Periodic orbits molecules

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... 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

In the analysis of the bulk periodic orbits, a simplification occurs for the bending oscillations. Because the Hamiltonian of a linear molecule depends quadratically on the angular momentum variable La, the time derivative of the conjugated angle given by = l2 c vanishes with La, in contrast to the time derivatives of the other angle variables, which are essentially equal to 0j - os j. Therefore, the subsystem La = 0 always contains bulk periodic orbits that are labeled by n, tr2,n-i). 

In tetra-atomic molecules that are linear, the number of degrees of freedom is F =7, which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. 

We focus on the nonlinear dynamics for the collinear configuration which we treat as part of the full multidimensional system. This is to be contrasted with two-degree-of-freedom models where the molecule is assumed to be frozen in some angular configuration, such that the bending degree of freedom is excluded from the dynamics. In our analysis, bending is taken into account in terms of linearized dynamics, which allows us to extend the results for the collinear situation to the full three-dimensional system. The restriction we must be aware of is that the three-dimensional system may have periodic orbits that are not of collinear type. 

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

Although unstable, this periodic orbit is an example of classical motion which leaves the molecule bounded. Other periodic and nonperiodic trajectories of this kind may exist at higher energies. The set of all the trajectories of a given energy shell that do not lead to dissociation under either forwarder backward-time propagation is invariant under the classical flow. When all trajectories belonging to this invariant set are unstable, the set is called the repeller [19, 33, 35, 48]. There also exist trajectories that approach the repeller in the future but dissociate in the past, which form the stable manifolds of the repeller Reciprocally, the trajectories that approach the... 

Figure 9. Supercritical antipitchfork bifurcation scenario for symmetric XYX molecules on the left, bifurcation diagram in the plane of energy versus position in the center, typical phase portraits in some Poincare section in the different regimes on the right, the fundamental periodic orbits in position space.

In the presence of reflection symmetry with respect to the diagonal of the potential-energy surface, as in symmetric molecules or in the four-disk scatterer, Burghardt and Gaspard have shown that a further symmetry reduction can be performed in which the symbolic dynamics still contains three symbols A = 0, +, - [10]. The orbit 0 is the symmetric-stretch periodic orbit as before. The orbit + is one of the off-diagonal orbits 1 or 2 while - represents a half-period of the asymmetric-stretch orbit 12. Note that the latter has also been denoted the hyperspherical periodic orbit in the literature. 

To conclude this section, let us add that the formulas (4.15)-(4.17) can be generalized in a straightforward way to repellers with more than three fundamental periodic orbits. In the following, these tools are applied to several dissociating molecules. 

In conclusion, quantum dynamical localization plays an important role in the excitation and ionization process of atoms and molecules. A question that remains open in connection with the previous talk is whether quantization via periodic orbits can account for this phenomenon. 

Dr. Gaspard, you discussed periodic orbits of a number of tri-atomic molecules. I would like to know how many degrees of freedom have been included in the analysis ... 

V. Engel You showed the plot of the Nal molecule, which is a system with a curve crossing. The treatment of classical mechanics in such a system is not well defined. What can be done to treat the nonadiabatic effects if you calculate periodic orbits that are then used for interpretation ... 

The interpretation of a spectrum from a dynamical point of view can also be applied to a spectrum containing a broad feature associated with direct and/or indirect dissociation reactions. From such spectra dynamics of a dissociating molecule can also be extracted via the Fourier transform of a spectrum. An application of the Fourier transform to the Hartley band of ozone by Johnson and Kinsey [3] demonstrated that a small oscillatory modulation built on a broad absorption feature contains information of the classical trajectories of the vibrational motion on PES, so-called unstable periodic orbits, at the transition state of a unimolecular dissociation. 

Figure 8.6 depicts the four simplest types of unstable periodic orbits for the model CO2 system. Despite the fact that the total energy is well above the dissociation threshold and despite the lack of an intermediate potential well, that could possibly trap the molecule at short distances,... 

Marston, C.C. and Wyatt, R.E. (1984a). Resonant quasi-periodic and periodic orbits for the three-dimensional reaction of fluorine atoms with hydrogen molecules, in Resonances in Electron-Molecule Scattering, van der Waals Molecules, and Reactive Chemical Dynamics, ed. D.G. Truhlar (American Chemical Society, Washington, D.C.). 

Schinke, R. and Engel, V. (1990). Periodic orbits and diffuse structures in the photodissociation of symmetric triatomic molecules, J. Chem. Phys. 93, 3252-3257. 

The purpose of this chapter is to review some properties of isomerizing (ABC BCA) and dissociating (ABC AB + C) prototype triatomic molecules, which are revealed by the analysis of their dynamics on precise ab initio potential energy surfaces (PESs). The systems investigated will be considered from all possible viewpoints—quanmm, classical, and semiclassical mechanics—and several techniques will be applied to extract information from the PES, such as Canonical Perturbation Theory, adiabatic separation of motions, and Periodic Orbit Theory. 

Figure 44 Poincare surfaces of section for a two-dimensional model of HOCl the HO bond distance is frozen in the classical calculations. 7 is the Jacobi angle and p is the corresponding momentum. Different symbols in square brackets denote different types of periodic orbits in the molecule. Reproduced, with permission of the American Chemical Society, from Ref. 258.

FIGURE 6.10 Correlation diagram for first-period diatomic molecules. Blue arrows indicate the electron filling for the H2 molecule. All of the atomic electrons are pooled and used to fill the molecular orbitals using the aufbau principle. In the molecules, electrons are no longer connected to any particular atom. 

FIGURE 6.16 Correlation diagrams for second-period diatomic molecules, (a) Correlation diagram and molecular orbitals calculated for N2. (b) Correlation diagram and molecular orbitals calculated for F2. 

New graphical representations of the exact molecular orbitals for Hj that make it easier to visualize these orbitals and interpret their meanings. These images provide a foundation for developing MO theory for the first- and second-period diatomic molecules. 

We have also used the periodic reduction method to predict with good accuracy the 3D structure of vibrationally bonded molecules. It should be stressed though, that in principle it is not necessary to use periodic reduction. As shown in Fig. 9 the RPO s of the IHI system are stable also in 3D, one can find bound quasi-periodic orbits and quantize them semiclassically directly without resorting to periodic reduction. 

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