Accuracy, of approximate methods

In this section, we will analyze an elementary problem in quantum mechanics, the square barrier. The purpose is twofold. First, such an analysis can provide physical insight into the process, to gain a conceptual understanding. Second, analytically soluble models are indispensable for assessing the accuracy of approximate methods, such as the MBA. 

Motion of two spheres along a line passing through their centers. In the Stokes approximation, an exact closed-form solution of the axisymmetric problem about the motion of two spheres with the same velocity was obtained in [463]. This solution is practically important and can be used for estimating the accuracy of approximate methods, which are applied for solving more complicated problems on the hydrodynamic interaction of particles. 

In the connnonly used atomic sphere approximation (ASA) [79], the density and the potential of the crystal are approximated as spherically synnnetric within overlapping imifiBn-tin spheres. Additionally, all integrals, such as for the Coulomb potential, are perfonned only over the spheres. The limits on the accuracy of the method imposed by the ASA can be overcome with the fiill-potential version of the LMTO (FP-LMTO)... 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

From the excellent agreement in measurements by the various methods Sair and Fetzer concluded that they have determined the true moisture content with an accuracy of approximately 0.1%. Though this claim is subject to further verification, it does indeed seem to be plausible. 

The Iodometric method has also been utilized in analyzing hydrogen sulfide in the air (EPA 1978). The method is based on the oxidation of hydrogen sulfide by absorption of the gas sample in an impinger containing a standardized solution of iodine and potassium iodide. This solution will also oxidize sulfur dioxide. The Iodometric method is suitable for occupational settings. The accuracy of the method is approximately 0.50 ppm hydrogen sulfide for a 30-L air sample (EPA 1978). 

In spite of the frequency-shifted excitation, the quantized PIP inevitably excites multiple sidebands located at n/At ( = 1, 2,...) from the centre band. An attempt was made16 to calculate the excitation profile of multiple bands created by a PIP of a constant RF field strength, using an approximate method based on the Fourier analysis. The accuracy of the method relies partly on the linear response of the spin system, which is, unfortunately, not true in most cases except for a small angle excitation. In addition, the spins inside a magnet consitute a quantum system, which is sensitive not only to the strengths but also to the phases of the RF fields. Any classical description is doomed to failure if the quantum nature of the spin system emerges. 

Since the unpaired electron in transition metal complexes is generally localized near the central ion and the ligand atoms in the first coordination sphere, summation in (5.5) over these nuclei is often sufficient. In this approximated form, the point-dipole model has frequently been applied in ENDOR studies of transition metal complexes to determine the proton positions from their hfs tensors (Sect. 6). In some cases the accuracy of this method has turned out to be significantly higher than that of an X-ray diffraction analysis62,130 131). 

Quantum mechanical methods can now be applied to systems with up to 1000 atoms 87 this capacity is not only from advances in computer technology but also from improvements in algorithms. Recent developments in reactive classical force fields promise to allow the study of significantly larger systems.88 Many approximations can also be made to yield a variety of methods, each of which can address a range of questions based on the inherent accuracy of the method chosen. We now discuss a range of quantum mechanical-based methods that one can use to answer specific questions regarding shock-induced detonation conditions. 

The statistical error determined for K is only a limited measure of the accuracy of the dilatometric measurements. Since the main errors will be similar for each measurement, the accuracy of the method is best determined by estimating the limits of error of each individual measurement.The main source of error and its approximate magnitude should be indicated. 

If a certified material is not available, as is usually the case in drug residue analysis, an approximation can be obtained by spiking the blank sample matrix to a nominal concentration. The accuracy of the method is then determined by assessing the agreement between the measured and nominal concentrations of the analytes in the spiked drug-free matrix sample. The spiking levels should cover the range of concern and should include one concentration close to the quantitation limit. 

Let us also notice that slow variations of K with Z imply that the gauge condition K may be treated as a semi-empirical parameter in practical calculations to reproduce, with a chosen K, the accurate oscillator strength values for the whole isoelectronic sequence. Thus, dependence of transition quantities on K may serve as the criterion of the accuracy of wave functions used instead of the comparison of two forms of 1-transition operators. In particular, the relative quantities of the coefficients of the equation fEi = aK2 + bK +c (the smaller the a value, the more exact the result), the position of the minimum of the parabola Kf = 0 (the larger the K value for which / = 0, the more exact is the approximation used, in the ideal case / = 0 for K = +oo) may also help to estimate the accuracy of the method utilized. 

Table 5.5. Comparison of the accuracies of the Method of Self-Organizing Models (MSOM) and differential approximation algorithms from results of retrieval of water level oscillations at the boundary of the Nyok Ngot lagoon (South Vietnam) with the South China Sea. From Bui (2001). Notation At is the time step, and e is the error (%).

Reactions (19.35), (19.36) have been studied theoretically for NA = X Xt on several occasions, and n-resolved cross-sections, obtained within the Grizinsky-Bauer-Bratky (GBB) model [59,60], are presented in [21]. (Total experimental cross-sections for these reactions for the X1T +(n = 0) initial state are given in [14].) The simplicity of GBB model permits to perform v—v resolved cross-section calculations also for transitions from excited (N > 2) states, but such calculations have not been published. It should be mentioned, however, that the accuracy of GBB method is (at best) within a factor of two, and more accurate quantal calculations of the cross-sections of processes (19.35), (19.36) are desirable. For transitions from the initial NA states with N > 3, the cross-sections of reactions (19.35), (19.36) within a given 1,3ylSj -series of states can be scaled (approximately as N2). 

The method works extremely well for calculations within the temperature and pressure range used to determine the equation constants. The average error in such a case is less than the experimental error of approximately 0.1% reported for volumetric measurements. The accuracy of this method for extrapolations is not known due to the lack of sufficient P-V-T data for polymers. 

Two kinds of property values are given in the certificate certified values and information values. The certified values are always quoted together with their uncertainty limits, whereas information values are quoted as numbers without the uncertainties. The information values should serve only to indicate the approximate level of analyte concentration and cannot be used for assessment of the accuracy of the method. 

But already at that point we realized that the accuracy of this method was not very good. Many physical and chemical quantities, like binding energy or chemical bond distances, could not be calculated directly with this method. When we finally also started calculations of real molecules at chemical distances this deficiency became even more obvious and we began to improve the method itself in order to calculate the total energy of a system within the Xa-approximation. 

The principal considerations in% choosing a finite-difference method for (7) are accuracy, stability, computation time, and computer storage requirements. Accuracy of a method refers to the degree to which the numerically computed temporal and spatial derivatives approximate the true derivatives. Stability considerations place restrictions on the maxi-... 

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