Calculations numerical methods

The mass transfer by convection and diffusion within the channel is precisely calculable. Numerical methods are used to fit measured concentrations at the detector electrode to the fluid flow in the cell, and to the reaction kinetics at the surface and in solution [24]. The technique was extensively developed during a study of the dissolution of calcite in dilute aqueous acid [25], and has latterly been applied to a number of organic reactions. 

Contributing group methods have been developed and published for calculating numerous other properties. However, for our purposes, it is not necessary to employ them. Moreover, there are some properties for which the method is not recommended. 

A yet more realistic cavity shape is that obtained from the van der Waals radii of the atoms of the solute. This is the approach taken in the polarisable continuum method (PCM) [Miertus et al. 1981], which has been implemented in a variety of ab initio and semi-empirical quantu/rt mechanical programs. Due to the non-analytical nature of the cavity shapes in the PCM approach, it is necessary to calculate numerically. The cavity surface is divided... 

The typical phase equiHbrium problem eacouatered ia distiHatioa is to calculate the boiling temperature and the vapor composition ia equiHbrium with a Hquid phase of specified composition at a givea pressure. If the Hquid phase separates, thea the problem is to calculate the boiling temperature and the compositions of the two equiHbrium Hquid phases plus the coexistiag vapor phase at the specified overall Hquid compositioa. Robust and practical numerical methods have been devised for solving this problem (95—97) and have become the recommended techniques (98,99). 

At this level of theory, the calculated equilibrium bond length is 110.47 pm, and the dipole moment changes sign around which may explain why one has to work so hard to achieve agreement with experiment. The dipole derivative can be found by numerical methods from the data points. 

In practice the finite-field calculation is not so simple because the higher-order terms in the induced dipole and the interaction energy are not negligible. Normally we use a number of applied fields along each axis, typically multiples of 10 " a.u., and use the standard techniques of numerical analysis to extract the required data. Such calculations are not particularly accurate, because they use numerical methods to find differentials. 

Many ab initio packages use the two key equations given above in order to calculate the polarizabilities and hyperpolarizabilities. If analytical gradients are available, as they are for many levels of theory, then the quantities are calculated from the first or second derivative (with respect to the electric field), as appropriate. If analytical formulae do not exist, then numerical methods are used. 

With a computer program, which solved Eqs. (5) and (7) for We by numerical method, Weber numbers were obtained from layer to layer for each injection molding operation. The calculated points were then connected to curves, depending on the normalized thickness. Figure 17 shows these theoretically deduced Weix)... 

Tycho, D. H., Thomas, L. H., and King, K.M., Phys. Rev. 109,369, Numerical calculation of the wave functions and energies of the PS and 23S states of He." Configurational expansion determined by numerical method angular terms up to l— 14 included. 

So, the calculation of the shape of an IR spectrum in the case of anticorrelated jumps of the orienting field in a complete vibrational-rotational basis reduces to inversion of matrix (7.38). This may be done with routine numerical methods, but it is impossible to carry out this procedure analytically. To elucidate qualitatively the nature of this phenomenon, one should consider a simplified energy scheme, containing only the states with j = 0,1. In [18] this scheme had four levels, because the authors neglected degeneracy of states with j = 1. Solution (7.39) [275] is free of this drawback and allows one to get a complete notion of the spectrum of such a system. 

The problem has already been solved for the normal state of the hydrogen molecule-ion (ZA = ZB = 1) by the use of numerical methods. A rather complete account of these calculations of Burrau (30) will be given here, since the journal in which they were published is often not available. 

In order to see how accurate this perturbation treatment actually is, we have substituted numerical values for the S s directly into the secular equation, and then solved it rigorously by numerical methods. The calculations are not given in detail, since they are quite straightforward and proceed along well-known lines. The results are shown in Table I. 

These two elements are treated as two capacitors in parallel. The capacity of a sphere relative to the sample can be calculated exactly by the method of images. Even then, however, a complicated expression is obtained that must be calculated numerically. Fortunately, when the tip-surface distance is sufficiently smaller than the tip radius, an approximate expression for the capacitance can be found by integrating the contributions of flat infinitesimal rings of spherical surface centered at the apex [36], The result is ... 

Danek and his group have independently proposed a quite similar model, which they call the dissociation modeV - For this model Olteanu and Pavel have presented a versatile numerical method and its computing program. However, they calculated only the electrical conductivity or the molar conductivity of the mixtures, and the deviation of the internal mobilities of the constituting cations from the experimental data is consequently vague. 

Two calculation procedures for steady, three-dimensional flows with recirculation, in Proceedings of the 3rd Int. Conf Numer. Methods Fluid Dyn., Paris (1972). 

Since analytic second derivatives are available for MP2 calculations, numerical difference calculations of CCSD(T) energies are only required for a relatively small basis set. This type of basis set correction approximation is also available in Grow. It is not possible to use some composite methods which, like the G2 and G3 schemes,66 involve adding non-differentiable corrections to the estimated electronic energy. However, there are other recently developed composite methods which might be effectively employed to construct this type of interpolated PES.67... 

The concentration of each chemical species, as a function of time, during cure can be calculated numerically from Equations 3-6 using the Euler-Romberg Integration method if the initial concentrations of blocked isocyanate and hydroxyl functionality are known. It is a self-starting technique and is generally well behaved under a wide variety of conditions. Details of this numerical procedure are given by McCalla (12). 

Simultaneous integration of these equations by numerical methods can provide the concentrations of A,B1,B2 and B3 as a function of time and the concentrations of o-methylol, p-methylol and methylene ether (note earlier definition) groups can be calculated as follows, where CONC is the initial monomer concentration. 

The second derivatives can be calculated numerically from the gradients of the energy or analytically, depending upon the methods being used and the availability of analytical formulae for the second derivative matrix elements. The energy may be calculated using quantum mechanics or molecular mechanics. Infrared intensities, Ik, can be determined for each normal mode from the square of the derivative of the dipole moment, fi, with respect to that normal mode. 

Horwitz claims that irrespective of the complexity found within various analytical methods the limits of analytical variability can be expressed or summarized by plotting the calculated mean coefficient of variation (CV), expressed as powers of two [ordinate], against the analyte level measured, expressed as powers of 10 [abscissa]. In an analysis of 150 independent Association of Official Analytical Chemists (AOAC) interlaboratory collaborative studies covering numerous methods, such as chromatography, atomic absorption, molecular absorption spectroscopy, spectrophotometry, and bioassay, it appears that the relationship describing the CV of an analytical method and the absolute analyte concentration is independent of the analyte type or the method used for detection. 

Recent polymer adsorption theories, such as those of Roe (3) and of Scheutjens and Fleer (h) allow the calculation of displacement isotherms, so that we could study the dependency of these isotherms on various parameters by numerical methods. However, all the essential features of displacement can also be demonstrated by means of a simple analytical expression for the critical point, which can be derived in a straightforward way. 

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