Operator phonon annihilation

Introducing the electron and phonon annihilation and creation operators, Oa, al bj, b], we can rewrite the Hamiltonian in the notation of second quantization after a canonical transformation... 

Here ak a ) is the annihilation (creation) operator of an exciton with the momentum k and energy Ek, operator an(a ) annihilates (creates) an exciton at the n-th site, 6,(6lt,) is the annihilation (creation) operator of a phonon with the momentum q and energy u) q), x q) is the exciton-phonon coupling function, N is the total number of crystal molecules. The exciton energy is Ek = fo + tfcj where eo is the change of the energy of a crystal molecule with excitation, and tk is the Fourier transform of the energy transfer matrix elements. 

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

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. 

Here a is a dimensionless constant, 5p(R) is the density fluctuation of the medium at the position R (the center of symmetry of the benzoic acid dimer), 0)D is the Debye frequency, and N is the number of acoustic modes, cot = 7 sound k, (bk) is the Bose operator of creation (annihilation of a acoustic phonon with the wave vector k). In the localized representation we have... 

In the Hamiltonian (205) the operator b] (q) ( , (q)) describes the creation the creation (annihilation) of an optical phonon with frequency 11, (q) and takes into account the polarizing influence of the charged fault on proton subsystem. The operator b2 (q) (b2 (q)) corresponds to creation (annihilation) of an acoustic phonon with frequency fl2(q) relative to new equilibrium site positions in this case the renormalized acoustic phonons become also polarizational ones. This is taken into account in the last term of the Hamiltonian H0 describing the ionic state interaction with the new optical and acoustical vibrations of the chain. 

Here Efn(0) refers to the /th excited state of a free molecule in the crystal a n + (aQ is the Bose operator of creation (annihilation) of an intramolecular vibrational excitation in the nth molecule M2(k) refers to the energy of an optical phonon with the wave vector k connected with proton oscillations in the O H O bridge (bk) is the Bose operator of phonon creation (annihilation) and is the coupling energy between the molecular excitation and phonons. 

The term M p,is the eph coupling constant, and ba is the annihilation operator of the mode a, whose frequency and normal mode coordinate are represented by Q,a and Qp, respectively. The sites for electrons i( T) coupled with phonons are restricted to the C region or a subpart of C. The focused modes should be sufficiently localized on the molecule in term of their definition. Practically, these internal modes can be calculated by means of a frozen-phonon approximation, where displaced atoms are atoms in the c region (or its subpart) denoted as a vibrational box though a check for convergence to the size of the vibrational box is necessary [90]. 

The first two tenns on the right describe the system and the bath , respectively, and the last tenn is the system-bath interaction. This interaction consists of terms that annihilate a phonon in one subsystem and simultaneously create a phonon in the other. The creation and annihilation operators in Eq. (9.44) satisfy the commutation relations ... 

Considering Equation 6.38 again, we need to transform the Hamiltonian expression. Thus, if cos(k) and ss(k) are the frequency and the polarization vector for the classic modes with polarization s and wave vector k, respectively, we can define the phonon creation (aks+) and annihilation ( /,s ) operators as... 

Expanding the quantity q in (3.90) with respect to deviations from equilibrium up to quadratic terms and introducing normal coordinates the Hamiltonian Hl can be written as a sum of Hamiltonians which correspond to harmonic oscillators in their normal coordinates. Then we use the phonon creation and annihilation operators, i.e. the operators 6 r and 5qr (q is the phonon wavevector and r indicates the corresponding frequency branch) and obtain the Hamiltonian Hl in the form... 

To take the interaction between phonons and photons into consideration, it is necessary to add to the Hamiltonian (6.32), the Hamiltonian Ho(a) of the free field of transverse photons and the Hamiltonian Hint for the interaction of the field of transverse photons with phonons. The linear transformation from the operators a and C to the polariton creation and annihilation operators, i.e. to the operators t(k) and p(k), diagonalizes the quadratic part of the total Hamiltonian. The two-particle states of the crystal, corresponding to the excitation of two B phonons, usually have a small oscillator strength and the retardation for such states can be neglected. In view of the afore-said, the quadratic part of the total Hamiltonian with respect to the Bose operators can be written in the form of the sum H0(B) + where... 

Here, ancj are the creation and annihilation operators, respectively, for phonons in mode q = (q, r), where q denotes the vector of the phonon and r is the branch label. The energy of these phonon modes is given by u>q. Furthermore, the single-molecule Hamiltonian as well as the intermolecular transfer interaction are still considered to be operators in phonon space. 

Here the superscript 0 represents the trace with respect to the non-interacting density matrix. The zeroth order Green functions are given in Eq. (55). The terms coming from the lead-molecule coupling (V. ) vanish because they are odd in creation and annihilation operators. Substituting Eq. (C34) in Eq. (C25) gives for the phonon contribution... 

The first two terms are diagonalized by introducing the phonon creation and annihilation operators (see (Cohen-Tannoudji et al. 1977)) ... 

The first two terms denote the reactant and the metal, the last term affects electron exchange between the metal and the reactant c denotes a creation and c an annihilation operator. Just like in Marcus (and polaron) theory, the solvent modes are divided into a fast part, which is supposed to follow the electron transfer instantly, and a slow part. The latter is modeled as a phonon bath after transformation to a single, normalized reaction coordinate q, with corresponding momentum p, the corresponding part of the Hamiltonian is... 

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