Cylindrical manifolds

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). 

Study of counterion condensation as a limiting property of the solutions of the Poisson-Boltzmann equation for arbitrary, charged cylindrical manifolds in H3 (see 2.3). 

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... 

Figure 27 illustrates reactive motion through cylindrical manifolds and construction of the manifolds. It shows that a trajectory initially trapped in conformer A eventually enters the interior of the cylinder W. By going through W, it reacts, and it goes to conformer B by entering manifold Wg. The cylinder Wj mediates all pre-reactive motion A B, and the cylinder Wg mediates all... 

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. 

Cylindrical Manifolds. There is one big advantage of looking at 2-DOF TS in phase space It puts emphasis on the existence of the tubes that determine the transport of classical probability in phase space. Existence of those tubes has been known for a long time [48]. These tubes are the set of trajectories that constitute the stable/unstable manifolds of PODS. Locally, in the vicinity of P, they immediatly generalize to higher dimensions. They are constructed as follows ... 

Figure 20 Schematic drawing of two cylindrical manifolds within isomer A in the weak-coupling limit (refer to Figure 8 for an explanation of the symbols). The two-dimensional cylinders will intersect each other along one-dimensional lines. These lines are two homoclinic orbits. The small, thin tube spanning both isomers corresponds to a reactive KAM torus. Note that although we have stopped drawing the cylinders beyond a certain point for clarity, in reality the cylinders continue to wind about and explore the entire accessible region of chaotic phase space. Reprinted with permission from Ref. 108.

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

M. A. Mehta, Ph.D. Dissertation, Yale University, New Haven, CT, 1990. Classical and Quantum Dynamics of Phase Space Cylindrical Manifolds. 

Vertical pre.s.sure leaf filters. These filters have vertical, paraUel, rec tangular leaves mounted in an upright cylindrical pressure tank. The leaves usually are of such different widths as to aUowthem to conform to the curvature of the tank and to fill it without waste space. The leaves often rest on a filtrate manifold, the connec tion being sealed by an O ring, so that they can be lifted individuaUy from the top of the fil-... 

Rotary Drum Filters The rotaiy drum filter is the most widely used of the continuous filters. There are many design variations, including operation as either a pressure filter or a vacuum filter. The major difference between designs is in the technique for cake discharge, to be discussed later. All the alternatives are characterized by a horizontal-axis drum covered on the cylindrical portion by filter medium over a grid support structure to allow drainage to manifolds. Basic materials of construc tion may be metals or plastics. Sizes (in terms of filter areas) range from 0.37 to 186 m (4 to 2000 ft ). 

Even at high n s one needs to follow the system for many orbital periods if one is to mimic the experimental results. The difficulty is compounded if one measures the time in units of periods of the core motion. This suggests that the time evolution be characterized using the stationary states of the Hamiltonian rather than propagating the initial state. We have done so, but our experience is that in the presence of DC fields of experimental magnitude (which means that Stark manifolds of adjacent n values overlap), and certainly so in the presence of other ions that break the cylindrical symmetry and hence mix the m/ values, the size of the basis required for convergence is near the limit of current computers. In our experience, truncating the quan-... 

In the open literature (Ref 17) is briefly described a continuous, remotely controlled, method of mixing ingredients for composite proplnts, which was recently developed at the US Naval Propellant Plant, Indian Head, Maryland. The pilot plant installation at Indian Head, represented schematically in Fig, consists of a mixer B (which is a long cylindrical vessel in which a porous pipe C is inserted) a tube A for introducing into the mixer the solid and liq ingredients and a series of manifolds, M, serving for injection of air thru the pores in C... 

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 ... 

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