Analytical methods comparison with numerical

In the preceding sections we show that, by postulating simple VB structures on a photochemical reaction path, one can deduce not only that a conical intersection may be involved but also the nature of the branching space of the conical intersection. For problems such as 3 orbitals with 3 electrons or 4 orbitals with 4 electrons it is simple to manipulate the VB matrix elements to make these deductions. By the time one gets to 6 orbitals with 6 electrons there are very many possibihties. So one has to leam " by extracting the VB structures from the ab initio data. For the 6 orbitals with 6 electron case, we use the MMVB method to do this. Once the more important structures are identified this way, we can perform the manipulations analytically to confirm the result by comparison with numerical data. Finally, for 8 orbitals with 8 electrons we were able to show that one may also extract the VB data from the MMVB method and come to understand the nature of the conical intersection. However, it is rather tedious to do the calculations analytically and this work has never been carried out. 

As mentioned previously, the energies of states in the linear Tiu hg JT system, which applies to our high spin states, have been calculated previously using a similar analytical approach to that presented in this paper [20]. More quantitative comparisons with other approaches are available for this system, with the agreement between analytical and numerical approaches being found to be good [20]. This therefore indicates that our method is a valid one. 

Example 5.3 The Semi-infinite Solid with Variable Thermophysical Properties and a Step Change in Surface Temperature Approximate Analytical Solution We have stated before that the thermophysical properties (k, p, Cp) of polymers are generally temperature dependent. Hence, the governing differential equation (Eq. 5.3-1) is nonlinear. Unfortunately, few analytical solutions for nonlinear heat conduction exist (5) therefore, numerical solutions (finite difference and finite element) are frequently applied. There are, however, a number of useful approximate analytical methods available, including the integral method reported by Goodman (6). We present the results of Goodman s approximate treatment for the problem posed in Example 5.2, for comparison purposes. 

Analytical methods should be precise, accurate, sensitive, and specific. The precision or reproducibility of a method is the extent to which a number of replicate measurements of a sample agree with one another and is expressed numerically in terms of the standard deviation of a large number of replicate determinations. Statistical comparison of the relative precision of two methods uses the variance ratio (F j or the F test. 

Nuclear polarization corrections for hydrogenlike Pb and ions have been reconsidered recently [66]. Recalculations were performed within two different numerical approaches the direct numerical integration [62 - 64] and the B-spline method [67]. Besides a comparison with analytical evaluations [65], an enlarged set of nuclear excitations has been taken into account in these calculations. 

In a paper by Nguen Van Kmet (1992), a difficult numerical integration of the Navier-Stokes equation was carried out in order to obtain information necessary for calculating collision efficiency. As usual, it is very difficult to evaluate the reliability of numerical calculations. Usually some judgement can be drawn when comparison with results tfom analytical formulas for known limiting cases or with numerical calculations based on substantially different methods are performed. Unfortunately such data are not available. 

Another difficulty arising from this comparison is connected with the mathematical complexity of the corresponding boundary problems even if only linear diffusion equations are used. The mathematical description of the adsorption kinetics from micellar solutions is essentially more complicated in comparison with the case of the adsorption process from sub-micellar solutions. Analytical solutions of the corresponding boundary problems using rather poor approximations have been obtained only for a small number of situations. A sufficiently general solution cannot be obtained analytically and the deficiency of the rather well elaborated numerical methods often compel experimentalists to apply approximate solutions. Therefore, it seems important to consider the main equations proposed for the description of kinetic dependencies of the surface tension and adsorption, and to elucidate the limits of their application before the discussion of experimental results. 

A check on the foregoing formulas is provided by a comparison with computations that have been laboriously (and carefully) carried out by six-dimensional numerical integration (A.V. Turbiner, private communication). The analytical and numerical methods are in agreement within the precision of the latter. 

The determination of the thermal conductivity of grain is based on the comparison of the temperature history data obtained by using the line heat source probe with the approximate analytical and numerical methods [35,54]. The analytical method has the advantage of being quick in calculating thermal conductivity. This method, however, requires a perfect line source and a small diameter tube holding the line heat source. In reality, this requirement is difficult to meet. Therefore, a time-correction procedure has been introduced [52,54,56]. Another objection to the analytical method is that it cannot easily be used to calculate the temperature distribution in the heated grain and to compare it with the measured one. Such a comparison can be easily accomplished by a numerical method, where the estimated accuracy for thermal conductivity is determined and the thermal conductivity of the device is taken into account [54]. 

The Dunham coefficients Yy are related to the spectroscopical parameters as follows 7io = cOe to the fundamental vibrational frequency, Y20 = cOeXe to the anharmonicity constant, Y02 = D to the centrifugal distortion constant, Yn = oie to the vibrational-rotational interaction constant, and Ym = / to the rotational constant. These coefficients can be expressed in terms of different derivatives of U R) at the equilibrium point, r=Re. The derivatives can be either calculated analytically or by using numerical differentiation applied to the PEC points. The numerical differentiation of the total energy of the system, Ecasccsd, point by point is the simplest way to obtain the parameters. In our works we have used the standard five-point numerical differentiation formula. In the comparison of the calculated values with the experimental results we utilize the experimental PECs obtained with the Rydberg-Klein-Rees (RKR) approach [58-60] and with the inverted perturbation approach (IPA) [61,62]. The IPA is method originally intended to improve the RKR potentials. 

Another method more intuitively understandable is the fitting in the time domain. Obviously in the cases when it is possible to obtain an analytical solution for the basic equations to describe a model, the comparison with the response curve is easily made. When a numerical solution is to be employed, the effect of each parameter on the calculated concentration curve cannot be easily visualized so the comparison needs repeated calculations for optimum parameter search. Anderssen and White (1970) introduced the error map method to show the deviation of the calculated curves and the response curves, which was later utilized by Wakao et al. (1979, 1981). 

Similar to quantitative methods, qualitative tests should also undergo a method validation. The measuring system of a qualitative test usually transforms a quantitative result into a negative or positive report or in some cases a semi-quantitative outcome however, the absolute numerical concentration of the analyte itself is not reported. Qualitative tests can be validated by using a series of samples with known concentrations of analyte that fall either side of the positive-negative cut-off. These known values may be assigned by an alternative method, or may be reference material. It is particularly important to assess reproducibility of results around a concentration of the analyte of interest that is clinically important such as around a diagnostic cut-off. A method comparison study can also be undertaken with a comparator. The new method can usually be implemented when predefined criteria are fulfilled. 

The elaborated in [R. V. Chepulskii, Analytical method for calculation of the phase diagram of a two-component lattice gas, Solid State Commun. 115 497 (2000)] analytical method for calculation of the phase diagrams of alloys with pair atomic interactions is generalized to the case of many-body atomic interactions of arbitrary orders and effective radii of action. The method is developed within the ring approximation in the context of a modified thermodynamic perturbation theory with the use of the inverse effective number of atoms interacting with one fixed atom as a small parameter of expansion. By a comparison with the results of the Monte Carlo simulation, the high numerical accuracy of the generalized method is demonstrated in a wide concentration interval. 

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]. 

The comparison between the finite element and analytical solutions for a relatively small value of a - 1 is shown in Figure 2.25. As can be seen the standard Galerkin method has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solution is expected to improve even further with mesh refinement. As Figmre 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh of 10 elements, for larger values of this coefficient the numerical solution produced by the standard... 

A detailed description of analytical techniques is given in a number of original articles and books [3]. We will focus our interest on comparison of capacities of the mentioned physical and chemical methods with those of semiconductor detectors (SCD) or semiconductor sensors (SCS). These detectors are growing popular in experimental studies. They are unique from the stand-point of their application in various branches of chemistry, physics, and biology. They are capable of solving numerous engineering, environmental and other problems. 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

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