Comparisons numerically exact results

A calculation of tunneling splitting in formic acid dimer has been undertaken by Makri and Miller [1989] for a model two-dimensional polynomial potential with antisymmetric coupling. The semiclassical approximation exploiting a version of the sudden approximation has given A = 0.9cm" while the numerically exact result is 1.8cm" Since this comparison was the main goal pursued by this model calculation, the asymmetry caused by the crystalline environment has not been taken into account. 

A different way, developed extensively by Schwartz and his coworkeis, - is to use approximate quantum propagators, based on expansions of the exponential operators. These approximations have been tested for a number of systems, including comparison with the numerically exact results of Ref 38 for the rate in a double well potential, with satisfying results. 

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... 

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 ... 

Two applications of numerically exact results of MD computations are considered. In the first case, MD results serve as reference data for theoretical approximations in the second, they serve as computed data for comparison with experiment. 

As a sample system we used the pyrazine molecule. Eortunately, we could use the MCTDH method to compute the dynamics of all the 24 modes, thus providing numerically exact results for comparison. Four different kinds of computations were performed and compared to each other. Based on these results we conclude that our QVC three-effective mode scheme well reproduces the short-time dynamics and the overall shape of the spectra. In particular, the autocorrelation function is more accurate up to the first 20 fs, compared to that obtained by the LVC three-effective mode approach. 

A general conclusion that can be drawn from this short survey on the many attempts to develop analytical theories to describe the phase behavior of polymer melts, polymer solutions, and polymer blends is that this is a formidable problem, which is far from a fully satisfactory solution. To gauge the accuracy of any such approaches in a particular case one needs a comparison with computer simulations that can be based on exactly the same coarse-grained model on which the analytical theory is based. In fact, none of the approaches described above can fully take into account all details of chemical bonding and local chemical structure of such multicomponent polymer systems and, hence, when the theory based on a simplified model is directly compared to experiment, agreement between theory and experiment may be fortuitous (cancellation of errors made by use of both an inadequate model and an inaccurate theory). Similarly, if disagreement between theory and experiment occurs, one does not know whether this should be attributed to the inadequacy of the model, the lack of accuracy of the theoretical treatment of the model, or both. Only the simulation can yield numerically exact results (apart from statistical errors, which can be controlled, at least in principle) on exactly the same model, which forms the basis of the analytical theory. It is precisely this reason that has made computer simulation methods so popular in recent decades [58-64]. 

In the past decade, vibronic coupling models have been used extensively and successfully to explain the short-time excited-state dynamics of small to medium-sized molecules [200-202]. In many cases, these models were used in conjunction with the MCTDH method [203-207] and the comparison to experimental data (typically electronic absorption spectra) validated both the MCTDH method and the model potentials, which were obtained by fitting high-level quantum chemistry calculations. In certain cases the ab initio-determined parameters were modified to agree with experimental results (e.g., excitation energies). The MCTDH method assumes the existence of factorizable parameterized PESs and is thus very different from AIMS. However, it does scale more favorably with system size than other numerically exact quantum... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

With G expanded in powers of a-1 [see Eq. (4.136)], this formula reproduces the asymptotic expression derived by Brown [88]. At G = 1 it reduces to his initial result [47] corresponding to Eq. (4.134) here. Function (a) from Eq. (4.147) is shown in Figure 4.9 for comparison with the exact result obtained by a numerical solution. Indeed, at ct 3 the results virtually coincide. 

A comparison of the numerical results with the exact results for temperature distribution in a cylinder would. show that the re.sults obtained by a numerical method are approximate, and they may or may not be sufficiently close to die exact (true) solution values. The difference between a numerical solution and the e.xacl solution is the error involved in the numerical solution, and it is primarily due to two. sources ... 

Fig. 4 Comparison of the model improvements on the Derjaguin approximation to the exact numerical computational results of the full Poisson-Boltzmann equation for two spheres with the scaled radius Rk — 0.1 and Rk — 15 and constant surface potential [j/ ez/(k-gT) = 1. The scaled energy, G h), on the vertical axis is defined by G(h) = (/,)/jsM...

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. 

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... 

From the comparison of the results of B approximation calculations and the exact solution (Fig. 1), one can see that the ults of this approximate method are dose to the exact ones for a wide range of ratios of the constants. This method is also rather simfde, and the numerical solution of Eq. (40) is not difficult, er wl long sequences must taken into consideration. 

Recently we have constructed a complete second-order QDT (CS-QDT), in which all excessive approximations, except that of weak system-bath interaction, are removed [38]. Besides two forms of CS-QDT corresponding to the memory-kernel COP [Eq. (1.2)] and the time-local POP [Eq. (1.3)] formulations, respectively, we have also constructed a novel CS-QDT that is particularly suitable for studying the effects of correlated non-Markovian dissipation and external time-dependent field driving. This paper constitutes a review of the three nonequivalent CS-QDT formulations [38] from both theoretical and numerical aspects. Concrete comparisons will be carried out in connection with the exact results for driven Brownian oscillator systems, so that sensible comments on various forms of CS-QDT can be reached. Note that QDT shall describe not only the evolution of p(t), but also the reduced thermal equilibrium system as p t oo) = peq(7 )-... 

For comparison, the exact numerical solution to Eqs (5.62) and (5.43) is shown by points in Figure 5.7. As can be seen, the anal d ical solution is in excellent agreement with the numerical result. 

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. 

There is a fundamental problem. The extent to which X-rays are absorbed as they travel through a crystal depends on the path they take and the mass of the atoms they meet. This therefore affects the diffraction intensities that we measure, and the problem is exacerbated for crystals that have a very anisotropic shape. If the composition of the crystal (the total formula) is known, then the contributions of the atoms of each element can be summed to calculate the absorption coefficient /x (typically quoted in mm ), which represents the amount of radiation absorbed by a crystal of a certain length. Ideally, then, crystals of heavily absorbing compounds should be small and isotropically shaped in order to avoid very different absorptions for different X-ray paths. However, if the three-dimensional geometry of the crystal is known, the absorption can be calculated exactly for the known X-ray path for every reflection, and so the measured diffraction intensity can be corrected. This is called the numerical absorption correction. More often, the crystal dimensions are not known, but the absorption can still be estimated from the total set of measured reflections, as any one reflection can be measured many times using different orientations of the crystal. Comparison of the resulting intensities and fitting to a suitable model then yields the corrections. 

Fig. 3.4. Time averaged kinetic energy distribution of an ion stored in a 16-pole and in a 32-pole trap. The results have been calculated using numerically exact trajectory calculations. For a better comparison the probability distributions are peak normalized.

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