Potential energy surfaces phase space

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

Figure 4 Two arbitrary potential energy surfaces in a two-dimensional coordinate space. All units are arbitrary. Panel A shows two minima connected by a path in phase space requiring correlated change in both degrees of freedom (labeled Path a). As is indicated, paths involving sequential change of the degrees of freedom encounter a large enthalpic barrier (labeled Path b). Panel B shows two minima separated by a barrier. No path with a small enthalpic barrier is available, and correlated, stepwise evolution of the system is not sufficient for barrier crossing.

Other theoretical studies discussed above include investigations of the potential energy profiles of 18 gas-phase identity S 2 reactions of methyl substrates using G2 quantum-chemical calculations," the transition structures, and secondary a-deuterium and solvent KIEs for the S 2 reaction between microsolvated fluoride ion and methyl halides,66 the S 2 reaction between ethylene oxide and guanine,37 the complexes formed between BF3 and MeOH, HOAc, dimethyl ether, diethyl ether, and ethylene oxide,38 the testing of a new nucleophilicity scale,98 the potential energy surfaces for the Sn2 reactions at carbon, silicon, and phosphorus,74 and a natural bond orbital-based CI/MP through-space/bond interaction analysis of the S 2 reaction between allyl bromide and ammonia.17... 

The basic assumption in statistical theories is that the initially prepared state, in an indirect (true or apparent) unimolecular reaction A (E) —> products, prior to reaction has relaxed (via IVR) such that any distribution of the energy E over the internal degrees of freedom occurs with the same probability. This is illustrated in Fig. 7.3.1, where we have shown a constant energy surface in the phase space of a molecule. Note that the assumption is equivalent to the basic equal a priori probabilities postulate of statistical mechanics, for a microcanonical ensemble where every state within a narrow energy range is populated with the same probability. This uniform population of states describes the system regardless of where it is on the potential energy surface associated with the reaction. 

Figure 3 A contour plot of the potential energy surface described in the text, as a function of z and Zu the distance of the incident and target atoms, respectively, above the surface plane. The distance between the atoms parallel to the surface is held fixed at the gas-phase H-H bond length. The marked contours are in eV, with a contour spacing of 0.2 eV. Taken from Ref. [83].

The analysis of the stability of isolated stationary points is different in the phase-space treatment from that in the coordinate space treatment. In the coordinate space treatment the slope of the potential energy surface gives the forces exerted on the system. Stationary points occur at extrema of the potential energy. Their stability is determined by the eigenvalues of the matrix of second derivatives evaluated at the extremum. Assuming the system has n DOFs, it will possess n... 

In the phase-space treatment the situation is very similar. However, rather than study the morphology of the potential energy surface, we must focus on the total energy surface. The geometry of this surface, which is defined on phase space instead of coordinate space, can also be characterized by its stationary points and their stability. In this treatment, the rank-one saddles play a fundamental result. They are, in essence, the traffic barriers in phase space. For example, if two states approach such a point and one passes on one side and the other passes on the other side, then one will be reactive and the other nonreactive. Once the stationary points are identified, then the boundaries between the reactive and nonreactive states can be constructed and the dynamical structure of phase space has been determined. As in the case of potential energy surfaces, saddles with rank greater than one occur, especially in systems with high symmetry between outcomes, as in the dissociation of ozone. 

As we noted above, the kinetic energy is positive definite. Furthermore, it is quadratic in the momenta. As a consequence, we can reduce the search for points of stationary flow in phase space to one of finding the stationary points of the potential energy surface. To see how this comes about, consider the Hamilton s equations for the three velocities... 

Transport. We need now to construct the NHIM, its stable/unstable manifolds, and the center manifold. Let P be the main relative equilibrium point. The first task is to find the short periodic orbits lying above P in energy. These p.o. are unstable. We did so by exploring phase space at energies 4, 10, and 14 cm above E (1 atomic unit = 2.194746 x 10 cm ). It is not possible to go much higher in E, since the center manifold disappears shortly above E + 14cm , because of the structure of the potential energy surface. 

The determination of the microcanonical rate coefficient k E) is the subject of active research. A number of techniques have been proposed, and include RRKM theory (discussed in more detail in Section 2.4.4) and the derivatives of this such as Flexible Transition State theory. Phase Space Theory and the Statistical Adiabatic Channel Model. All of these techniques require a detailed knowledge of the potential energy surface (PES) on which the reaction takes place, which for most reactions is not known. As a consequence much effort has been devoted to more approximate techniques which depend only on specific PES features such as reaction threshold energies. These techniques often have a number of parameters whose values are determined by calibration with experimental data. Thus the analysis of the experimental data then becomes an exercise in the optimization of these parameters so as to reproduce the experimental data as closely as possible. One such technique is based on Inverse Laplace Transforms (ILT). 

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