Repulsive manifold

Assume that is repulsive. Since Tv-i is an invariant surface, motion precisely on Xy-i can never fall away. However, speaking somewhat loosely for the moment, a slight push along q will cause motion initially on tv, to roll away from the barrier top. The set of motions that will roll away most slowly are motions that are asymptotic to the repulsive manifold and the surface formed by these motions constitutes the multidimensional version of a sep-aratrix. As motion asymptotic to Tn,. i falls away, it will generate a surface embedded in the full phase space whose geometry is the direct product of the sphere and the real line, x that is, a hypercylinder. The dimension of this hypercylinder is thus IN - As we will see, this (2N - 2)-dimen-... 

Figure 7. Potential energy diagram for HI, showing the two lowest ionization states (2n3//2 and 2 IT j, ) coupled to a neutral dissociative continuum (3Ao) at the three-photon (3

II. The molecular version of Hund s rule in order to minimize electron repulsions in pi-bonding, each added pair of electrons in the degenerate p manifold should be assigned... 

As will be elaborated later, to deal with the spectra of transition metal ions with more than one electron in the empty (or hole in the filled) d10 shell, it is necessary to take into account inter-electronic repulsion energies within the d manifold. On the basis of CFT, of course, the parameters... 

The coupling to the dissociative part of the Hamiltonian (represented by the repulsive diabatic potential curve in Figure 7.1) broadens the discrete energy levels with the widths depending on the coupling strength between the two manifolds. 

Other systems which have been simulated with these types of model are now manifold. King [321] has extended his work to hydrogen desorption from various tungsten planes. On the 110 plane, co was found to be repulsive to the extent of 6 kJ mole-1, but for the 100 and 111 planes the results could not be fitted exactly, possibly due to the presence of different binding energy sites. Lateral interaction effects have been inferred to explain the presence of two j3 states in the desorption of H2 from Pt lll [303]. Other examples of systems exhibiting these effects are CO/Ru 1010 [307] and Ru 001 [308], O2/W 100 [309], H2/Ni 100 [310] and Ni lll [306] and CO/Mo 100 [311]. 

As the unperturbed Hamiltonian, we choose the same HF Hamiltonian as was employed in the above EP development, and we use a basis set of real orthonormal spin-orbitals. We develop an approximation to the PP that yields the primary excitation energies and the corresponding transition moments (and the frequency-dependent polariziibility) consistent through second order in the residual electronic repulsion (Nielsen et ai, 1980). To determine the poles belonging to the principal excitation energies, the corresponding transition moments, and the frequency-dependent polarizability through second order, it proves sufficient to consider the truncated projection manifold... 

Armed as we are now with the KAM theorem, the Center Manifold theorem, and the Stable Manifold theorem, we can begin to visualize the phase space of reaction dynamics. Returning to our original system (see Uncoupled Reaction Dynamics in Two Degrees of Freedom ), we now realize that the periodic orbit that sews together the half-tori to make up the separatrix is a hyperbolic periodic orbit, and it is not a fixed point of reflection. From our previous visualization of uncoupled phase-space dynamics, we know that the separatrix is completely nontwisted. In the terminology of Poliak and Pechukas, the hyperbolic periodic orbit is a repulsive PODS. ... 

Although the De Leon-Berne Hamiltonian is apparently the only system for which these fairly complex surface plots have been made, the reactive islands Poincare map structure (which is a unique signature of the cylindrical geometry) has been observed in models of 3-phospholene as well as in a symmetric triple-well prototype. 24 ij has also been observed in the bi-molecular reactions H + H2 - H2 + H52,i24 anj fn (he unimolecular decomposition of the (He I2) cluster - and of HNSi. 24 These studies have shown that even in a strong-coupling limit, where the repulsive PODS wanders away from the barrier to some extent or even bifurcates into multiple P0DS, > - 124 cylindrical separatrix manifolds mediate the pre- and postreaction dynamics. 

The surface t v-1 is a particular example of what Wiggins has referred to as a hyperbolic manifold and what De Leon and Ling have termed a normally invariant hyperbolic manifoldd Hyperbolic manifolds are unstable and constitute the formal multidimensional generalization of unstable periodic orbits. Hyperbolic manifolds, like PODS, can be either repulsive or attractive. - If motion near a hyperbolic manifold falls away without recrossing it in configuration space, the hyperbolic manifold is said to be repulsive. On the other hand, it is often the case that motion near a hyperbolic manifold will cross it several times in configuration space as it falls away, and in this case it is said to be attractive. 

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