Periodic orbit structure

Schinke R, Weide K, Heumann B and Engel V 1991 Diffuse structures and periodic orbits in the photodissociation of small polyatomic molecules Faraday Discuss. Chem. Soc. 91 31... 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

Corrected DFT methods the standard DFT calculation is supplemented by an empirical atom-atom term that reproduces correlation effects. They perform well because the missing effect in pure DFT is exactly pinpointed and circumscribed. Examples are a comparison of calculated lattice energies with experimental thermochemical data [46], including periodic orbital treatment, or the accurate reproduction of organic crystal structures without distortion [47]. 

The PIXEL code and documentation (OPiX package) can be downloaded at the author s site http //users.unimi.it/gavezzot. Computing times on an ordinary PC are of the order of a few seconds for a molecular dimer, to a few minutes for a complete crystal structure. PIXEL has been repeatedly shown to give results of accuracy comparable to ab initio MO methods [51]. For an example. Fig. 5 [52] shows a comparison between MP2, PIXEL, uncorrected and corrected periodic-orbital... 

Note that complex periodic orbits generally account for non-classical effects. The most conspicuous example is provided by the coupling between electronic surfaces beyond the Bom-Oppenheimer approximation, at an avoided crossing or conical intersection [14]. The analysis of such problems has recently seen remarkable advances due to the discovery of the gauge structure induced by the separation of the quantum system into the subsets of electronic and nuclear degrees of freedom [52-54],... 

The periodic-orbit quantization can be used to calculate not only the resonances but also the full shape of the photoabsorption cross section using (2.26) and (2.27). This semiclassical formula for the cross section separates in a natural way the smooth background from the oscillating structures due to the periodic orbits. In this way, the observation of emerging periodic orbits by the Fourier transform of the vibrational structures on top of the continuous absorption bands can be explained. 

In a more recent work, Joens [158] has assigned the structures of the Hartley band using a Dunham expansion, that is equilibrium point quantization. The lifetime predicted by his analysis is extremely short, equal to 3.2 fs, while the symmetric stretching period is of 30 fs. Recall, however, that the interpretations in terms of equilibrium point expansions and in terms of periodic orbits are strictly complementary only for regular regimes. 

Finally a question for Prof. Gaspard. You have shown the beautiful structure of periodic orbits in H3. Why, beyond a preliminary work of... 

IN the past twenty years the electronic structures of many organic molecules, particularly benzene and related compounds, have been discussed in toms of the molecular orbital and valence bond methods.1 During the same period the structures of inorganic ions have been inferred from the bond distances f a bond distance shorter than the sum of the conventional radii has been attributed to the resonance of double bonded structures with the single bonded or Lewis structure. 

Because of the superposition of three distinct types of periodic motion with different periods the wavepacket by itself does not reveal a clear picture in the present case, i.e., the classical skeleton is hardly visible through the quantum mechanical flesh . The perfect agreement between the recurrence times of the quantum mechanical wavepacket and the periods of the classical periodic orbits, however, provides convincing evidence that the structures in the absorption spectrum are ultimately the consequence of the three generic unstable periodic orbits. This correlation is... 

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