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Lattice relaxation in the 1B u state for the finite polyenes
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24.Note that the two different parametrization of the electron-electron interaction do not give exactly the same qualitative trend as a function of scaling in U and ε for many properties, such as the potential surface which we will show later. In the U parametrization, the Hückel limit is recovered by setting while electron-electron interaction always exists in the ε parametrization. We get the Hubbard model as since the on-site electron repulsion is U for all ε.
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26.As pointed out by Shuai et al. (Ref. 20) the Huang-Rhys factor, defined by tends to decrease first with the increasing of the chain length but starts to increase after some critical length in the Huckel model, which has been been used to explain the photoabsorption experiments on short polyenes (Ref. 27) and β-carotene (Ref. 28). In the presence of electron correlation, the same experimental results can be qualitatively explained by the chain length dependence of the bipolaron formation discussed here (Ref. 29). However, the calculation performed here is for polyene chains with periodic boundary condition, further calculation for the open chains need to be performed.
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