Cylindrical manifolds theory

N. De Leon, M. A. Mehta, and R. Q. Topper, Cylindrical manifolds in phase space as mediators of chemical reaction dynamics and kinetics. I. Theory, J. Chem. Phys. 94, 8310 (1991). 

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. 

N. De Leon, /. Chem. Phys., 96, 285 (1991). Cylindrical Manifolds and Reactive Island Kinetic Theory in the Time Domain. 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

Oped a so-called reactive island theory the reactive islands are the phase-space areas surrounded by the periodic orbits in the transition state, and reactions are interpreted as occurring along cylindrical invariant manifolds through the islands. Fair et al. [29] also found in their two- and three-dof models of the dissociation reaction of hydrazoic acid that a similar cylinderlike structure emerges in the phase space as it leaves the transition state. However, these are crucially based on the findings and the existence of (pure) periodic orbits for all the dof, at least in the transition states. Hence, some questions remain unresolved, for example, How can one extract these periodic orbits from many-body dof phase space and How can the periodic orbits persist at high energies above the saddle point, where chaos may wipe out any of them ... 

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