The SSH Hamiltonian

The total energy will then be the sum of tlie nuclear kinetic energy plus an effective potential energy 

For P(Ac) here, a=1.222 A is the average distance between adjacent sites along the chain axis, d=2.4 A is the perpendicular distance of the impurity from the chain, V3d is the perpendicular distance between adjacent chains. The lattice site opposite to an impurity is denoted and a lattice site on an adjacent chain n .P . . denotes the charge density on the n -th site of an adjacent chain. This quantity is taken to be identical to the self-coiisistently calculated charge density on the chain treated quantum mechanically, shifted however, in order to match the soliton sites on the adjacent chains. 

The electron-electron repulsion term including diagonal elements only, often referred to as the extended Hubbard term, is  

this interaction is treated within the self-consistent field Hartree-Fock approximation which gives the following mean-field spin-polarized electron-electron interaction Hamiltonian  

is the charge density at the m-th site and Um,. the effective Coulomb repulsion integrals between 2p orbitals attached to sites n and m along the polymer chain. We adopt the screened Ohno expression [33] for these integrals  

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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 ... 

Stafstrom [235] has employed a method that includes the SSH Hamiltonian with dopant and e-e correlation terms, which is discussed in the next section. 

Evidently, the lattice portion is identical to that in the SSH Hamiltonian. 

Outline the major features, strong points and drawbacks of the following theoretical methods EH VEH methods using the SSH Hamiltonian methods using Pariser-Parr-Pople type Hamiltonians methods using Peierls-Hubbard type Hamiltonians. 

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