Electron-phonon operator

The electron-phonon operator is a tensor product between the electronic dipole and the nuclear dipole operators. A mixing between the AA and BB via the singlet-spin diradical AB state is possible now. A linear superposition of identical vibration states in AA and BB is performed by the excited state diradical AB. If the system started at cis state, after coupling may decohere by emission of a vibration photon in the trans state furthermore, relaxation to the trans... 

It is clear that the standard molecular approach and the Q C model are correlated in the sense that their corresponding Hamiltonians are isometric. However, within the Q C model, the handling of the separation between electrons and nuclei is conceptually different. In the latter, the electrons and a PCB subsystem endowed with mass are coupled via electron-phonon operators usually employed for descrihing Jahn-Teller effects... 

It is also shown that the electron-phonon interaction is operative in the polymerization process of the one-dimensional conjugated polymeric crystals a simple dynamical model for the polymerisation in polydiacetylenes is presented that accounts for the existing observations. 

The term H e is the electron correlation operator, the term H p corresponds to phonon-phonon interaction and H l corresponds to electron-phonon interaction. If we analyze the last term H l we see that when using crude approximation this corresponds to such phonons that force constant in eq. (17) is given as a second derivative of electron-nuclei interaction with respect to normal coordinates. Because we used crude adiabatic approximation in which minimum of the energy is at the point Rg, this is also reflected by basis set used. Therefore this approximation does not properly describes the physical vibrations i.e. if we move the nuclei, electrons are distributed according to the minimum of energy at point Rg and they do not feel correspondingly the R dependence. The perturbation term H) which corresponds to electron-phonon interaction is too large... 

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 ... 

Thus if u, is the displacement of an ion at site j arising from phonons, then the one-electron density operator is approximated by... 

In terms of the creation-annihilation electron and phonon operators the Hamiltonian can be cast as follows ... 

Here d ,dl and airaj are annihilation and creation operators for the QD electrons and phonons, respectively. As in case (1), Mq is a semiconductor electron-phonon constant and a>fD is a phonon frequency. A-D is the energy of noninteracting electrons and 3 is a Coulomb integral. 

It should be noted that some mechanism other than the electron-phonon coupling of BCS theory must be operative for organic superconductors (Tc < 13 K) and for high-temperature cuprate superconductors (Tc< 125 K). 

As a result of this type of Hamiltonian transformation the linear electron-phonon interaction disappears, an effective interaction between the JT centers is created, but the Hamiltonian may become very complicated. The transfer of the mixing of the phonon and electron operators to the other Hamiltonian terms is the price for the accuracy of the canonical transformation. Of course, in this case also the problem can not be solved exactly and some approximations should be applied. 

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

As easily checked, this operator commutes with Hj, leading to the local conservation law of pseudospin angular momentum in the electron-phonon coupled system. If (4) is assumed, the total pseudospin rotation operator T = j) conserved in... 

The commutator and anticommutator operations in Hilbert space can thus be implemented with a single multiplication by a and + superoperator, respectively. We further introduce the Liouville space-time ordering operator T. This is a key ingredient for extending NEGFT to superoperators when applied to a product of superoperators it reorders them so that time increases from right to left. We define (A(t)) = Tr A(f)Peq where peq = p(t = 0) represents the equilibrium density matrix of the electron-phonon system. It is straightforward to see that for any two operators A and B we have... 

The system-reservoir interaction operators are products of the type Vak = AR, where A represents a molecule operator (aj, a, B, B) and R a reservoir operator b, b ). Because of the BO approximation the electronic operators commute with the phonon operators. Moreover, the pseudolocalized phonon operators commute with those of the baind phonons because of the boson commutation rules. This amounts to the statement that all molecule variables commute with the reservoir variables, [A,R] = 0. 

Influence of the doping concentration on the electron-phonon coupling in silicon is an important issue for Si-based nanoscale devices. The electron-phonon coupling is weak at low teiiqterature and electrons and phonons can attain different temperatures even when a small heat flow introduced into the system. In some circumstances, strong hot electron effects can restrain the operation of nanoscale devices, such as microbolometers and microcoolers. 

The electron-phonon coupling constant as a function of the doping level in silicon is presented in Fig. 2. The coupling is approximately directly proportional to the electron carrier concentration in the heavily doped silicon, but the electrical resistance of the silicon only slightly depends on the carrier concentration i n this range. This can be used for optimization of thermal characteristics of different microdevices operating at low tenqieratures. 

The term V(X) Is called the operator for electron-phonon coupling. We can see that If It Is zero the llneshape not far from the maximum Is Lorenzlan... 

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