Comparisons numerically exact

Fig. 1. Comparison between the CID-CSP, CSP, TDSCF, and the numerically exact autocorrelation functions.

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. 

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... 

There exist a fairly large number of numerical tests of VTST (including tunneling) for three center reactions, the tests being the comparison with exact quantum me-... 

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. 

Figure 4.26. Comparison of the numerically exact solution (solid lines) and the effective time approximation (circles) with respect to the quadratic SNR e = 0.1 (1), 0.3 (2), 0.5 (3). Low-frequency case flio = 1CT4.

The validity of the CSP method for sub-picosecond processes in systems which exhibit moderate quantum effects was established by comparison with numerically exact quantum dynamical calculations for small modell systems [27,49], Good correlation with experiments for large realistic systems is also encouraging [50, 51], Currently, limitations of the CSP scheme given by its separable nature are being overcome by extending the method in the direction of configuration interaction [53],... 

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 ... 

Figure 6.4 shows the comparison between the composite asymptotic solution and the numerically exact solution for e = 0.5. It is useful to note that the asymptotic solution agrees quite well with the exact solution even when e = 0.5. 

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. 

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.

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

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]. 

M. S. Child and M. Baer, A model for reactive non-adiabatic transitions Comparison between exact numerical and approximate analytical results, J. Chem. Phys., in press. 

Fig. 7. Comparison of various transport schemes for advecting a cone-shaped puff in a rotating windfield after one complete rotation (a), the exact solution (b), obtained by an accurate numerical technique (c), the effect of numerical diffusion where the peak height of the cone has been severely tmncated and (d), where the predicted concentration field is very bumpy, showing the effects of artificial dispersion. In the case of (d), spurious waves are...

Compressors have numerous forms, the exact configuration being based on the application. For comparison, the different types of compressors can be subdivided into two broad groups based on compression mode. There are two basic modes intermittent and continuous. The intermittent mode of compression is cyclic in nature, in that a specific quantity of gas IS ingested by the compressor, acted upon, and discharged, before the cycle is repeated. The continuous compression mode is one in which the gas is moved into the compressor, is acted upon, moved through the compressor, and discharged without interruption of the tlnv. at any point in the process. 

In Eq. (3.66) the sign + is chosen to provide the decay in time of the spectrum correlation function. When the approximate solution (3.66) is used for the back iterations in Eq. (3.58) from bN = 1 + bN up to ho and subsequent calculation of ao(co) the error does not accumulate. This was proved by comparison of approximate numerical calculations of limiting cases 2 and 3 with exact formulae (3.61) and (3.62). 

The semiempirical methods represent a real alternative for this research. Aside from the limitation to the treatment of only special groups of electrons (e.g. n- or valence electrons), the neglect of numerous integrals above all leads to a drastic reduction of computer time in comparison with ab initio calculations. In an attempt to compensate for the inaccuracies by the neglects, parametrization of the methods is used. Meaning that values of special integrals are estimated or calibrated semiempirically with the help of experimental results. The usefulness of a set of parameters can be estimated by the theoretical reproduction of special properties of reference molecules obtained experimentally. Each of the numerous semiempirical methods has its own set of parameters because there is not an universial set to calculate all properties of molecules with exact precision. The parametrization of a method is always conformed to a special problem. This explains the multiplicity of semiempirical methods. 

First, we have applied the ZN formulas to the DHj system to confirm that the method works well in comparison with the exact quantum mechanical numerical solutions [50]. Importance of the classically forbidden transitions has been clearly demonstrated. The LZ formula gives a bit too small results... 

The numerical value of S is listed in Table 9.1. The simple variation function (9.88) gives an upper bound to the energy with a 1.9% error in comparison with the exact value. Thus, the variation theorem leads to a more accurate result than the perturbation treatment. Moreover, a more complex trial function with more parameters should be expected to give an even more accurate estimate. 

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