Shape invariance region

Since TD-DFT is applied to scattering problems in its QFD version, two important consequences of the nonlocal nature of the quantum potential are worth stressing in this regard. First, relevant quantum effects can be observed in regions where the classical interaction potential V becomes negligible, and more important, where p(r, t) 0. This happens because quantum particles respond to the shape of K, but not to its intensity, p(r, t). Notice that Q is scale-invariant under the multiplication of p(r, t) by a real constant. Second, quantum-mechanically the concept of asymptotic or free motion only holds locally. Following the classical definition for this motional regime,... 

The line shapes were calculated for the flipping motion with the two-site jump model described above, and the calculated spectra are shown in Fig. 11 for the higher temperature region. The experimental line shapes at 20 and 30° C are well reproduced showing the motional mode and rates obtained by T analysis are reasonable at least around these temperatures. Above 40°C the calculated line shapes are invariable and remain in the powder pattern undergoing a rapid flipping motion, while the experimental ones... 

Mak L, Grandison S, Morris RJ (2008) An extension of spherical harmonics to region-based rotationally invariant descriptors for molecular shape description and comparison. J Mol Graph Model 26 1035-1045... 

Although the shape of the Visual-Empirical Region-of-Influence (VERI) mask is invariable, it size scales automatically according to the properties of the cluster (Fig. 10.10b). The VERI algorithm requires preprocessing of the data and for that purpose PCA or PCR preprocessing is routinely used. It is relatively immune to the presence of unknowns, and nonlinearity and nonadditivity of sensor responses (Osbourn et al., 1998). It has been used successfully to determine the optimum... 

Figure 17-7 is a diagrammatic representation of tRNA folded into the typical cloverleaf structure, containing a number of stems (base-paired) and loops. While the sequences of the different tRNAs are different, there are regions that remain invariant. Most of these are in the loops, within which the unusual bases are concentrated, and at the 3 end of the molecule contained within the acceptor stem. The sequence at this end is always CCA, and it is to the 3 OH that the appropriate amino acid is attached through its carboxyl group. The three nucleotides complementary to the codon for the amino acid make up what is known as the anticodon (shaded part of Fig. 17-7). The three-dimensional structure of tRNA is known. In this structure, there are additional H bonds, which stabilize the cloverleaf in a more elongated L-shaped structure, with the acceptor sequence at one end and the anticodon loop at the other.

Titration curves for conductometric titrations take a variety of shapes, depending on the chemical system under investigation. In general, they are characterized by straight line portions with dissimilar slopes on either side of the equivalence point, as shown previously in Fig. 3. To establish a conductometric endpoint, after correcting for volume changes, the conductance data are plotted as a function of titrant volume. The two linear portions are then extrapolated, and the point of intersection is taken as the equivalence point. Frequently, reactions fail to proceed to absolute completion, and the conductometric titration curves invariably show departures from strict linearity in the region of the equivalence point. 

Qualitatively the slope, a, changes as ACzg varies, even though the shapes of the parabolas are assumed to be invariant, because the cross-over point has slopes which alter (see Appendix 1, Section A 1.1.3). The Marcus equation, although simple in concept, is remarkably successful, probably because parabolas are good approximations to the shapes of energy surfaces over small regions of space. 

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