Periodic orbit bending

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). 

Moreover, the Poincare mappings of (3.14) at values of Pc) fixed by the existence of (5, 6) and I4 + L5 = 0 show the presence of classically chaotic motions with a bifurcation at E - 6900 cm-1 (see Fig. 6). At this bifurcation, the periodic orbit (5, 6) becomes unstable because one of its Lyapunov exponents turns positive, as shown in Fig. 7. The periodic orbit (6, 7) destabilizes by a similar scenario around E - 7200 cm-1. These results show that the interaction between the bending modes leads to classically chaotic behaviors that destabilize successively the periodic orbits. For the bulk peri-... 

Figure 7. Lyapunov exponents of the periodic orbits (n4, n ) = (5,6), (6,7) in the bending subsystem of Hamiltonian (3.14). The numerical error is -5 x 10-4 fs-1. (From Ref. 114.)...

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. 

Figure 16. Scattering resonances of the full rotational-vibrational Hamiltonian describing the dissociation of CO2 on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with 7 = 0,..., 10 are given by dots. Their close vicinity explains the formation of hyphens , i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with V2 = 0,. .. 5. The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

Fig. 8.9. Contour plot of the potential energy surface of H2O in the BlA state as a function of the H-OH dissociation bond Rh-oh and the HOH bending angle a the other O-H bond is frozen at the equilibrium value in the ground electronic state. The energy normalization is such that E = 0 corresponds to H(2S ) + OH(2E, re). This potential is based on the ab initio calculations of Theodorakopulos, Petsalakis, and Buenker (1985). The structures at short H-OH distances are artifacts of the fitting procedure. The cross marks the equilibrium in the ground state and the ellipse indicates the breadth of the ground-state wavefunction. The heavy arrow illustrates the main dissociation path and the dashed line represents an unstable periodic orbit with a total energy of 0.5 eV above the dissociation threshold.

As in the cases discussed in Section 8.1 an unstable periodic orbit, illustrated by the broken line in Figure 8.9, guides the indirect classical trajectories and likewise the temporarily trapped part of the quantum mechanical wavepacket. It essentially represents large-amplitude bending motion, that is strongly coupled, however, to the stretching coordinate... 

In this chapter we have explored the structure of organic compounds. This is important since structure determines reactivity. We have seen that weak bonds are a source of reactivity. Strong bonds are made by good overlap of similar-sized orbitals (same row on periodic table). Bends or twists that decrease orbital overlap weaken bonds. Lewis structures and resonance forms along with electron flow arrows allow us to keep track of electrons and explain the changes that occur in reactions. VSEPR will help us predict the shape of molecules. Next we must review how bonds are made and broken, and what makes reactions favorable. Critical concepts and skills from this chapter are ... 

We compare the results in Table IV with the approximate resonant periodic orbit (RPO) calculations of Poliak and Wyatt (147) and with accurate quantal calculations (61). In the RPO calculations, the bending degrees of freedom are Included using... 

Using these methods, several resonant quaslperlodlc and periodic orbits were computed and plotted In the Internal coordinate space. These orbits exhibit resonant energy transfer between local (dressed) vlbra-tlon-bend oscillations In the entrance and exit regions of the collision complex. Frequencies and actions from the periodic orbits were then used in the arbitrary-trajectory semlclasslcal quantization scheme (19). The lowest resonance energy predicted for the J=0 reaction was In good agreement with all available quantal and adiabatic results. Further properties of both types of orbit, including those obtained from a stability analysis, will be presented elsewhere (21). 

E. Poliak, Periodic orbit analysis of bend level structure of resonances in 3D H + H2 reactive scattering, Chem. Phys. Lett. 137 171 (1987). 

At this point one may make an adiabatic or a sudden assumption. If the translational motion is faster than the bend motion, one must first solve for the u motion in Eq. (40) and only then return to the full Hamiltonian, This sudden approximation will be derived later. If translational motion is slow relative to the bend then one inserts h(y) back in the full Hamiltonian and averages the full Hamiltonian over the period of the angle dependent periodic orbit. The adiabatic Hamiltonian that emerges from this process is... 

One of the theoretical shortcomings of the periodic reduction method is the rather arbitrary definition of the bend angle. Although, as was the case to date, one may a posteriori verify that the adiabatic reduction is justified by comparing bend frequencies with stretch frequencies, it is still desirable to construct a method which a priori does not have this ambiguity. One possibility is using the quasiperiodic reduction method outlined in this section. However, as we shall point out a much simpler method may be based on stability analysis of periodic orbits in 3D. 

ABSTRACT. The mechanisms for energy flow from overtone excited HC and HO local modes have been elucidated in two mode model Hamiltonians of benzene and trihalomethanes and in a six mode model of HOOH molecule. Intramolecular vibrational relaxation (IVR) from the excited 2 1 Fermi resonance is shown to be very sensitive to the stretch-bend potential energy coupling in connection with the stability of the HC stretch periodic orbit. The overtone induced dissociation of HOOH, which is a slow process in comparison with the initial HO overtone relaxation, is explained in terms of the details of the potential energy surface. 

---