dynamically. In this section we briefly describe the predicted dynamics of the Su-Schrieffer-Heeger model (introduced in Section 2.8.2). 

The Su-Schrieffer-Heeger model alone is too simplistic to realistically model excited states in conjugated polymers, as electron-electron interactions lead to significantly different predictions. The study of breathers within an interacting electron model has been performed by Takimato and Sasai (1989) and Tretiak et al. (2003). 

The BLA in conjugated polymers can be very well described by an improved Hiickel-type method, the sem-iempirical Longuet-Higgins-Salem (LHS) model [7, 8]. The corresponding solid-state physics approximation is the Su-Schrieffer-Heeger model [9, 10]. The relationship between the LHS and the SSH models was pointed out in Ref. [11]. 

So far we have outlined the conceptual framework in which we discuss charge transfer in organic semiconductors. It is based on a molecular picture where the molecular unit is considered central, with interactions between molecular units added afterwards. For amorphous molecular solids and for molecular crystals this approach is undisputed. In the case of semiconducting polymers, a conceptually different view has been proposed that starts from a one-dimensional (ID) semiconductor band picture, and that is generally known as the Su-Schrieffer-Heeger (SSH) model [21-24]. 

In this chapter we describe the consequences of electron-phonon coupling in the absence of electron-electron interactions. The celebrated model for studying this limit is the so-called Su-Schrieffer-Heeger model (Su et al. 1979, 1980), defined in Section 2.8.2. In the absence of lattice dynamics this model is known as the Peierls model. We begin by describing the predictions of this model, namely the Peierls mechanism for bond alternation in the ground state and bond defects in the excited states. Finally, we reintroduce lattice dynamics classically and briefly describe amplitude-breathers. 

Initial experimental studies [32] [using UV-Vis-NIR and electron spin resonance (ESR) techniques] of highly doped polythiophene showed the bipolaron structure to be dominant. Later, this finding was supported by theoretical studies [33]. However, the majority of the early theoretical studies used the Su-Schrieffer-Heeger (SSH) Pariser-Parr-Pople (PPP) model Hamiltonians and or low-level ab initio calculations such as HF/STO-3G. Semiempirical calculations predicted that two polarons on the oligothio-phene chain would be more stable than a bipolaron for oligomers longer than the dodecamer [34], and many... 

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