The bilinear interaction model

Consider first the model described by Eqs (13.6), (13.7), (13.10), and (13.11) where the hannonic oscillator under study is coupled bi-linearly to the harmonic bath. The relevant correlation functions ( F(Z),.F(0) ) and (F(Z)F(0))c should be calculated with F = and its classical countei part, where Uj are coordin- 

Problem 13.2. For a harmonic thermal bath show that 

Solution-. Using the fact that different normal modes are uncorrelated, the classical correlation function is computed according to 

Note that the bath spectral density associated with the coupling coefficient A was defined as (cf. Eqs (6.90), (6.92)) 

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 

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Equations (13.6), (13.7), (13.9), and (13.10) provide the general stracture of our model. The relaxation process depends on details of this model mainly through the form and the magnitude of T/sb- We will consider in particular two fonns for this interaction. One, which leads to an exactly soluble model, is the bilinear interaction model in which the force F( ry ) is expanded up to first order in the deviations rj of the solvent atoms from their equilibrium positions, F( ry ) = F ( rJ )5ry,... 

Here, I denotes the lattice sites, while and Xfo are the a components of the n-dimensional momentum and displacement vectors, respectively. M denotes the mass of particle A, C is the model parameter, and h is a homogeneous external field. The bilinear interaction is restricted to the nearest neighbor only. For the anharmonic part of the interaction ... 

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... 

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. 

To conclude this section let us note that already, with this very simple model, we find a variety of behaviors. There is a clear effect of the asymmetry of the ions. We have obtained a simple description of the role of the major constituents of the phenomena—coulombic interaction, ideal entropy, and specific interaction. In the Lie group invariant (78) Coulombic attraction leads to the term -cr /2. Ideal entropy yields a contribution proportional to the kinetic pressure 2 g +g ) and the specific part yields a contribution which retains the bilinear form a g +a g g + a g. At high charge densities the asymptotic behavior is determined by the opposition of the coulombic and specific non-coulombic contributions. At low charge densities the entropic contribution is important and, in the case of a totally symmetric electrolyte, the effect of the specific non-coulombic interaction is cancelled so that the behavior of the system is determined by coulombic and entropic contributions. 

Recent theoretical treatments of the soft-mode behaviour include a detailed study by Onodera using classical mechanics, and a theory of hydrogen-bond mechanics, including tunnelling effects, by Stamenkovic and Novakovic. ° Onodera assumes a quartic potential function for his individual oscillators, with a bilinear interaction which reduces to c x, where x is the displacement, under the Weiss-molecular-field approximation. The model is soluble without further approximation (in series of elliptic functions), yielding the temperature variation of frequency and damping. If the quartic potential has a central hump larger than kTc,... 

The bilinear model has been used to model biological interactions in isolated receptor systems and in adsorption, metabolism, elimina- tion, and toxicity studies, although it has a few limitations. These include the need for at least 15 data points (because of the presence of the additional disposable parameter jS and data points beyond optimum LogP. If the range in values for the dependent variable is limited, unreasonable slopes are obtained. 

Figure 7.10. Left panel shows the intrasite exchange interactions used in the generalized Heisenberg model. Shown on the right hand panel are the (bilinear) intersite exchange interactions.

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. 

However, despite the simplicity of the analyses and the good correlations obtained in these studies, a ligand interaction-based model like the CoMFA method should not be used to model nonlinear effects arising from transport and distribution no reasonable results can be expected for sets of compounds which are no homologous series. Better and theoretically more consistent alternatives would be the addition of suitably weighted log P values to the CoMFA table, the use of lipophilicity similarity matrices (chapter 9.4), or the correlation with log P values in the classical manner, applying either the parabolic or the bilinear model. 

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. 

The underlying notion in bilinear modeling is that something causes the systematic variabilities in the X data. But we may not correctly know what it is there may be surprises in the data due to unexpected interferents, chemical interactions, nonlinear responses, etc. An approximate model of the subspace spanned by these phenomena in X is created. This X model is used for stabilizing the calibration modeling. The PLS regression primarily models the most dominant and most y-relevant of these X phenomena. Thus neither the manifest measured variables nor our causal assumptions about physical laws are taken for granted. Instead we tentatively look for systematic patterns in the data, and if they seem reasonable, we use them in the final calibration model. 

Besides the nonlinear models and, specifically, the parabolic model, other models were proposed for nonlinear dependence of the biological response from hydrophobic interactions. Among them, the most important are the Hansel bilinear models [Kubinyi, 1977 Kubinyi, 1979] such as ... 

In addition, bilinear lipophilicity-activity and/or molar rcfractivity- activity relationships are frequently observed for ligand-enzyme interactions (cf. Sec. 6) [60,64,66 68]. In these relationships, they model the limited size of a lipophilic binding pocket. 

A possibility to overcome this limitation of the above conical-intersection models, at least in a quahtative manner, is to consider anhar-monic couplings of the active degrees of freedom of the conical intersection with a large manifold of spectroscopically inactive vibrational modes. The effect of such a couphng with an environment has been investigated for the pyrazine model in the weak-coupling limit (Redfield theory) in Ref. 19. The simplest ansatz for the system-bath interaction, which is widely employed in quantum relaxation theory assumes a coupling term which is bilinear in the system and bath operators... 

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