Linear coupling model

Eq. [33] according to the assumption of the classical character of this collective mode. Depending on the form of the coupling of the electron donor-acceptor subsystem to the solvent field, one may consider linear or nonlinear solvation models. The coupling term - Si -V in Eq. [32] represents the linear coupling model (L model) that results in a widely used linear response approximation. Some general properties of the bilinear coupling (Q model) are discussed below. 

It has been noted [95] that 2 in Eq. 34 plays two rather distinct roles 2 in the quadratic expression is a solvation energy which defines (with AC°) the vertical gap at equilibrium, whereas the other 2 s control the width of the Gaussian distribution characteristic of the linear coupling model. 

In this Section we apply the general formalism developed in Section 13.3 together with the interaction models discussed in Section 13.2 in order to derive explicit expressions for the vibrational energy relaxation rate. Our aim is to identify the molecular and solvent factors that determine the rate. We will start by analyzing the implications of a linear coupling model, than move on to study more realistic nonlinear interactions. 

We have seen that vibrational relaxation rates can be evaluated analytically for the simple model of a hannonic oscillator coupled linearly to a harmonic bath. Such model may represent a reasonable approximation to physical reality if the frequency of the oscillator under study, that is the mode that can be excited and monitored, is well embedded within the spectrum of bath modes. However, many processes ofinterest involve molecular vibrations whose frequencies are higherthan the solvent Debye frequency. In this case the linear coupling rate (13.35) vanishes, reflecting the fact that in a linear coupling model relaxation cannot take place in the absence of modes that can absorb the dissipated energy. The harmonic Hamiltonian... 

To verify effectiveness of NDCPA we carried out the calculations of absorption spectra for a system of excitons locally and linearly coupled to Einstein phonons at zero temperature in cubic crystal with one molecule per unit cell (probably the simplest model of exciton-phonon system of organic crystals). Absorption spectrum is defined as an imaginary part of one-exciton Green s function taken at zero value of exciton momentum vector... 

Of particular interest is the model of a bath as a set of harmonic oscillators qj with frequencies cOj, which are linearly coupled to the tunneling coordinate... 

A transition linearly coupled to the phonon field gradient will experience, from the perturbation theory perspective, a frequency shift and a drag force owing to phonon emission/absorption. Here we resort to the simplest way to model these effects by assuming that our degree of freedom behaves like a localized boson with frequency (s>i. The corresponding Hamiltonian reads... 

Zhu and Nakamura treated the two-state time-independent linear potential model (two linear diabatic potentials coupled by a constant diabatic coupling)... 

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

Instead of this nonlinear coupling, a linear coupling provides an exactly solvable model with the Hamiltonian (S.P. Kim et.al., 2000 2002 2001)... 

The strategy, usually adopted to achieve a theoretical description of this complex dynamics, is to describe the influence of the solvent environment on the electron-transfer reaction within linear response theory [5, 26, 196, 197] as linear coupling to a bath of harmonic oscillators. Within this model, all properties of the bath enter through a single function called the spectral density [5, 168]... 

Zimberg, M. J., S. H. Prankel, J. P. Gore, and Y. R. Sivathanu. 1998. A study of coupled turbulence, soot chemistry and radiation effects using the linear eddy model. Combustion Flame 113 454-69. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

In Reference [35], numerical examples of perturbative Sq - S2 excitation and the S2 IC dynamics for the / -carotene are discussed, too. The absence of reliable potential surfaces for this system motivated the use of a minimal two-dimensional model [66], which utilizes a Morse potential in each dimension. All three electronic surfaces Sq, and S2 involved in this example assume the same 2D potential form however, these potentials are shifted to each other. More importantly, in Ref. [35], each potential has 396 bound states in each electronic state within this model, while additionally the S2 and electronic states are coupled by linear coupling. Thus, the Q-space and P-space, as introduced in the context of the QP-algorithm in Section 1.3.1, consist of the S2 and 5 bound states, respectively. 

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