SSH Hamiltonian

In the one-dimensional limit, the coupling to acoustic phonons is at the heart of the Su-Schrieffer-Heeger (SSH) Hamiltonian [21] ... 

The Hamiltonians of relevance here are the molecular crystal [Eq. (4)] and the SSH Hamiltonians [Eq. (5)]. The electron-phonon interactions have the following forms [45] ... 

On the other hand, the SSH Hamiltonian introduced in 1979 [59] assigns both facts to the Peierls instability. It is a one-electron tight-binding Hamiltonian in which the electron-phonon interaction is explicitly included, such that... 

The cause or causes of the opening of a gap in the band structure of trans-PA has been the subject of many theoretical papers and of much debate (see Chapter 11, Section IV.A and reviews and discussions in [17,146,147,181]). It would seem that electron-phonon and electron-election interactions are of comparable importance. If electron correlations are treated by adding a Hubbard on-site interaction term to the SSH Hamiltonian, the available experimental results for tram-PA are best accounted for by taking about equal values for the electron-phonon coupling X and for the Hubbard U. It might be that in other CPs the importance of electron correlations is greater. Note, however, that a U term (on-site interactions) is not enough to treat the correlations correctly, especially if excitons are to be studied (see the discussion of the PDA case above). 

Therefore the electronic part of the SSH Hamiltonian contains two adjustable parameters, namely B and the so-called electron-phonon coupling constant a. In original SSH-theory they are chosen such, that for an infinite ideally dimerized chain the n-band width and the single particle gap of tPA are reproduced, leading to B =-2.5 eV and a = 4.1 eV/A [1]. 

The SSH Hamiltonian with an additional term describing the external electric field has the form ... 

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,... 

For polyacetylene, many authors apply the so-called SSH Hamiltonian introduced by Su, Schrieffer, and Heeger (1980), which is an alternative to the Longuet-Higgins model. The difference is only in the handling of the a core potential function f and in the parametrization of the model. 

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