Parametric geometry

D CAD/modeler Free-form surfaces, parametric geometry, trim curves, surface/material properties, assembly information Tessellation (static dynamic), data reduction (LOD), siuface properties — > textures, materials, reflection mapping... 

K. Sertel and J.L. Volakis, Method of moments solution of volume integral equations using parametric geometry modeling. Radio Sci. 37, 1 (2002)... 

J.L. VolaJsis, K. Sertel, E. Jorgensen, R.W. Kindt, Hybride finite element and volume integral methods for scattering using parametric geometry, CMES 1, 11 (2000)... 

La sensibilite aux defauts et autres parametres de controle peut etre amelioree par le choix optimal de la sonde. II apparait, apres etudes des differents types de sondes (ferritiques, acier doux, isolant) avec des geometries differentes (plate, conique,. ..), necessaires de souligner que le succes d une recherche de faisabilite depend en grande partie de la bonne definition des capteurs de mesure, de telle sorte que ceux-ci soient adaptes au probleme considere. 

Since the form of the electronic wave functions depends also on the coordinate p (in the usual, parametric way), the matrix elements (21) are functions of it too. Thus it looks at first sight as if a lot of cumbersome computations of derivatives of the electronic wave functions have to be carried out. In this case, however, nature was merciful the matrix elements in (21) enter the Hamiltonian matrix weighted with the rotational constant A, which tends to infinity when the molecule reaches linear geometry. This means that only the form of the wave functions, that is, of the matrix elements in (21), in the p 0 limit are really needed. In the above mentioned one-elecbon approximation... 

Table I-l lists the various theoretical treatments published on the thiazole molecule for each the type of approximation, the mode of parametrization. and, eventually, the geometry employed are given net charges and bond orders for various theoretical calculations are listed in Tables 1-2 and 1-3.

ZINDO/1 and ZINDO/S are Dr. Michael Zerner s INDO versions and used for molecular systems with transition metals. ZINDO/1 is expected to give geometries of molecules, and ZINDO/S is parametrized to give UV spectra. 

Construction of Alignment Charts. Of the ways to constmct alignment charts, the bmte force method, which requires some idea of the geometry for the chart, is the easiest method to use. The mathematical method, which uses parametric equations of scale to determine the placement and scale of each axis, is the most accurate, but the most difficult to apply. 

Step 1 of the parametrization process is the selection of the appropriate model compounds. In the case of small molecules, such as compounds of pharmaceutical interest, the model compound may be the desired molecule itself. In other cases it is desirable to select several small model compounds that can then be connected to create the final, desired molecule. Model compounds should be selected for which adequate experimental data exist, as listed in Table 1. Since in almost all cases QM data can be substimted when experimental data are absent (see comments on the use of QM data, above), the model compounds should be of a size that is accessible to QM calculations using a level of theory no lower than HE/6-31G. This ensures that geometries, vibrational spectra, conformational energetics, and model compound-water interaction energies can all be performed at a level of theory such that the data obtained are of high enough quality to accurately replace and... 

These features are illustrated for H2O in Figure 2.5, where the exact form is taken firom a parametric fit to a large number of spectroscopic data. The simple harmonic approximation (P2) is seen to be accurate to about 20° from the equilibrium geometry and the cubic approximation (P3) up to 40°. Enforcing the cubic polynomial to have a zero derivative at 180° (P3 ) gives a qualitative correct behaviour, but reduces the overall fit, although it still is better than a simple harmonic approximation. 

CNDO/2 with DelBene-Jaffe parametrization [for geometry, see (91)] open-shell method of Longuet-Higgins and Pople eq. (90). 

One might add that the failure of CNDO/2 is probably mainly due to the method of parametrization. If a semiempirical method is to be used to estimate heats of formation and molecular geometries, the parameters in it should be chosen accordingly rather than to mimic the results of an approximation known to give unsatisfactory estimates of energies. Recent studies suggest that CNDO/2 may in fact prove useful if properly parametrized. u>... 

Although the geometry can be expressed entirely mathematically in terms of formulas and relations, to do so may be awkward for those proficient in mathematics and quite incomprehensible for those who are not. It is often the case that curves and surfaces must be specified functionally, parametrically, or as piecewise sections, all of which add burden and potential error in specification. 

No one of the equations introduced here are defined as in the standard Bom-Oppenheimer approach. The reason is that electronic base functions that depend parametrically on the geometry of the sources of external potential are not used. The concept of a quantum state with parametric dependence is different. This latter is a linear superposition the other are objects gathered in column vectors. 

Unlike the curves you may have seen in geometry books (such as bullet-shaped paraboloids and saddle surfaces) that are simple functions of x and y, certain surfaces occupying three dimensions can be expressed by parametric equations of the form x = f(u,v), y = g(u,v), z = h(u,v). This means that the position of a point in the third dimension is determined by three separate formulas. Because g, and h can be anything you like, the remarkable panoply of art forms made possible by plotting these surfaces is quite large. For simplicity, you can plot projections of these surfaces in the x-y plane simply by plotting (x,y) as you iterate u and in a... 

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