Unstable periodic orbits theory

The term scar was introduced by Heller in his seminal paper (Heller, 1984), to describe the localization of quantum probability density of certain individual eigenfunctions of classical chaotic systems along unstable periodic orbits (PO), and he constructed a theory of scars based on wave packet propagation (Heller, 1991). Another important contribution to this theory is due to Bogomolny (Bogomolny, 1988), who derived an explicit expression for the smoothed probability density over small ranges of space and energy... 

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal iepeller. As we mentioned in Section II, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodic-orbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. 

A beautiful classical theory of unimolecular isomerization called the reactive island theory (RIT) has been developed by DeLeon and Marston [23] and by DeLeon and co-workers [24,25]. In RIT the classical phase-space structures are analyzed in great detail. Indeed, the key observation in RIT is that different cylindrical manifolds in phase space can act as mediators of unimolecular conformational isomerization. Figure 23 illustrates homoclinic tangling of motion near an unstable periodic orbit in a system of two DOFs with a fixed point T, and it applies to a wide class of isomerization reaction with two stable isomer... 

De Leon and co-workers [34—37] established an elegant reaction theory for a system with two DOFs, the so-called reactive island theory to mediate reactions through cylindrical manifolds apart from the saddles. Their original algorithm depends crucially on the existence of pure unstable periodic orbits in the nonreactive DOFs in the region of the saddles and did not extend to systems with many DOFs. 

Why has this elegant reaction theory not been applied to general reactive systems with many DOFs It was because their algorithm depends crucially on finding pure unstable periodic orbits in the nonreactive degrees of freedom. As pointed out previously [44], it is always possible to find this regulatory object in... 

Among the diatomic molecules of the second period elements are three familiar ones, N2,02, and F2. The molecules Li2, B2, and C2 are less common but have been observed and studied in the gas phase. In contrast, the molecules Be2 and Ne2 are either highly unstable or nonexistent. Let us see what molecular orbital theory predicts about the structure and stability of these molecules. We start by considering how the atomic orbitals containing the valence electrons (2s and 2p) are used to form molecular orbitals. 

A meaningful difference in activity of lanthanides (31, 41), apart from a qualitative similarity among them, in giving cis-poly-diolefins should be related to their periodicity, since a minimum of activity corresponds to a maximum of tendency in attaining the divalent state. Wang and his colleagues ventured to explain the differences in catalytic activity in terms of molecular orbital theory and attributed the inactivity in diolefin polymerization of Sm and Eu to their unstable +3 oxidation state. 

This experiment established the nuclear model of the atom. A key point derived from this is that the electrons circling the nucleus are in fixed stable orbits, just like the planets around the sun. Furthermore, each orbital or shell contains a fixed number of electrons additional electrons are added to the next stable orbital above that which is full. This stable orbital model is a departure from classical electromagnetic theory (which predicts unstable orbitals, in which the electrons spiral into the nucleus and are destroyed), and can only be explained by quantum theory. The fixed numbers for each orbital were determined to be two in the first level, eight in the second level, eight in the third level (but extendible to 18) and so on. Using this simple model, chemists derived the systematic structure of the Periodic Table (see Appendix 5), and began to... 

As accurately as these calculations can be made, however, the behavior of celestial bodies over long periods of time cannot always be determined. For example, the perturbation method has so far been unable to determine the stability either of the orbits of individual bodies or of the solar system as a whole for the estimated age of the solar system. Studies of the evolution of the Earth-Moon system indicate that the Moon s orbit may become unstable, which will make it possible for the Moon to escape into an independent orbit around the Sun. Recent astronomers have also used the theory of chaos to explain irregular orbits. 

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