Bilinear coupling terms

Here, a topology-adapted representation [55] was chosen, where (Xi,X2) lift the degeneracy at the intersection and thus span the branching plane [74], These modes are obtained by orthogonalizing the modes (X, Xa) of Eq. (10). The third mode Xg is in turn orthogonal to (Xi, ) and carries information on the intersection space, i.e., the X+ component of Eqs. (9)-(10). Alternative construction schemes are possible in particular, the bilinear coupling terms can be eliminated within the three-mode subspace [54,72]. 

By making use of the form of the Landau energy proposed by de Gennes [17], all coefficients in this expression have been evaluated from experimental data [4]. Using these data to calculate U, one obtains a value that is considerably smaller than the one determined from the linearized analysis outlined above. As has been noted, however, a consistent analysis should not only include terms quadratic in the strains and bilinear coupling terms of the order parameter and the strain, but rather also nonlinear effects as well as nonlinear coupling terms between strain and the nematic order parameter [4]. 

We included the bilinear coupling term fj Mi Pj (which exists for 58 magnetic classes in bulk samples and for almost all surface magnetic classes inherent to nanosystems (see Sect. 4.3.3)). The higher terms MiMjPk and MiMjPtPi are typically small in comparison with the terms linear in magnetization, which are included into Eq. (4.39). 

The situation may change, however, when second-order coupling terms are included. Then there exist also bilinear coupling terms, involving the products QxQg or QyQg in the Hamiltonian (3). Formally these are obtained by the substitution... 

The first two terms reproduce the ground state Hamiltonian, but located on the diagonal of the electronic 2x2 matrix. The two energies ei and C2 are the vertical excitation energies. The next four sums describe the linear and the bilinear coupling terms. 

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. 

In Refs. [51,53], we used an explicit representation of the bath in terms of a collection of oscillators which are bilinearly coupled to the last three members of the HEP hierarchy,... 

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. 

Hence, for the transfer of polarization between two energy-matched spins, the (effective) coupling term must contain both orthogonal bilinear operators, since = 0 if = Q or = 0, that is, if Ia qI = adqI-... 

Homonuclear isotropic coupling terms are not invariant in this toggling frame and can be transformed into nonisotropic bilinear terms ... 

Bilinear coupling in stishovite, as described by the term, 2( 1 - e2)Q in Equation (23), leads to a change in the transition pressure from Pc to P where... 

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... 

The Hamiltonian of Eq. (9.2) couples the reaction coordinate to the environmental oscillator degrees of freedom by terms linear in both reaction coordinate and bath degree of freedom. This is derived in Zwanzig s original approach by an expansion of the full potential in bath coordinates to second order. This innocuous approximation in fact conceals a fair amount of missing physics. We have shown [16a] that this collection of bilinearly coupled oscillators is in fact a microscopic version... 

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 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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