Vibrational subsystem

Equation (9) was obtained using the assumption that the vibrational subsystem is in the state of thermal equilibrium corresponding to the initial electron state. The expression for the effective frequency a>eff has the form5... 

Equation (106) shows that the interaction of the proton with the motion of the center of mass, described by the terms proportional to fx, is formally of the same form as the interaction with the medium atoms, and the first three terms in the Hamiltonian in Eq. (106) are equivalent to addition of one more degree of freedom to the vibrational subsystem. Thus, this problem does not differ from that for the process of tunnel transfer of the particles stimulated by the vibrations which were discussed in Section IV. So we may use directly the expressions obtained previously with substitution of the appropriate parameters. 

First, we will consider the symmetric transition and will assume that proton transfer occurs between unexcited vibrational states, the other part of the vibrational subsystem being described by classical mechanics. Then we obtain67... 

Here, He(j) is Hamiltonian of a free electron, V,-(r) is Coulomb s interaction of the electron with the donor ion residue, Hlv( q ) is Hamiltonian of the vibration subsystem depending on the set of the vibration coordinates qj that corresponds to the movement of nuclei without taking into account the interaction of the electron with the vibrations. The short-range (on r) potential Ui(r, q ) describes the electron interaction with the donor ion residue and with the nuclear oscillations. The wave function of the system donor + electron may be represented in MREL in the adiabatic approach (see Section 2 of Chapter 2) ... 

Here JAqq t) is a nonequilibrium function of the excitations distribution in vibrational subsystem at the moment of time t matrix elements pkk(t) determine... 

Thus the energy passed into the vibrational subsystem is equal to hVUV trei) 80 /t B7 that is, virtually all the total energy of the absorbed photon (85 k T) has passed on to the vibrational subsystem by the time of t = frei. 

Such a form of presentation is convenient when the equilibrium values of normal coordinates at the initial or final states and the form of interaction energy between the electron and the vibrational subsystem at initial and final states Vj Kq) and VjP(q) are known. Indeed, since the free energy surfaces Ui and Uf can be written as... 

For symmetric reactions AF(R)=0, and the reorganization energy equals the difference between the energies of interaction of an electron with a medium at initial (or final) equilibrium values of coordinates of effective oscillators. As seen from Eq. (22), in this case it is sufficient to know only the type of interaction between an electron and a vibrational subsystem at the initial and final states, and it may seem at a glance that no additional data on the properties of a vibrational subsystem itself are required. 

However, in this approach the main calculational problem is merely transferred to calculating equilibrium coordinates oi or q f, since this calculation requires a knowledge of the response function of a vibrational subsystem [in the case of a polar dielectric this function is s k, co)] to the external field. 

Equations of type (20) and (22) are more suitable, however, in the cases where there are no data on force constants or vibration frequencies for a vibrational subsystem, which are necessary for calculating by Eq. (9), and the equilibrium values of / or qkoi coordinates are experimentally known. Such an approach may be used in the most effective way for calculating the contribution of intramolecular degrees of freedom into the reorganization energy (see below). In this case, however, one must be convinced that the experimental methods for determining equilibrium positions of atoms (the X-ray method, for example) yield the same values of coordinates as in Eqs. (20) and (22). 

Here feo is the pre-exponential factor proportional to the effective frequency coq of a vibrational subsystem and to the transmission coefficient of the reaction x, AFq is the reaction free energy, Vi is the free energy of approaching of reactants from the infinity to the reaction configuration corresponding to some effective distance R between the reactants, Vf is the free energy of separating the reaction products and Eg is determined by Eq. (12). 

This argument can be generalized to any number of subsystems and energy levels. For the case of a molecular system in a given electronic state, the factorization into translational, vibrational, and rotational contributions gives... 

The phrase full Fast Fourier Transform (FFT) signature is usually applied to the vibration spectrum that uniquely identifies a machine, component, system, or subsystem at a specific operating condition and time. It provides specific data on every frequency component within the overall frequency range of a machine-train. The typical frequency range can be from 0.1 to 20,000 Hz. 

Let us assume that all the nuclear subsystems may be separated into several subsystems (R, q9 Q, s,...) characterized by different times of motion, for example, low-frequency vibrations of the polarization or the density of the medium (q), intramolecular vibrations, etc. Let (r) be the fastest classical subsystem, for which the concept of the transition probability per unit time Wlf(q, Q,s) at fixed values of the coordinates of slower subsystems q, Q, s) may be introduced. 

The brief review of the newest results in the theory of elementary chemical processes in the condensed phase given in this chapter shows that great progress has been achieved in this field during recent years, concerning the description of both the interaction of electrons with the polar medium and with the intramolecular vibrations and the interaction of the intramolecular vibrations and other reactive modes with each other and with the dissipative subsystem (thermal bath). The rapid development of the theory of the adiabatic reactions of the transfer of heavy particles with due account of the fluctuational character of the motion of the medium in the framework of both dynamic and stochastic approaches should be mentioned. The stochastic approach is described only briefly in this chapter. The number of papers in this field is so great that their detailed review would require a separate article. 

The book thus embraces an extended study on a variety of issues within the theory of orientational ordering and phase transitions in two-dimensional systems as well as the theory of anharmonic vibrations in low-dimensional crystals and dynamic subsystems interacting with a phonon thermostat. For the sake of readability, the main theoretical approaches involved are either presented in separate sections of the corresponding chapters or thoroughly scrutinized in appendices. The latter contain the basic formulae of the theory of local and resonance states for a system of bound harmonic oscillators (Appendix 1), the theory of thermally activated reorientations and tunnel relaxation of orientational... 

Valence-deformation vibrations of a molecular subsystem in condensed phase... 

The second equality in Eq. (4.3.2) demonstrates that the harmonic component of the Hamiltonian of the molecular subsystem is diagonalized by the Fourier transform in terms of wave vectors K of vibrational excitations ... 

Control of the lattice vibrations in crystals has so far been achieved only through classical interference. Optical control in solids is far more complicated than in atoms and molecules, especially because of the strong interaction between the phononic and electronic subsystems. Nevertheless, we expect that the rapidly developing pulse-shaping techniques will further stimulate pioneering studies on optical control of coherent phonons. 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

Atomic subsystem transitions nuclear spins electron spins rotation vibration... 

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