Non-local coupling

In addition to the entropy term we assume that there is an extra local coupling between the fields via and, in addition to the coulombic coupling which is long range, we assume the existence of a short-range non-local coupling via We can choose several functional forms to... 

Slinko, M.M., Ukharskii, A.A., and Jaeger, N.I., Global and non-local coupling in oscillating heterogeneous catalytic reactions the oxidation of CO on zeolite-supported palladium, Phys. Chem. Chem. Phys., 3, 1015-1021, 2001. 

Bearing in mind that the connectivity of monomeric units along the backbone of the polymer is an essential ingredient of the single chain dynamics it is clear that a non-local coupling should lead to a better description. In the Rouse model forces acting on a monomer caused by the chain connectivity are additionally taken into account [88,89]. This leads to a kinetic factor that is proportional to the intramolecular pair-correlation function [90-92], P(r,r )... 

X is defined as x s q R.l/6 and g is the Debye fimction. Another model for non-local coupling is the reptation model [89,93] which is appropriate for polymer melts with very long chains, i.e., entangled chains. 

The next significant development in the history of the geomebic phase is due to Mead and Truhlar [10]. The early workers [1-3] concenbated mainly on the specboscopic consequences of localized non-adiabatic coupling between the upper and lower adiabatic elecbonic eigenstates, while one now speaks... 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

Hooke s law relates stress (or strain) at a point to strain (or stress) at the same point and the structure of classical elasticity (see e.g. Love, Sokolnikoff) is built upon this linear relation. There are other relationships possible. One, as outlined above (see e.g. Green and Adkins) involves the large strain tensor Cjj which does not bear a simple relationship to the stress tensor, another involves the newer concepts of micropolar and micromorphic elasticity in which not only the stress but also the couple at a point must be related to the local variations of displacement and rotation. A third, which may prove to be very relevant to polymers, derives from non-local field theories in which not only the strain (or displacement) at a point but also that in the neighbourhood of the point needs to be taken into account. In polymers, where the chain is so much stiffer along its axis than any interchain stiffness (consequent upon the vastly different forces along and between chains) the displacement at any point is quite likely to be influenced by forces on chains some distance away. 

In all of the above cases, a strong non-linear coupling exists between reaction and transport at micro- and mesoscales, and the reactor performance at the macroscale. As a result, the physics at small scales influences the reactor and hence the process performance significantly. As stated in the introduction, such small-scale effects could be quantified by numerically solving the full CDR equation from the macro down to the microscale. However, the solution of the CDR equation from the reactor (macro) scale down to the local diffusional (micro) scale using CFD is prohibitive in terms of numerical effort, and impractical for the purpose of reactor control and optimization. Our focus here is how to obtain accurate low-dimensional models of these multi-scale systems in terms of average (and measurable) variables. 

Newton s second law, 2, 383 NMR shielding, 239 Non-adiabatic coupling elements, 55 Non-bonded energy, in force held methods, 18 Non-bonded list, in force held methods, 43 Non-local DFT methods, 184 Norm-extended Hessian ophmizahon method, 320... 

As was shown in numerous articles, [10-11] and we briefly repeat it here, the non-local effects are properly treated if and only if the matrix, x. that contains the NACTs is quantized. We will not elaborate on this issue now because it will be extensively discussed in the next section however, for the sake of completeness, we just say the following The ADT yields the diabatic PES matrix elements but there is no a-priori guarantee that tiiese potentials are singlevaliied. In fact if the non-local effects are not properly treated these potentials are most likely to become imdtivaliied. In the numerical part that relates to the NACTs of the H-tH2 system, we show that the ad-initio electronic eigenfunctions form quantized nonadiabatic coupling matrices (NACM). 

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