Phonon Hamiltonian

To describe the effect of the change of the elastic springs on the optical spectrum of an impurity center, we use the adiabatic approximation. In this approximation, phonons are described by different phonon Hamiltonians in different electronic states. The optical spectrum, which corresponds to a transition between different electronic states is determined by the expression /( >) = const X oj1 1 I(oj) [28], where the — sign corresponds to the absorption spectrum and the + sign stands for the emission spectrum,... 

While the electronic structure calculations addressed in the preceding Section could in principle be used to construct the potential surfaces that are a prerequisite for dynamical calculations, such a procedure is in practice out of reach for large, extended systems like polymer junctions. At most, semiempirical calculations can be carried out as a function of selected relevant coordinates, see, e.g., the recent analysis of Ref. [44]. To proceed, we therefore resort to a different strategy, by constructing a suitably parametrized electron-phonon Hamiltonian model. This electron-phonon Hamiltonian underlies the two- and three-state diabatic models that are employed below (Secs. 4 and 5). The key ingredients are a lattice model formulated in the basis of localized Wannier functions and localized phonon modes (Sec. 3.1) and the construction of an associated diabatic Hamiltonian in a normal-mode representation (Sec. 3.2) [61]. 

This Hamiltonian is similar to the usual electron-phonon Hamiltonian, but the vibrations are like localized phonons and q is an index labeling them, not the wave-vector. We include both diagonal coupling, which describes a change of the electrostatic energy with the distance between atoms, and the off-diagonal coupling, which describes the dependence of the matrix elements tap over the distance between atoms. 

Since the phonon energy quantum hu> does not explicitly appear in both Eqs. 35 and 36, these formulas for the rate constant in the semiclassical regime should be derived without the single-mode model being relied on. That is, we should not neglect that various phonon modes contributing to the reaction coordinate Q have, in reality, various energy quanta. In this case, the phonon Hamiltonian in the reactant state (where the electron to be transferred is at the donor) consists of various normal modes. 

The rate-constant formula (Eq. 20) in the non-adiabatic limit is applicable irrespective of whether the phonon Hamiltonians Hr and Hn for the reactant and the... 

The 40 intramolecular Hg modes are expected to be nearly dispersionless in the solid state. For this case, the electron-phonon Hamiltonian is particularly simple, and may be written as... 

When obtaining Eq. (357), we let fIt > flt at least by several times. Hence, instead of the exciton-phonon Hamiltonian (341), one can turn to the exciton-polariton Hamiltonian ... 

The lattice electron-phonon Hamiltonian has such a view... 

The electron-phonon Hamiltonian used for the explanation of magnetoelastic effects, eq. (14), has to be generalized for RAI2 because of the strain-optic-phonon couphng mentioned above. This leads to an additional mixed term in the lattice potential... 

We are interested in a situation where the extra particles in the lattice are described by a single band Hubbard Hamiltonian coupled to the acoustic phonons of the lattice as given in Equation 12.12 [ 128]. In the latter equation, the first and second terms describe the nearest-neighbor hopping of the extra-particles with hopping amplitudes J, and interactions V, computed for each microscopic model by band-structure calculations for Uj = 0, respectively. The third term is the phonon Hamiltonian. The fourth term is the phonon coupling obtained in lowest order in the displacement... 

Here, the molecular vibrational motions are explicitly involved. He represents the carrier Hamiltonian of the sites and it is given by constant 8y in eqn (12.69). The phonon Hamiltonian Hph is written as a collection of the harmonic oscillators in mass weighted coordinates as follows ... 

In order to investigate the dynamics of the charge carrier with the fluctuations caused by the harmonic oscillator bath, one can translate the total Hamiltonian eqn (12.72) into the interaction representation of the phonon Hamiltonian Hp. ... 

We next draw the analogy between the phonon hamiltonian and that of a collection of independent harmonic oscillators. We begin with the most general expression for the displacement of ions, Eq. (6.22). In that equation, we combine the amplitude c with the time-dependent part of the exponential exp(—i t) into a new variable Q t), and express the most general displacement of ions as ... 

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