The parabolic approximation

Figure 4 demonstrates that in order to variationally describe a realistic barrier shape (Eckart potential) by an effective parabolic one, the frequency of the latter, should drop with decreasing temperature. At high temperatures, T > T, transitions near the barrier top dominate, and the parabolic approximation with roeff = is accurate. 

In Eq. 7.16,0 can be approximated using Eq. 7.18 the integral s value is dominated by 5rji 0, and therefore its limits can be taken to be oo, even though the parabolic approximation is valid only near equilibrium,... 

The terms represent successive approximations to a function at small values of its argument by the value of the function at the origin, the straight-line approximation of the function at the origin, the parabolic approximation of the function at the origin, and so forth. 

Fk,. 2.—Validity of the parabolic approximation for the total interaction potential in the evaluation of the integral in eqn (12). Solid curves are exact values, while points represent quantities determined... 

Except for K, all the integrals are centered at extrema. This allows the use of the parabolic approximation for the interaction potential. One obtains... 

Substituting the parabolic approximation (14) into the integrand yields ... 

A simple procedure consists of using the parabolic approximation of the enthalpy in the vicinity of its minimum at... 

Figure 10.7 Comparison of the results of numerical calculations of the relaxation and of the parabolic models with experimental results. Solid lines experimental results by Seidel-Morgenstern [41]. Dot-dashed line profiles calculated with the relaxation model. Dotted lines profiles calculated with the parabolic approximation. The figure on each subfigure indicates the duration of the rectangular injection. Reproduced from F. James, M. Pastel, M. Sepulveda, Physica D, 138 (2000) 316.(Fig. 8).

James et al. [37] discussed the numerical solutions of fwo above mentioned models in defail and compared the numerical solution of fwo mefhods for the same value of e for various initial and boimdary values. The differences between the solution of fhe relaxation method and the solution obtained using the parabolic approximation increases with increasing the value of e, of the injection voltune and of the concentration of the solute injected into the system. They also showed that the parabolic approximation is valid in a relatively wide range of injected amounts and of relaxation parameters. The parabolic approximation has the big advantage of requiring a much shorter computation time than the calculation of a solution of the complete relaxation model. 

Figure 3.6. Frozen solvent reaction coordinate potentials for aqueous C1 +CH3C1 reaction near standard conditions versus y = x — xK As in Figure 3.5A, the instantaneous potential V(S, x) is the sum of the gas phase potential U x) and the cage potential v S x) U x) is that of Figure 3.2 and v S x) is estimated [23] from the molcular dynamics results of ref. 14a. The parabolic approximation to V S x) of Eq. (3.48) is also shown.

In the simple case in which the parabolic approximation holds, F(k) = h (k +kj + kl)/2m = h k /2m for a conduction band with a constant effective mass m of the electrons, we have... 

A more sophisticated approach to using the parabolic approximation was developed by Poulsen (1990) who pointed out that within defined mathematical limits the error function expression for the diffusion coefficient can be approximated to the simple parabolic function. He used the following expression for Pick s second law ... 

We see that the exact path deviates markedly from the straight-line but does closely resemble the parabolic approximation. This result is readily understood... 

This formula follows from the parabolic approximation of the Morse potential (which is valid for small displacements x), while... 

While in the one-dimensional case, the new function evaluation can be exploited at each iteration to satisfactorily upgrade the parabolic approximation in the multidimensional case, we have useful information only on the search direction. 

In a number of systems, two geometries are possible for a molecule, even without the involvanent of any electron. There are systans with inversion center, where two equivalent states are present and the energy is exactly the same after inversion. Depending on the size of the activation barrier, this type of molecule may have a spectrum different from the parabolic approximation due to the interaction between two parabolic PES. 

This model is widely used in the calculation of the probabilities of vibrational excitation. As for explicit r dependence of Ubc and Uabj it is controlled by the model accepted. In particular, the parabolic approximation for Ubc is frequently assumed (the free BC is simulated by the harmonic oscillator) and the exponential function is used for Uab- However, whatever the actual form of U, the approximation (10.6) permits the construction of a general pattern of the potential... 

In-spite of the assumptions employed to calculate the normal approach component, namely the parabolic approximation outside the Hertzian region and constant film thickness within it, it is believed that the analysis brings the mathematical representation o unsteady elastoh-ydrodynamic a step nearer to the situation encountered in such important applications as gear teeth, cams and rolling contact bearings. ... 

As a matter of fact the parabolical approximation for P can be regarded as the first 2 symmetrical terms of a Taylor series expansion, whereas the Reynolds equation is weighted with a... 

A parabolic potential results in harmonic oscillation. At high vibrational excitation energies the parabolic approximation is poor. 

Fig. 4.6 Orientational fluctuation relaxation time as a function of sample thickness for nematic liquid crystal 5CB on rubbed Nylon with strong anchoring at the boundaries. The squares are the measured data and the solid line best fit of the parabolic approximation. Reprinted with permission from [8]. Copyright by the American Physical Society...

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