Overlap visualization

Figure Al.6.30. (a) Two pulse sequence used in the Tannor-Rice pump-dump scheme, (b) The Husuni time-frequency distribution corresponding to the two pump sequence in (a), constmcted by taking the overlap of the pulse sequence with a two-parameter family of Gaussians, characterized by different centres in time and carrier frequency, and plotting the overlap as a fiinction of these two parameters. Note that the Husimi distribution allows one to visualize both the time delay and the frequency offset of pump and dump simultaneously (after [52a]).

The space filling model developed by Corey, Pauling, and Koltun is also known as the CPK model, or scale model [197], It shows the relative volume (size) of different elements or of different parts of a molecule (Figure 2-123d). The model is based on spheres that represent the "electron cloud . These atomic spheres can be determined from the van der Waals radii (see Section 2.10.1), which indicate the most stable distance between two atoms (non-bonded nuclei). Since the spheres are all drawn to the same scale, the relative size of the overlapping electron clouds of the atoms becomes evident. The connectivities between atoms, the bonds, are not visualized because they are located beneath the atom spheres and are not visible in a non-transparent display (see Section 2.10). In contrast to other models, the CPK model makes it possible to visualize a first impression of the extent of a molecule. 

Each filled cell in Exhibit 2-9 shows where PSM or ESH programs and elements meet a particular ISO 9000 requirement. Exhibit 2-9 provides a powerful visual impression of the overlap between PSM, ESH, and Quality Management. Further, it is possible to quickly see where the greatest benefits may arise. By examining Exhibit 2-9 initial priorities for attention in our example might be ... 

The above pression is often visualized (and computed) (9) in terms of the Fourier transform of the overlap of two tune-propagated wavefimctions ... 

To visualize bond formation by an outer atom other than hydrogen, recall the bond formation in HF. One valence p orbital from the fluorine atom overlaps strongly with the hydrogen 1 S orbital to form the bond. We can describe bond formation for any outer atom except H through overlap of one of its valence p orbitals with the appropriate hybrid orbital of the inner atom. An example is dichloromethane, CH2 CI2, which appears in Figure 10-11. We describe the C—H bonds by 5 -I S overlap, and we describe the C—Cl bonds by 5 - 3 p... 

In triethylaluminum, each A1—C bond can be visualized as an. y p hybrid on aluminum overlapping with an S p hybrid on a carbon atom. Figure 10-13 shows this bonding representation, with three equivalent A1—C bonds and the unused 3 p orbital on the aluminum atom. 

The plot of pH against titrant volume added is called a potentiometric titration curve. The latter curve is usually transformed into a Bjerrum plot [8, 24, 27], for better visual indication of overlapping pKiS or for pffjS below 3 or above 10. The actual values of pKa are determined by weighted nonlinear regression analysis [25-27]. 

A useful tool in the interpretation of SIMCA is the so-called Coomans plot [32]. It is applied to the discrimination of two classes (Fig. 33.18). The distance from the model for class 1 is plotted against that from model 2. On both axes, one indicates the critical distances. In this way, one defines four zones class 1, class 2, overlap of class 1 and 2 and neither class 1 nor class 2. By plotting objects in this plot, their classification is immediately clear. It is also easy to visualize how certain a classification is. In Fig. 33.18, object a is very clearly within class 1, object b is on the border of that class but is not close to class 2 and object c clearly belongs to neither class. 

Visual overlap test spectrum (t) and reference spectrum (r) to compare spectral shapes for similarity... 

The hybridization parameter X can be estimated at each R by expressions such as (3.1 lb), which in turn are related to visual plots of the s-pz overlap. A procedure for numerically estimating X will be given at the end of this section. For present purposes, we need only recognize from the percentage s characters that the relative energies of the starting atomic hybrids satisfy... 

The strength of the Fajb interaction and its variations with distance and orientation can be conveniently visualized in terms of the overlap of 7ra and 7tb NBOs, on the basis of a Mulliken-type approximation (cf. Eq. (1.34)). As an example, the top two panels of Fig. 3.38 compare the overlapping 7ta-7tb orbital contours for trails 1 and cis 2 isomers of butadiene. As shown in Fig. 3.38, the overlap in the cis isomer 2 (S = 0.2054) is slightly weaker than that in the trans isomer 1 (S = 0.2209), due to the unfavorable orientation of the 7ta across the nodal plane of the 7tb in the latter case. Consistently with the weaker 7ta-7tb overlap, the JtA F nh ) interaction is less, namely 0.0608 in 1 versus 0.0564 in 2. The delocalization tail of the 7fa NEMO is correspondingly less than its value in the trans isomer... 

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