Setting Contour Levels

Figure 3. Differences among water dimer densities p, as obtained with the theoretical models explored in this work. Same symmetry plane, geometry, basis set and contour levels as that of Figure 2. 

This problem is shown in Figure 4.5. The feasible region is defined by linear constraints with a finite number of comer points. The objective function, being nonlinear, has contours (the concentric circles, level sets) of constant value that are not parallel lines, as would occur if it were linear. The minimum value of/corresponds to the contour of lowest value having at least one point in common with the feasible region, that is, at xx = 2, x2 = 3. This is not an extreme point of the feasible set, although it is a boundary point. For linear programs the minimum is always at an extreme point, as shown in Chapter 7. 

This section describes the function of the most important buttons in the button panel (Fig. 4.21). Further options available within the Display pull-down menu of 2D WIN-NMR will be discussed in section 4.8.3. Two procedures to set contour levels for a 2D spectrum displayed as a contour plot are discussed in section 4.8.2. 

Contour mode The currently selected 2D spectrum is displayed as a contour plot, the usual mode of display. At the same time corresponding local mode buttons appear in the button panel (Fig. 4.21) which allow to set contour levels. See section 4.8.2 or use the Help tool for more information concerning these options. 

With a 2D spectrum displayed in the Contour mode, the number, intensitiy and color of contour levels have to be defined. Two procedures to set contours exist ... 

As an alternative contour levels may be defined using suitable rows or columns of a 2D spectrum. This is especially useful if very weak cross peaks with intensities close to the noise level should be detected. To set levels in this way the Scan mode i.s activated and the option to freeze a suitable row or column is exploited. 

Colors... Allows you to set the screen colors (contour levels, background). 

Choose the Page Setup... option in the Output pull-down menu and select the plotting parameters and projections to be included in the plot. Adjust the sizes of all windows (title, parameter, spectrum) to the size of the paper sheet. Set up a title and select the parameters to be plotted. Load the defined levels with the Load button, choose different colors for the levels if you have a multicolor output device. Click the Preview button to inspect the final layout in the Preview window. Repeat these steps, load the defined levels with the Load button as before, but add additional contour levels with the Fill button. 

Similar to the theoretical model of Palm et al. [54], based on dynamic surface properties for the prediction of drug absorption into human intestinal Caco-2 cell lines the authors used their molecular dynamics/GRID approach to correlate the absorption of the same set of six P-adrenoceptor antagonists with the coefficients obtained by the water probe at contour level -2 and -3 kcal/mol. 

Level sets F(a) [as well as the closely related density domains DD(a), as we shall see in the next section] provide a representation of formal molecular bodies. A similar definition gives a useful concept of a formal molecular surface the concept molecular isodensity contour surface (MIDCO). For any formal nuclear configuration K, it is possible to define a surface by choosing a small value a for the electronic density, and by selecting all those points r in the 3D space where the density p(r) happens to be equal to this value a, that is, where equation (2.3) is fulfilled. For an appropriate small value a, this contour surface may be regarded as the surface of the essential part of the molecule and, in short, it may be referred to as the molecular surface. These surfaces, the molecular isodensity contour surfaces, or MIDCO s, are denoted by G(a) and are defined as... 

For a continuous function, such as the electronic density p(r), all points r fulfilling equation (2.3) do form a continuous surface. Consequently, the terms contour surface and isodensity surface are appropriate for G(a). For the study of the 3D shape properties of molecular bodies, represented by level sets F(a) of electronic charge densities, it is sufficient to study the shape of their boundaries these boundaries are the MIDCO s G(a). 

The concept of density domains is related to the concept of MIDCO in a simple way. A maximum connected part of an isodensity contour surface G(a) and the corresponding part of the level set F(a) enclosed by it is called a density domain, DDj(a) [109]. Below we shall give a more formal definition and describe the most essential properties of density domains. 

A family of molecular surfaces can be defined in terms of the MEP using methods similar to those applied in the case of the electronic density. Since the MEP is a continuous function of the three-dimensional position variable r, it can also be analyzed in terms of level sets F(a) and their contour surfaces G(a) for a selected constant MEP value a, defined analogously to those of charge density ... 

A variety of molecular surfaces can be defined in terms of the molecular orbitals Ti(r) of a molecule. A contour surface of an MO 4 j(r) is defined in terms of the level sets F(a) and its boundary surface G(a), with respect to a threshold a of the function value Ti(r). Similar to the case of MEPCO s, these threshold values may take both positive and negative values. 

The definitions of level sets F(a) and contour surfaces G(a) of an individual molecular orbital Tj(r) are analogous to those of the charge density and electrostatic potential ... 

Since this program is in the early stages of development, this chapter will discuss recent progress associated with a planned flight test of a military aircraft to acquire narrowband acoustics. The majority of full-scale acoustic studies have been performed to establish the environmental impact on a community by estimating ground contours to set Duration Index Levels. As a result, these studies are generally performed in Vs-octave bands. 

Figure 5. Effect of model on the interaction Laplacian densities A (V p) = (V p dimer - V p monomers) in water dimer (a plane, experimental geometry, Millot-Stone basis set). Same contours as in Figure 2, but with n beginning at -3. Dashed contours denote negative, solid contours positive values of A(V p). A negative contour level means that locally the dimer has less charge concentration than the pro-dlmer.

Fig. 9 Effect of re-initialization for two merging bubbles. Left columns show the interface and right columns show contours of the level set function . (a) Initial configuration, (b) just before merging (time step n = 100) with no re-initialization, and (c) just before merging (n = 100) with re-initialization back to a distance function [24]...

On the level set, the interface at any instant corresponds to the contour (x, t) = 0. Since the interface moves with the fluid particles, the evolution of <1> is then given by the following convection equation ... 

The 3D shape properties of molecular bodies, represented by level sets F a) of electronic charge densities, can be described by isodensity contour surfaces, denoted by G(a) ... 

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