Bilinear Coupling Model

The Q model allows an exact formulation for F,(X) for classical solvent modes.The instantaneous energy in this case is given by the bilinear form 

The calculations of the diabatic (no off-diagonal matrix elements) free energy surfaces in Eq. [18] can be performed exactly for Ei(q) given by Eq. [65]. This procedure yields the closed-form, analytical expressions for the free energies P,(X). It turns out that the solution exists only in a limited, one-sided band of the energy gaps Specifically, an asymptotic expansion of the exact solution leads to a simple expression for the free energy 

The parameter Xq establishes the boundary of the energy gaps for which a finite solution E,(X) exists. The band definition and its boundary 

The other model parameters entering Eq. [66] are the nuclear reorganization energies defined through the second cumulants of the reaction coordinate 

The two sets of parameters defined for each state are not independent because of the following connections between them 

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Clearly, the MFI description does not capture all possible complicated mechanisms of ET activation in condensed phases. The general question that arises in this connection is whether we are able to formulate an extension of the mathematical MH framework that would (1) exactly derive from the system Hamiltonian, (2) comply with the fundamental linear constraint in Eq. [24], (3) give nonparabolic free energy surfaces and more flexibility to include nonlinear electronic or solvation effects, and (4) provide an unambiguous connection between the model parameters and spectroscopic observables. In the next section, we present the bilinear coupling model (Q model), which satisfies the above requirements and provides a generalization of the MH model. 

Equations (13.26) and (13.29b) now provide an exact result, within the bilinear coupling model and the weak coupling theory that leads to the golden rule rate expression, for the vibrational energy relaxation rate. This result is expressed in terms of the oscillator mass m and frequency ca and in tenns of properties of the bath and the molecule-bath coupling expressed by the coupling density A ((a)g (a) at the oscillator frequency... 

Equation (13,35) is the exact golden-rule rate expression for the bilinear coupling model. For more realistic interaction models such analytical results cannot be obtained and we often resort to numerical simulations (see Section 13.6). Because classical correlation functions are much easier to calculate than their quantum counterparts, it is of interest to compare the approximate rate ks sc, Eq. (13.27), with the exact result kg. To this end it is useful to define the quantum correction factor... 

Problem 13.3. Show that for the bilinear coupling model, the classical limit rate, and the exact quantum result are identical, and are both given by ... 

Equation (13.39) implies that in the bilinear coupling, the vibrational energy relaxation rate for a quantum hannonic oscillator in a quantum harmonic bath is the same as that obtained from a fully classical calculation ( a classical harmonic oscillator in a classical harmonic bath ). In contrast, the semiclassical approximation (13.27) gives an error that diverges in the limit T 0. Again, this result is specific to the bilinear coupling model and fails in models where the rate is dominated by the nonlinear part of the impurity-host interaction. 

The role of two-phonon processes in the relaxation of tunneling systems has been analyzed by Silbey and Trommsdorf [1990]. Unlike the model of TLS coupled linearly to a harmonic bath (2.39), bilinear coupling to phonons of the form Cijqiqja was considered. In the deformation potential approximation the coupling constant Cij is proportional to (y.cUj. There are two leading two-phonon processes with different dependence of the relaxation rate on temperature and energy gap, A = (A Two-phonon emission prevails at low temperatures, and it is... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

We consider the same reaction model used in previous studies as a simple model for a proton transfer reaction. [31,57,79] The subsystem consists of a two-level quantum system bilinearly coupled to a quartic oscillator and the bath consists of v — 1 = 300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly to the two-level quantum system. In the subsystem representation, the partially Wigner transformed Hamiltonian for this system is,... 

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. 

In this model, a two-level system is coupled to a classical nonlinear oscillator with mass Mo and phase space coordinates (Pq,Pd)- This coupling is given by hyoRo- The nonlinear oscillator, which has a quartic potential energy function Vn Ro) = aPg/4 — Moco Ro/", is then bilinearly coupled... 

This rate has two remarkable properties First, it does not depend on the temperature and second, it is proportional to the bath density of modes g(ct>) and therefore vanishes when the oscillator frequency is larger than the bath cutoff frequency (Debye frequency). Both features were already encountered (Eq. (9.57)) in a somewhat simpler vibrational relaxation model based on bilinear coupling and the rotating wave approximation. Note that temperature independence is a property of the energy relaxation rate obtained in this model. The inter-level transition rate, Eq. (13.19), satisfies (cf. Eq, (13.26)) k = k (l — and does depend on temperature. 

We explain here the operation principles of simple molecular devices, a thermal rectifier [20] and a heat pump [21]. First we present the heat current in the anharmonic (TLS) model. Figure 12.2 demonstrates that the current ino-eases monotonicaUy with AT, then saturates at high tanperature differences. It can be indeed shown that dJ/dAT > 0, which indicates that negative differential thermal conductance (NDTC), a decrease of J with increasing AT, is impossible in the present (bilinear coupling) case. As shown in Ref [19], NDTC requires nonlinear system-bath interactions, resulting in an effective temperature-dependent molecule-bath coupling term. 

A high symmetry of the molecule does not only help to (sometimes dramatically) reduce the number of parameters, it also provides a solid basis for the vibronic coupling model Hamiltonian. When the two interacting electronic states are of different symmetry (as assumed here), the interstate coupling must be an odd function of the couphng coordinate. Hence, there can be no constant or quadratic terms, only linear or bilinear ones are allowed. The vibronic coupling Hamiltonian was first derived by Cederbamn et and is more fully described in a review article by Kbppel et and in Chapter 7 of this book. 

Chain-type configuration of the residual bath (see Fig. 15.1, center panel). In this scheme, the bilinear coupling matrix is cast into a band-diagonal form, such that only the first layer of the residual bath is coupled to the effective-mode subspace. We have referred to this model as a hierarchical electron-phonon (HEP) model [38]... 

The model Hamiltonian (37) obtained from Eq.(32) contains solute oscillators linearly perturbed by its coupling with the solvent as well as bilinear terms that break down a total separability between solute and solvent ... 

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