Saddle point features

FigurelO. LEEM images of 2-D gratings following annealing at 980C and at 1060C. The980C sample shown in a) is symmetrical with respect to maxima and minima and the saddle point features( bow-tie shapes) are approximately 4-fold symmetric hypocycloids. In the 1060C sample the minima have extensive (001) facets and the saddle point features are elongated in the directions between maxima note also that a particular reconstructed domain is preferred at the saddle points[31,34].

The Saddle Point Features of the 2-D Gratings For an ideal 2-D sinewave the saddle point features should appear to have 4-fold symmetry when viewed in LEEM images. From the sketch of figure 13 it can be seen that the hypocycloid shaped terrace at the saddle point has the same type of monoatomic step on all four sides due to the difference in the free energies of the two steps, Sa and Sb, on Si(OOl) there should be a strong preference for Sa steps and hence each maximum would prefer to be flanked by two white domains and two "black ones as is the case in figure 10. (A similar conclusion follows if the the saddle point terrace is surrounded by two Sa steps and two double steps of Db type[31]). 

At a temperature of 1060C the saddle point features assume an elongated bow-tie shape with the short dimension along a line coimecting adjacent minima. We believe this to be the result of the evaporation of islands from above the initial saddle point so that along the line between maxima there are few steps between saddle points compared with the number along the line between minima interstep repulsion would then produce the observed elongation. 

A study of Table 1.1 reveals interesting features as to the mobility of the adsorbed atoms. Thus, for an argon atom on the (100) face, the easiest path from one preferred site S to the next is over the saddle point P, so that the energy barrier which must be surmounted is (1251 — 855) or 396 X 10 J/molecule. Since the mean thermal energy kT at 78 K is only 108 J/molecule, the argon molecule will have severely limited mobility at this temperature and will spend nearly all of its time in the close vicinity of site S its adsorption will be localized. On the other hand, for helium on the... 

Although intrinsic reaction coordinates like minima, maxima, and saddle points comprise geometrical or mathematical features of energy surfaces, considerable care must be exercised not to attribute chemical or physical significance to them. Real molecules have more than infinitesimal kinetic energy, and will not follow the intrinsic reaction path. Nevertheless, the intrinsic reaction coordinate provides a convenient description of the progress of a reaction, and also plays a central role in the calculation of reaction rates by variational state theory and reaction path Hamiltonians. 

Potential energy surfaces show many fascinating features, of which the most important for chemists is a saddle point. At any stationary point, both df/dx and df /Sy are zero. For functions of two variables f(x, y) such as that above, elementary calculus texts rarely go beyond the simple observation that if the quantity... 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

With the identification of the TS trajectory, we have taken the crucial step that enables us to carry over the constructions of the geometric TST into time-dependent settings. We now have at our disposal an invariant object that is analogous to the fixed point in an autonomous system in that it never leaves the barrier region. However, although this dynamical boundedness is characteristic of the saddle point and the NHIMs, what makes them important for TST are the invariant manifolds that are attached to them. It remains to be shown that the TS trajectory can take over their role in this respect. In doing so, we follow the two main steps of time-independent TST first describe the dynamics in the linear approximation, then verify that important features remain qualitatively intact in the full nonlinear system. 

Figure 3 displays the molecular partition of the fragments for the three states previously discussed in the quantum mechanical section, at d= 6 k. Figure 3 A and 3 B respectively display the and 2 covalent states, and Figure 3 C shows the ionic 82 Charge Transfer state. It is worth examining the striking features of the molecular partitions in each case. In the A1 molecular partition, the disynaptic basin V(Cli, CI2), indicated by an arrow, corresponds to the Cl—Cl bond [22]. Two basins are found around Li, one corresponding to its core C(Li), and the second one, V(Li), to its valence odd electron (L shell). The 82 covalent state is characterized by two monosynaptic basins, Vi(Li) and V2(Li), located on both sides of the C(Li) basin in the molecular plane. They correspond to the half-filled 2p AO of Li. As when dealing with the previous state, the Cl atoms are bonded through a disynaptic basin, still noted V(Cli, CI2). In the ionic state, the Cl atoms are linked by a (3, -I) saddle point, or.

One common feature of processes 3 to 13 is that the final state is higher in energy than the initial state. The saddle point is typically late, i.e. close to the final state. 

If we examine a potential energy surface there are several features which play an important role in the interpretation of kinetic processes. These are minima (stable configurations of all the atoms), valleys (separate stable groups of atoms which we identify as reactants and products) and saddle points (transition states). However, before we give a more formal definition of these features we have to consider the coordinate system that is used. 

This leads to die third and final main topic, the use of Heff model u-scaling predictions to detect the changes in the resonance structure that occur near chemically important topographic features of a potential surface. Such features include an isomerization saddle point or a sharp bend in the minimum-energy isomerization path. The key feature of this matrix model is... 

The fractionation patterns exhibited % successive members of a progression of polyads (along 02, CC stretch, or along v4, trans-bend) provide a surveyor s map of IVR. One can look at the 1VR trends and see whether the multiresonance model expressed in the H nres (1 polyads provides a qualitative or quantitative representation of the fractionation patterns. The dynamics of even a four-atom molecule is so complicated that, unless one knows what to look for, one can neither identify nor explain trends in the dynamics versus V2 or u4 or Evib- Moreover, by defining the pattern of the IVR and how this pattern should scale with V2, v4, or EVib, the H res / polyad model may make it possible to detect a disruption of the pattern. Such disruptions could be due to a change in the resonance structure of the exact H near some chemically interesting topographic feature of the V(Q), such as an isomerization saddle point. 

Stimulated emission pumping spectra of HCP X1 have been recorded via the A lA" [12] and C lA states [5]. These 0.05-cm-1 resolution spectra sample eigenstates rather than feature states, extending to Vib - 25,315 cm-1, above the X state zero-point level, about 1300 cm-1 above the ab initio predicted linear HPC saddle point [5, 13]. 

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