Comparison with Exact Results

These results confirm and complement the earlier work of Beswick and Jortner who compared DWBA widths with those from collinear coupled-channel scattering calculations(33) where, as here, good agreement was observed. In the next section the DWBA widths are calculated for the chemically bonded HCO radical to give H+CO. Based on the present comparisons with exact results we are optimistic that the DWBA will provide realistic widths for this system. 

The analytical forms derived by Dresner (11) for rectangular channels were converted for tubular geometry (, 10). By comparison with exact results obtained by infinite series methods, a semi-empirical form was derived (12) to extend Dresner s analytical result beyond the entrance region. For our experiments the following results are Important ... 

The approximation is very good as can be seen by comparison with exact results or with precise numerical results obtained with the strip method.16 Let us give two examples... 

Unfortunately, the result (13.1.167) is not as good as could be expected, for reasons already explained in connection with the calculation of the radius of gyration (Chapter 4, Section 2.1.2). In fact, eqns (13.1.164) and (13.1.165) represent reality only very crudely (see Section 1.5.6 and Chapter 10, Section 7.2). Actually, the equality (13.1.166) can be tested by comparison with exact results. 

Comparison with Exact Results. It is not unreasonable to suspect that truncation errors in the numerical approximation of first and second derivatives might accumulate in the computational scheme used to integrate the mass transfer equation. One check for accuracy involves a comparison between numerical results and exact analytical solutions. Of course, only a limited number of analytical solutions are available. For example, the following solutions have been obtained analytically for catalytic duct reactors ... 

Another problematic system for the ALDA is the streched H2 molecule [61,62]. Prom a comparison with exact results it was found that the ALDA fails to reproduce even qualitatively the shape of the potential curves for the and states. A detailed analysis of the problem shows that the failure is related to the breakdown of the simple local approximation to the kernel. 

Equation 13.5-2 is the segregated-flow model (SFM) with a continuous RTD, E(t). To what extent does it give valid results for the performance of a reactor To answer this question, we apply it first to ideal-reactor models (Chapters 14 to 16), for which we have derived the exact form of E(t), and for which exact performance results can be compared with those obtained independently by material balances. The utility of the SFM lies eventually in its potential use in situations involving nonideal flow, wheic results cannot be predicted a priori, in conjunction with an experimentally measured RTD (Chapters 19 and 20) in this case, confirmation must be done by comparison with experimental results. 

In Fig. 2 we compare results using e = 0.4 for the two mixed quantum-classical methods outlined in this chapter with exact results obtained from MCTDH wavepacket dynamics calculations. To make a reliable comparison the approximate finite temperature calculations were performed at very low temperatures (/ = 25), though a product of ground state wave functions for the independent harmonic oscillator modes could have been used to make the initial conditions identical to those used in the MCTDH calculations. 

However, apparently the value of Cs is not universal in LES either. In practice, Cs is adjusted to optimize the model results. Deardorff [28] [29] quoted several values of Cg based on Lilly s estimates. The exact value chosen depends on various factors like the filter used, the numerical method used, resolution, and so forth, but they are generally of the order of Cg = 0.2. However, from comparison with experimental results, Deardorff concluded that the constant in the Smagorinsky effective viscosity model should be smaller than this, and a value of about Cg = 0.10 was used. In addition, for the case of an anisotropic resolution (i.e., having different grid width Ax, Ay and Az in the different co-ordinate directions), the geometry of the resolution has to be accounted for. 

Table 4.8 Equilibrium lattice constant a, cohesive energy and bulk modulus B of FCC Al Exact exchange in comparison with LDA results.

A comparison of exact results for H + H2 with semiclassical ones (Wu and Levine, 1973c) was done for reactive 0 - 0 and 0 - 1 transitions between = 12-5 and 30kcal/mo e. Comparisons were made with the primitive semiclassical approximation, a simplified version of the uniform semiclassical approximation, and a near-classical approximation (Levine and Johnson, 1970). These approximations were only qualitatively correct. 

Gas phase experiments are ideally suited for this since the perturbations induced by the local environment are completely eliminated, providing an ideal tool for exact measurements. Studies of coherent proton transfer of isolated molecules in the gas phase are able to yield absolute numbers on proton transfer tunneling without being influenced by solvent effects or the interaction of a crystal surrounding. Moreover, high resolution spectroscopy can yield state specific tunneling frequencies. These studies allow direct comparison with the results of ab initio studies and will therefore provide a sensitive test for recently developed theoretical methods. 

TABLE 3. Results for the local Nusselt number, Nu(Z), for the parallel-plates case with Bi = CO, =0.1 and fS = h Comparison of exact results. Ref. [6], and recovered results from estimated parameters. 

The idea of an effective Hamiltonian for diatomic molecules was first articulated by Tinkham and Strandberg (1955) and later developed by Miller (1969) and Brown, et al., (1979). The crucial idea is that a spectrum-fitting model (for example Eq. 18 of Brown, et al., 1979) be defined in terms of the minimum number of linearly independent fit parameters. These fit parameters have no physical significance. However, if they are defined in terms of sums of matrix elements of the exact Hamiltonian (see Tables I and II of Brown, et al., 1979) or sums of parameters appropriate to a special limiting case (such as the unique perturber approximation, see Table III of Brown, et al., 1979, or pure precession, Section 5.5), then physically significant parameters suitable for comparison with the results of ab initio calculations are usually derivable from fit parameters. 

Comparison with the results of numerical calculations of Ei by means of exact formulae, Eq. (12.30), has shown that discrepancies between the values of E obtained and the respective values of Ei do not exceed 2% in the region A < 0.1, and... 

On the other hand, the exact formulation of a theo3 y usually represents an untractable expression which is often not useful from a practical point of view. This is the reason for preference of a simpler formulation even when its results are not very accurate. A way out of this situation is to elaborate a theory as simple as possible and estimate its accuracy through a comparison with the results of the exact theory, if possible, at least in some particular cases. Another and more practical possibility is to find simple criteria permitting the determination of the limits of validity of the approximate theory considered. This is especially desirable in the theoretical study of the chemical reactivity which is the subject of this book. 

In this section, we lay the general foundation of the idea of the TSM for liquids, and particularly for liquid water. We explore both the usefulness and the limitations of this approach. Of course, because of its generality, we cannot expect to pursue the study to the point where comparison with experimental results is possible. This must be done by invoking a specific ad hoc model for the system. The latter may be viewed as an approximate version of the exact TSM, a topic which will concern us in the next section. 

Detailed numerical studies of the Born-Oppenheimer approximation have been performed in the context of studies of baryons with double charm [7,73]. The method works quite well for ccq configurations, as expected, but also for the ssq or even qqq cases. In table 7.1, we display a comparison of the extreme and uncoupled adiabatic approximations with exact results for the mmm system with masses m = l and m = 0.2,0.5 and 1, bound by the smooth 2 S r potential. The quality of the approximation is impressive for both the energy of the first levels and the short-range correlation. 

Besides benchmark comparisons with exact simulation results, model calculations have been performed to numerically explore additional Issues. 

In eonelusion, it is not clear which is the most exact theoretical approach to reproduce values of the isosteric heat of 2D L-J fluids on perfectly flat surfaces following the Steele approach. None of the theories or equations used can be considered fliUy exact, and some discrepancies are found, especially at low temperatures and high densities (near the monolayer coverage). The CM equation, although clearly the simplest approach, gives adequate values for the isosteric heat, at least for densities below the monolayer coverage. A comparison with experimental results will be described next. 

This approach is amenable to carrying out Monte Carlo simulations that employ the relative free energy A< > to build the Markov chain. With some care when extracting the bulk properties of the system, this method also provides reliable results for a range of conditions. However, these results deteriorate with increasing diffraction effects, as shown by comparison with exact PIMC results [103]. The reader is referred to Ref. 145 for specific details about this free-energy-based method. 

Based on this decomposition, in the cumulant expansion of the autocorrelation function, the exact cumulants are recovered up to the second order. To demonstrate the capability of our method and for comparison with other results, the pyrazine molecule is chosen as a numerical example. 

Calculated results by DLT and FLT in comparison with exact solution. 

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