Pseudo-three-dimensional representations

Figure 12.3 is a pseudo-three-dimensional representation of a portion of the three-dimensional experiment space associated with the system shown in Figure 12.1. The two-dimensional factor subspace is shaded with a one-unit grid. The factor domains are again 0 < Xj < +10 and 0 < < +10. The response axis ranges from 0...

Figure 12.4 is a pseudo-three-dimensional representation of a response siuface showing a system response, plotted against the two system factors, x, and X2. The... 

Figure 11.3 is a pseudo-three-dimensional representation of a portion of the three-dimensional experiment space associated with the system shown in Figure 11.1. The two-dimensional factor subspace is shaded with a one-unit grid. The factor domains are again 0 < xl < +10 and 0 < x2 < +10. The response axis ranges from 0 to +8. The location in factor space of the single experiment at xn = +3, x2l = +7 is shown as a point in the plane of factor space. The response (yn = + 4.00) associated with this experiment is shown as a point above the plane of factor space, and is connected to the factor space by a dotted vertical line.

There are many different ways to plot a 2D IR correlation spectrum. A pseudo-three-dimensional representation (so-called fishnet plot) as shown in Figure 1-1 is well suited for providing the overall features of a 2D IR spectrum. Usually, however, 2D IR spectra are more conveniently displayed as contour maps to indicate clearly the location and intensity of peaks on a given spectral plane. In the following section, basic properties of and information extracted from 2D IR spectra are reviewed by using schematic contour maps (Figures 1-10 and 1-11). 

FIG. 4.2. Dependence of dimensionless pseudo-steady-state intermediate concentration and temperature excess on reactant concentration for the model with the exponential approximation (a) three-dimensional representation of universal locus (b) projection showing dependence of a5S on n (c) projection showing... 

Figure 4.3 Pseudo-three-dimensional, iconographic, and linear representations of calix[ 6 arene conformations...

Non-pharmaceutical applications tend to be aimed at accurate stmcture prediction together with properties such as electrostatic potential maps and surface hydrophobic-ity. One consequence of this is a desire for easily understood graphical representations of both the molecules and the calculated properties. An example of this can be seen in Figure 1, which shows a geometry-optimized computed structure of the potassium complex of the macrocycle 18-crown-6. A representation of the complex s electrostatic potential on a van der Waals surface is overlaid on top of this. A cutaway view is depicted so that the connectivity and identity of each atom can be seen clearly and correlated with the local electrostatics. In the example, blue represents an electropositive region, red is electronegative, and green is neutral. As the complex can be rotated on a computer screen, the structure can be considered to be pseudo three dimensional. The representation of physical properties adds a further dimension to this. Snapshots of molecules and associated properties, perhaps modeled at precise intervals... 

The familiar set of the three t2g orbitals in an octahedral complex constitutes a three-dimensional shell. Classical ligand field theory has drawn attention to the fact that the matrix representation of the angular momentum operator t in a p-orbital basis is equal to the matrix of — if in the basis of the three d-orbitals with t2g symmetry [2,3]. This correspondence implies that, under a d-only assumption, l2 g electrons can be treated as pseudo-p electrons, yielding an interesting isomorphism between (t2g)" states and atomic (p)" multiplets. We will discuss this relationship later on in more detail. 

Response surfaces in more than one dimension (more than one parameter) are hard to visualize. Two representations are common for two-dimensional optimization problems, where the response surface as a function of the two parameters forms a three-dimensional picture. Figure 5.2 shows a pseudo-isometric three-dimensional plot of such a surface (figure 5.2a) as well as a contour plot (figure 5.2b). 

Figure 5.2 (a) Pseudo-isometric three-dimensional response and (b) iso-response contour plot for a two-parameter optimization problem. Parameters (in triangular representation) quaternary mobile phase composition. Criterion normalized resolution product (see section 4.3.2). O, is the location of the optimum. For further details see section 5.5.2. Figure taken from ref. [502]. Reprinted with permission. 

Fig. 11 Representation of lowest adiabatic potential of singlet (S = 0) and triplet (S = 1) Fe(CO)4 around T Jahn-Teller conical intersection at tetrahedral (7 ) geometry. There are three equivalent two-dimensional troughs in the space spanned by each pair-wise selection of equal L-M-L angles (boxed vs unboxed). The topological connectivity where the troughs intersect is indicated. There are two non-equivalent epikemel distortion directions E[ 2(Td,h) leading to 6 equivalent C2v minima ( ), and 12 equivalent Cs(x) saddle-points respectively. The non-Berry pseudo-rotation barrier is very small ( 5kcal mol ). CASSCF optimised geometrical parameters for singlet and triplet states are shown at the top left...

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