Minimum energy path system

In the chapter on reaction rates, it was pointed out that the perfect description of a reaction would be a statistical average of all possible paths rather than just the minimum energy path. Furthermore, femtosecond spectroscopy experiments show that molecules vibrate in many dilferent directions until an energetically accessible reaction path is found. In order to examine these ideas computationally, the entire potential energy surface (PES) or an approximation to it must be computed. A PES is either a table of data or an analytic function, which gives the energy for any location of the nuclei comprising a chemical system. 

Fig. 20. A resonance state for the H + HD system at Ec = 1.2 eV at a total angular momentum of J = 25. In (a) the wavefunction is shown in the H + HD Jacobi coordinates for the collinear subspace. In (b) the wavefunction has been sliced perpendicular to the minimum energy path and is plotted in symmetric stretch and bend normal mode coordinates.

Fig. 16. Potential energy contours for the H + D2O system as a function of the OH and one OD bond length. In each panel, the energy has been minimized with respect to the remaining degrees-of-freedom in the vicinity of the minimum energy paths. In (a) the saddle point for the abstraction reaction, and in (b) the shallow < >, minimum for the exchange reaction are marked with X.

Figure 1. (Left) Potential energy surface for the LiNC/LiCN isomerizing system drawn as a contours plot. The minimum energy path connecting the two stable linear isomers, LiNC (0 = 180°) and LiCN (6 = 0), is shown as a dotted line. 

A very perceptive treatment of chemical reaction dynamics, called the reaction path Hamiltonian analysis, states that the reactive trajectory is determined as the minimum energy path, and small displacements from that path, on the potential-energy surface [64-71]. The usual analysis keeps the full dimensionality of the reacting system, albeit with a focus on motion along and orthogonal to the minimum energy path. It is also possible to define a reaction path in a reduced dimensionality representation. 

In reactions of mechanistic borderline, the reaction pathway may not follow the minimum energy path, but the reaction proceeds via unstable species on the PES. In other cases, the reacting system remains on the IRC but does not become trapped in the potential energy minimum. In some cases, intermediates are formed in reactions that should be concerted, whereas in other reactions a concerted TS gives an intermediate. Thus, the question of concerted versus stepwise appears too simple and the definition of concerted and stepwise reactions becomes unclear. In some reactions, the post-TS dynamics do not follow IRCs, and path bifurcation gives two types of products through a common TS. 

Figure 4.2 depicts snapshots of the evolving wavepacket for the two-dimensional model system that we discussed in Section 2.5 to illustrate vibrational excitation (see Figure 2.3). Because the PES is quite steep in the proximity of the ground-state equilibrium, the molecule dissociates at once. Immediately after the wavepacket is released it slides down the potential slope and disappears into the asymptotic channel for good. Since 4>/(0) starts significantly displaced from the minimum energy path of the PES, the evolving wavepacket oscillates along the r-direction, which in-...

The photodissociation of symmetric triatomic molecules of the type ABA is particularly interesting because they can break apart into two identical ways ABA — AB + A and ABA — A + BA. Figure 7.18(a) shows a typical PES as a function of the two equivalent bond distances. It represents qualitatively the system IHI which we will discuss in some detail below. We consider only the case of a collinear molecule as illustrated in Figure 2.1. The potential is symmetric with respect to the C -symmetry line 7 IH = i HI and has a comparatively low barrier at short distances. The minimum energy path smoothly connects the two product channels via the saddle point. A trajectory that starts somewhere in the inner region can exit in either of the two product channels. However, the branching ratio ctih+i/cti+hi obtained by averaging over many trajectories or from the quantum mechanical wavepacket must be exactly unity. 

The calculation of the thermal stability requires the estimation of transition rates between stable equilibrium states of the magnet. The calculation of transition rates needs a detailed characterization of the energy landscape along the most probable path which is taken by the system from its initial state to a final state. Henkelman and Jonsson [20] proposed the nudged elastic band method to calculate minimum energy paths. Starting from an... 

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