Multiquanta states derived from Davydov's mod D1) ansatz: I. Equations of motion for the Su-Schrieffer-Heeger Hamiltonian

Abstract

We present equations of motion for the Su-Schrieffer-Heeger (SSH) Hamiltonian derived with the help of ansatz states similar to Davydov's so-called mod D1) state for soliton dynamics in proteins. Such an ansatz state allows for quantum effects in the lattice and goes beyond previous calculations which mostly apply adiabatic models. In the most general case, called mod Phi 0), which is treated here in detail, we assume that the coherent-state amplitudes for the lattice depend on the site and the molecular orbital of the electrons. The equations of motion are derived from the Lagrangian of the system, a method which is equivalent to the time-dependent variational principle. In the resulting equations we find that, although the SSH Hamiltonian is a one-particle operator, indirect electron-electron interactions are present in the system which originate from the electron-phonon interactions. Inclusion of direct electron-electron interactions, will give insight into the interplay between electron-electron and electron-phonon interactions which can lead effectively to an attractive force between the electrons in systems other than polyacetylene, where bipolarons are known to be unstable. Further with our time-dependent wavefunction also vibrational details of absorption spectra can be computed. From the equations of motion several approximations can be derived. In a further approximation, mod Phi 2), the dependence of the coherent-state amplitudes on the lattice site is neglected. This mod Phi 2) ansatz state consists of a simple product of the electronic and the lattice wavefunctions; however, the electrons are not constrained to follow the lattice dynamics instantaneously as in the adiabatic case. Finally the classical adiabatic case is discussed on which soliton-dynamics simulations are usually based. Further we discuss how to include temperature effects in our model. Applications to soliton dynamics are discussed for the example of the mod Phi 2) model with emphasis on the dependence of the results on soliton width and temperature. We found that in contrast to results reported in the literature, where a similar ansatz is used, but only one electron is treated explicitly, the solitons remain stable also for small soliton widths. This indicates that the interactions of the electrons not occupying the soliton level with the lattice have a stabilizing effect on the soliton. Further our results indicate that the temperature model using random forces and dissipation terms to introduce temperature effects has to be applied with extreme care in this case due to the strong electron-lattice interactions.