Contour lines

Contour lines show equidistant energies at wave number dilFerenees of 3000 em up to 30 000 em The upper eiirves are one-dimensional outs along O ft) and (right). The dashed ourves in the two upper... 

Figure B3.5.1. Contour line representation of a quadratic surface and part of a steepest descent path zigzagging toward the minimum.

Fig. 10. Conformational flooding accelerates conformational transitions and makes them accessible for MD simulations. Top left snapshots of the protein backbone of BPTI during a 500 ps-MD simulation. Bottom left a projection of the conformational coordinates contributing most to the atomic motions shows that, on that MD time scale, the system remains in its initial configuration (CS 1). Top right Conformational flooding forces the system into new conformations after crossing high energy barriers (CS 2, CS 3,. . . ). Bottom right The projection visualizes the new conformations they remain stable, even when the applied flooding potentials (dashed contour lines) is switched off.

You can specify the number and values of visible contour lines. You specify the total number of contour lines to be shown by simple stating the number, n>0. You normally specify the values of the contour lines as default values. For this case, HyperChem computes the maximum and minimum values on the grid and then draws contours at these values plus n-2 contour lines evenly spaced in between these maximum and minimum values. If you need non-default values, you can specify the starting value and then an increment to define the other n-1 evenly spaced contour lines. If default values were computed previously, HyperChem suggests the starting value and increment of the previous default computation for the new non-default option. 

Example of a two-factor response surface displayed as (a) a pseudo-three-dimensional graph and (b) a contour plot. Contour lines are shown for intervals of 0.5 response units. 

If we use a contour map to represent a three-dimensional surface, with each contour line representing constant potential energy, two vibrational coordinates can be illustrated. Figure 6.35 shows such a map for the linear molecule CO2. The coordinates used here are not normal coordinates but the two CO bond lengths rj and r2 shown in Figure 6.36(a). It is assumed that the molecule does not bend. 

Isotherm A line in a flow system or on a graph connecting points of equal temperature, or a mathematical or graphical relationship between two variables at constant temperature. Or a display using lines on a drawing to show constant-temperature contour lines, as from thermal imaging with infrared techniques. 

FIGURE 9.5. The potential surface for the 0"C = 0— 0-C-0" step in amide hydrolysis in solution, where the surface is given in terms of the angle 0 and the distance b. The heavy contour lines are spaced by fi (at room temperature) and can be used conveniently in estimating entropic effects. The figure also shows the regions (cross hatched) where the potential is less than for the corresponding reaction in the active site of subtilisin. 

The contour lines represent in section places of relative density 1.0, 0.9, 0.8,. . 0.1. The density at points along a line drawn through the nuclei is given above. 

The contour lines represent points of relative density 1.0, 0.9, 0.8,..0.1 for a hydrogen atom. This figure, with the added proton 1.06 A from the atom, gives the electron distribution the hydrogen molecule-ion would have (in the zeroth approximation) if the resonance phenomenon did not occur it is to be compared with figure 6 to show the effect of resonance. 

Figure 1. Departure from spherical on the inner ring of segments before correcting for tilt (left), after correcting for tilt (middle), and after correcting for tilt on the outer ring (right). Contour lines are approximately 1 mwm surface error.

Fig. 11 —Isometric views and associated contour line plots of film shape for k=2.9 and rolling speed of 0.021 and 0.042 ms" (Ref. [56]).

The availablility of an estimate of the conditional cdf F CzjCN)) at each nodal joint x allows an assessment of the risks a(x) or p(x) of making wrong decisions. Consider the contour map of a particular estimate p (x), x (see Figure 3a). Suppose that the threshold value 500 has been selected to declare any sub-area of A hazardous. The contour line 500 on Figure 3 delineates the zones which are candidates for cleaning. Within these zones, the probability that the concentration is actually under 500, i.e. the risk a(x) of cleaning unduly, can also be mapped ... 

Fig. 44.20. (a) The trained logical OR. The contour lines in the xl, x2-plane around the centres denote the 20, 40, 60 and 80 percent confidence limits of the Gaussian kernels, (b) The trained logical OR with increased width factors. 

The computation of the above surface in the parameter space is not trivial. For the two-parameter case (p=2), the joint confidence region on the krk2 plane can be determined by using any contouring method. The contour line is approximated from many function evaluations of S(k) over a dense grid of (k, k2) values. 

ELF can be visualized with different kinds of images. Colored sections through a molecule are popular, using white for high values of ELF, followed by yellow-red-violet-blue-dark blue for decreasing values simultaneously, the electron density can be depicted by the density of colored points. Contour lines can be used instead of the colors for black and white printing. Another possibility is to draw perspective images with iso surfaces, i.e. surfaces with a constant value of ELF. Fig. 10.2 shows iso surfaces with ELF = 0.8 for some molecules from experience a value of ELF = 0.8 is well suited to reveal the distribution of electron pairs in space. 

Two variable problems can be plotted as shown in Figure 1.17. The values of the objective function are shown as contour lines, as on a map, which are slices through the three-dimensional model of the function. Seeking the optimum of such a function can be... 

Figure 4. Momentum density anisotropy of Cu the solid line marks the boundary of the first Brillouin zone solid and dashed contour lines mark positive and negative anisotropies, respectively.

Figure 6.4 shows a third and commonly used way of representing electron density profiles the two-dimensional contour map. This map for the SCI2 molecule corresponds to the relief map in Figure 6.2. Although this map is able to show very detailed information, we are restricted to a particular choice of plane, or to a selection of planes. To obtain an approximately equally dense distribution of contour lines, contour values used in this book increase in the nearly geometrical sequence, I0 3, 2 X 10 3, 4 X 10 3, 8 X 10-3. 2 X 10-3.

Figure 6.4 Contour plot of the electron density in the molecular plane of SCI2. The outer contour line corresponds to 0.001 au and the next contour lines correspond to values increasing according to the pattern 2 X 10", 4 X 10", 8X10" where n varies from —3 to 2.

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