Decoupling approximations

Green S., Monchik L., Goldflam R., Kouri D. J. Computational tests of angular momentum decoupling approximations for pressure broadening cross sections, J. Chem. Phys. 66, 1409-12 (1977). 

Yasuda [55] obtained a correlation energy functional Ec[ D] from the first-and second-order density equations together with the decoupling approximations for the 3- and 4-reduced density matrices. The Yasuda functional is capable of properly describing a high-density HEG [56] and encouraging results were reported for atoms and molecules [55]. Some shortcomings of this functional were also pointed out [57]. 

It remains now to solve Eq. (2.3). Here, there are various approaches, depending on the conditions. When a non-steady-state solution is required, one can introduce the decoupling approximation of Sumi and Marcus, if there is the difference in time scales mentioned earlier. Or one can integrate Eq. (2.3) numerically. For the steady-state approximation either Eq. (2.3) can again be solved numerically or some additional analytical approximation can be introduced. For example, one introduced elsewhere [44] is to consider the case that most of the reacting systems cross the transition state in some narrow window (X, X i jA), narrow compared with the X region of the reactant [e.g., the interval (O,Xc) in Fig. 2]. In that case the k(X) can be replaced by a delta function, fc(Xi)A5(X-Xi). Equation (2.3) is then readily integrated and the point X is obtained as the X that maximizes the rate expression. The A is obtained from the width of the distribution of rates in that system [44]. 

We now turn to the case T < C t. Then a decoupling approximation [26] yields... 

To derive Eq. (9-44), a decoupling approximation was used so that a fourth rank tensor could be replaced by a product of second rank tensors. 

Using the Green function method and some decoupling approximations corresponding to the RPA and the Zeeeman reduced splitting smaller than Debye phonon quantum, it is possible to get the dispersion equation... 

For Eqs. (15)-(17) to be a self-contained rheological theory, the fourth-order moment Q has to be related to the second-moment 5 by a closure approximation. Such a closed theory describes the LCP orientation by the second-rank tensor S, and the director can be identified as the eigenvector for the largest eigenvalue. Doi ° introduced a decoupling approximation 5 Q = S.SS, which turns out to be unsatisfactory as it artificially suppresses director tumbling.More sophisticated closure models have since appeared, and their impact on the theory s prediction has been... 

This equation underlines the need to know the local field topography in the material if one aims at determining the acmal complex value of the effective x . It can be further simplified if the field inside particles is homogeneous (mean field, or decoupling , approximation) [82, 83]. Indeed, in this case the normalized volume averaged product (i.e. the quotient in Eq. (22)) reduces to give... 

Another exciting development gets rid of the limitation (i) from the start by basing the approach on theoretical developments in the theory of fluids [44-49, 52, 224-225]. These theories all are based on the concept of the direct correlation function [226] related to the direct interaction between monomers the total correlation function is expressed approximately in terms of the intramolecular structure factor and this direct correlation function, which needs a suitable decoupling approximation, which can be motivated by suitable... 

The exact solution will then involve the propagation of a large number of coupled 3D wave packets. How many depends of course upon the energy, but around 1000 would, for most purposes, be a minimum. Hence a reduction of the complexity is required. This can be achieved either by decoupling approximations or by introducing a quantum-classical description. 

Vugmeyster L, Raleigh DP, Palmer AGr, Vugmeister BE (2003) Beyond the decoupling approximation in the model free approach for the interpretation of NMR relaxation of macromolecules in solution. J Am Chem Soc 125 8400-8404... 

More recently KUBPERMANN, SCHATZ and BAER /77c/ developed a method for an accurate treatment of complanar collisions of an atom A with a diatomic molecule BC This method was then extended by SCHATZ and KUPPERMANN /77d/ to atom-diatom collisions in a three-dimensional physical space, making use of the "tumbling-decoupling approximation /41b/ In this treatment the collision is conveniently described in a body-fixed coordinate system. Together with the quantum number v of BC-vibration, two quantum numbers j and J are introduced for the rotation of BC-molecule and the overall rotation of the system ABC, respectively. Two corresponding qiiantum numbers m and n. are associated with the projections of the BC-angular momentum... 

After the early stage has passed, the sum temi in eq 2.16 cannot be neglected, and the equation becomes non-linear. Langer et al. [15] tried to circumvent the arising difficulty by a decoupling approximation to p2, which allows Sn(k,t) to be replaced by ((u ) / (tt ))5(fc,<). Then, eq 2.16 is transformed into... 

Figure 10-7 are in qualitative agreement with Monte Carlo simulations and experimental results [10] in the intermediate stage, suggesting theit the decoupling approximation is reasonable for this stage. 

On the other hand the correct value of S pis, +0) is Qap( given by eqn (7.115). Thus in the case of stepwise deformation, the lA approximation amounts to a decoupling approximation... 

Other decoupling approximations are possible but here we shall only examine this approximation since it is mathematically simple. 

In this flow, the diffusion equation can be solved rigorously, so that it is possible to check the accuracy of the decoupling approximation. Since k... 

In Fig. 8.6, eqns (8.153) and (8.157) are compared. Though the decoupling approximation is not correct for small the difference becomes smaller as becomes larger. 

Fig. 8.6. Effect of the elongational flow on the orientational order parameter. For small , the decoupling approximation gives S -1/9, as compared to the rigorous results — /1S.

Unlike the case of flexible polymers the constitutive equation cannot be written in a simple closed form unless we use the decoupling approximation. Since the equation for S p is given by the same equation as that in dilute solution, eqns (8.149) and (9.57) give a closed equation for a,... 

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