Localization properties of the wave

The division of the molecular volume into atomic basins follows from a deeper analysis based on the principle of stationary action. The shapes of the atomic basins, and the associated electron densities, in a functional group are very similar in different molecules. The local properties of the wave function are therefore transferable to a very good approximation, which rationalizes the basis for organic chemistry, that functional groups react similarly in different molecules. It may be shown that any observable... 

For Ef near a band extremity, we find, using (49) for a , that directional properties of the wave functions. Thus [Pg.38]

There are two extreme views in modeling zeolitic catalysts. One is based on the observation that the catalytic activity is intimately related to the local properties of the zeolite s active sites and therefore requires a relatively small molecular model, including just a few atoms of the zeolite framework, in direct contact with the substrate molecule, i.e. a molecular cluster is sufficient to describe the essential features of reactivity. The other, opposing view emphasizes that zeolites are (micro)crystalline solids, corresponding to periodic lattices. While molecular clusters are best described by quantum chemical methods, based on the LCAO approximation, which develops the electronic wave function on a set of localized (usually Gaussian) basis functions, the methods developed out of solid state physics using plane wave basis sets, are much better adapted for the periodic lattice models. 

After demonstrating the effect of disorder on the energy-band structure of a chain and on the localization properties of the corresponding wave functions, we shall describe in the subsequent sections several methods to determine the eneigy-level distribution (density of states) and wave functions of an aperiodic chain. 

If one tries to develop a perturbation theory proceeding directly from the reaction-diffusion equations this meets with serious difficulties. They arise because translation and rotation perturbation modes for the spiral wave are not spatially localized. We bypass such difficulties by using the quasi-steady approximation formulated in the previous section. In this approximation the trajectory of the tip motion can be calculated by solving a system of ordinary differential equations which depend only on the local properties of the medium in the vicinity of the tip. The perturbations which originate outside a small neighbourhood of the end point propagate quickly to the periphery and do not influence the motion of the tip. The evolution of the entire curve can then be calculated in the WR approximation using the known trajectory of the tip motion as a dynamic boundary condition. 

It follows from (74) that the motion of curves is sensitive to the local properties of the surface only if its Gaussian curvature is nonvanishing. Since the Gaussian curvature is zero for such surfaces as cylinders and cones (because one of the two principal curvatures vanishes) the motion of curves and the properties of spiral waves on these surfaces are identical to those on the plane. 

A detonation shock wave is an abrupt gas dynamic discontinuity across which properties such as gas pressure, density, temperature, and local flow velocities change discontinnonsly. Shockwaves are always characterized by the observation that the wave travels with a velocity that is faster than the local speed of sound in the undisturbed mixtnre ahead of the wave front. The ratio of the wave velocity to the speed of sound is called the Mach number. 

Everything considered, sonoelastography is a very challenging approach for characterization of mechanical waves. To become the gold standard in non-invasive elastography, US methods should provide good anatomic localization of the viscoelastic properties as well as the 3D assessment of the wave pattern. But such improvements would possibly make sonoelasticity methods slower and less convenient. 

There is a fundamental difference between the properties of the molecular bond length in the STIRAP and APLIP processes that is overlooked by the average bond distance. In APLIP the wave function is always localized around the average internuclear distance, since it is constrained by the electronic forces exerted by Uup(x,t) x(t) is also stable since it corresponds to the minimum of a potential, and it is quasi-static, since the rate of change of Uup(x,t) can be arbitrarily controlled. Therefore, it truly corresponds to the classical notion of a bond length. 

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