Coupling bilinear

As shown by Zwanzig the GLE, Eq. 1, may be derived Ifom a Hamiltonian in which the reaction coordinate q is coupled bilinearly to a harmonic bath ... 

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... 

Just like the Zeeman interaction (S B), the hyperfine interaction (.S /) is a bilinear term and its coupling to strain (T S I), which we will call A-strain (also, -strain ), should be formally similar to the g-strain (T S B) just discussed. In the early work of Tucker on the effective S = 1/2 system Co2+ in the cubic host MgO, a shift in central hyperfine splitting was found to be proportional to the strain-induced g-shift given by Equation 9.22 (Tucker 1966). 

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 ... 

In this section analytical expressions for ENDOR transition frequencies and intensities will be given, which allow an adequate description of ENDOR spectra of transition metal complexes. The formalism is based on operator transforms of the spin Hamiltonian under the most general symmetry conditions. The transparent first and second order formulae are expressed as compact quadratic and bilinear forms of simple equations. Second order contributions, and in particular cross-terms between hf interactions of different nuclei, will be discussed for spin systems possessing different symmetries. Finally, methods to determine relative and absolute signs of hf and quadrupole coupling constants will be summarized. 

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. 

The bilinear coupling case (i.e. position-independent friction) corresponds to G (s) = 1, or equivalently, to Y s=s . The position-dependent friction Eq. (39) can then be rewritten as... 

Figure 2 The coupling Junction g (s) defined in Eq. (36). The deviation from a straight line is the deviation from bilinear coupling. The positions of the transition state, the reactant and product wells are also shown by the dashed vertical lines.

M. Rupprecht and T. Probst, Development of a method for the systematic use of bilinear multivariate calibration methods for the correction of interferences in inductively coupled plasma-mass spectrometry. Anal. Chim. Acta, 358, 1998, 205-225. 

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