Electron-phonon Hamiltonian

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. 

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... 

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... 

The very simplest theoretical approach, with linear electron-phonon coupling, is in terms of a two-center (a,b) one-electron Hamiltonian (27), with just one harmonic mode, u>, associated with each center. This is (in second quantized notation, with H = 1) ... 

Since electrons are much faster than nuclei, owing to Wg Mj, ions can be considered as fixed and one can thus neglect the //ion-ion contribution (formally Mion-ion Hee, where Vion-ion is a Constant). This hrst approximation, as formulated by N. E. Born and J. R. Oppenheimer, reflects the instantaneous adaptation of electrons to atomic vibrations thus discarding any electron-phonon effects. Electron-phonon interactions can be a-posteriori included as a perturbation of the zero-order Hamiltonian Hq. This is particularly evident in the photoemission spectra of molecules in the gas phase, as already discussed in Section 1.1 for nJ, where the 7T state exhibits several lines separated by a constant quantized energy. 

The symbol V(q,Q) stands for a kinematic operator containing spin-orbit terms, electron-phonon couplings and, eventually, a coupling to external fields. The molecular Hamiltonian is given by ... 

Consider now the equality Hoj> n Jm=8j>j. Thus, in this model, preparing the system in the ground state of the Coulomb Hamiltonian, no time evolution can be expected if we do not switch on the kinematic couplings. We take a simple case where the electron-phonon coupling is on. The matrix elements of H in this base set look like ... 

The magnitude of the off-diagonal Hamiltonian (i.e. the energy transfer rate) thus depends on the strengths of the electron-phonon and Coulombic couplings and also the overlap of the two exciton wavefunctions[53]. Energy transfer rates from states to state jx, are calculated via the golden rule [54] and used as inputs to a master equation calculation of the excitation transfer kinetics in PSI, in which the dynamical information is included in the matrix K. 

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,... 

In this section, a model which gives the basis of the present study is introduced to investigate the electronic properties of A,Cfio [17]. First, the one-electron part of the Hamiltonian, which describes the itinerant motion of the flu electrons in terms of the electron transfer T, is given. Next, the electron-electron interaction U and the electron-phonon interaction S are examined U represents the Coulomb repulsion between the t u electrons and S represents the coupling of the fiu electrons to the intramolecular phonons of the Cdynamical aspect of S is pointed out. 

The electronic Hamiltonian He is now augmented by the electron-phonon interaction [61],... 

Given the lattice Hamiltonian Eq. (5), which casts the interactions in terms of site-specific and site-site interaction terms, a complementary diabatic representation can be constructed which diagonalizes the Hamiltonian excluding the electron-phonon interaction, Hq = He + f7ph. This leads to the form... 

Following the analysis of Refs. [54,55,72], we now make use of the fact that the nuclear modes of the Hamiltonian Eq. (8) produce cumulative effects by their coupling to the electronic subsystem. From Eq. (9), the electron-phonon interaction can be absorbed into the following collective modes,... 

To calculate the distortion due to the Pieirls instability and determine the bandgap Hamiltonian including the electron-phonon interaction is diagonalized and its eigenvalues are calculated. The electron-electron interaction is neglected. The eigenvalues are given by... 

A QD Hamiltonian includes both Coulomb and electron-phonon interactions. Apparently, the phonon modes (denoted as QD) in the quantum dot are different from the semiconductor ones. The electron-phonon interaction determines relaxation processes in quantum dot (hot electrons or excitons). Thus, the QD Hamiltonian yields... 

In this subsection, we include electron-electron and electron-phonon interactions into the QD Hamiltonian. Thus, the QD Hamiltonian yields... 

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... 

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