The Minimum Energy Path

The minimum energy path in mass-weighted Cartesian coordinates is the path through configuration space traced by a hypothetical trajectory initiated at the saddle point with all inertia effects removed. It is the path that molasses would follow flowing downhill. The MEP satisfies the differential equation. 

Euler integration of Eq. [8] requires only a calculation of the gradient at each step. 

The calculation of force constants at points along the MEP is often done separately from the determination of the path by numerical integration of Eq. [8], but these two problems can be profitably combined. Methods recently have been proposed ° i that efficiently use the available force constants to better follow the path. To understand these methods and the relationship between them, consider two different Taylor series expansions about a point on the MEP. The first is the familiar expansion of the energy in the mass-weighted Cartesian coordinates, 

Here gg, Fg, and Gg are, respectively, the first (gradient), second (force constants), and third energy derivatives evaluated at Xg. The square brackets indicate that the three-dimensional array of third derivatives is contracted with the vector of coordinate changes to yield a square matrix. If Xg is a stationary point, i.e., a minimum, then the usual theory of small vibrations applies the gradient term vanishes, and truncation after the second-order term leads to separable, harmonic normal modes of vibration. However, on the MEP, the gradient term generally is not zero. The second relevant expansion is the Taylor series representation of the path (of the solution to Eq. [8]) in the arc length parameter, s, about the same point, Xg  

These coefficients can also be interpreted. The first-order coefficient, Vg(0), is the reaction path curvature it is the vector that describes the change in direc- 

Given the definition of the geometry of the transition states in TST as the highest energy point in the minimum energy pathway from reactants to products, the formal definition of MEP is as follows. The MEP is, in one direction, the path of steepest descents from the transition state to reactants while, in the other direction, it is the path of steepest descents from transition state to products. For reasons which will not be discussed here, the formal definition of MEP includes the statement that the pathway is expressed in mass scaled Cartesian coordinates of the position of the atoms (introduced in Chapter 3, e.g. x is replaced by x = ). This simplifies 

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Figure B2.5.22. Potential V along the minimum energy path for the stereonuitation of hydrogen peroxide. Adapted from [103].

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

In order to define how the nuclei move as a reaction progresses from reactants to transition structure to products, one must choose a definition of how a reaction occurs. There are two such definitions in common use. One definition is the minimum energy path (MEP), which defines a reaction coordinate in which the absolute minimum amount of energy is necessary to reach each point on the coordinate. A second definition is a dynamical description of how molecules undergo intramolecular vibrational redistribution until the vibrational motion occurs in a direction that leads to a reaction. The MEP definition is an intuitive description of the reaction steps. The dynamical description more closely describes the true behavior molecules as seen with femtosecond spectroscopy. 

FIGURE 18.1 Illustration of how the steepest descent algorithm follows a path that oscillates around the minimum energy path. 

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. 

After 28 years the perepoxide quasi-intermediate was supported by a two-step no intermediate mechanism [71, 72]. The minimum energy path on the potential energy surface of the reaction between singlet molecular oxygen ( A and dg-teramethylethylene reaches a vaUey-ridge inflection point and then bifurcates leading to the two final products [73]. 

In most cases, the observables measured in the study of a chemical reaction are interpreted under the following (often valid) assumptions (1) each product channel observed corresponds to one path on the PES, (2) reactions follow the minimum energy path (MEP) to each product channel, and (3) the reactive flux passes over a single, well-defined transition state. In all of the reactions discussed in this chapter, at least one, and sometimes all of these assumptions, are invalid. 

One disadvantage of statistical approaches is that they rely on two of the assumptions stated in the introduction, namely, that reactions follow the minimum energy path to each product channel, and that the reactive flux passes through a transition state. Several examples in Section V violate one or both of these assumptions, and hence statistical methods generally cannot treat these instances of competing pathways [33]. 

G. OH + CH3F Avoiding the Minimum Energy Path The reaction... 

Figure 11. The minimum energy path of the OH + CH3F reaction, not including zero-point energy. The four labeled structures are (A), the central barrier TS (B), the nearly collinear backside well complex [HOCH3 F] (C) the transition of the F atom toward the OH moiety (D) the hydrogen-bonded [CH3OH F ] structure. Reprinted from [63] with permission from the American Association for the Advancement of Science.

The surface in Fig. 12 demonstrates that there is little coupling between the C—F translation coordinate and the bending coordinate of the complex. Stated another way, the time scale for intramolecular vibrational redistribution between these coordinates is slow compared to the time scale for breaking the C—F bond. These conclusions are not obvious upon examination of the minimum energy path shown in Fig. 11, and indeed such diagrams, while generally instructive, can lead to improper conclusions because they hide the multidimensional nature of the true PFS. A central assumption of statistical product distribution theories... 

The SQ method extracts resonance states for the J = 25 dynamics by using the centrifugally-shifted Hamiltonian. In Fig. 20, the SQ wavefunc-tion for a trapped state at Ec = 1.2 eV is shown. The wavefunction has been sliced perpendicular to the minimum energy path and is plotted in the symmetric stretch and bend normal mode coordinates. As anticipated, the wavefunction shows a combination of one quanta of symmetric stretch excitation and two quanta of bend excitation. The extracted state is barrier state (or quantum bottleneck state) and not a Feshbach resonance. 

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.

In some cases orbital symmetry rules can label the least-motion approach of two reacting fragments as forbidden. Semi-empirical MO calculations, such as EHT ones, can then be used to pick out the minimum-energy path, as outlined in the foregoing section. Another example is given by the reactions 17) ... 

In Fig. 18 the transition coordinates (Section 6.6.) of the three calculated transition states are shown for illustration of the above discussion. These eigenvectors give a quantitative picture of the atomic motions (towards the minima linked by the transition states) when crossing the respective barriers along the minimum energy path. As expected the transition coordinates of the Cs- and C2 -conformations are symmetric with respect to the mirror plane and the twofold axis, respectively, indicating the conservation of these symmetry elements during the associated transitions. (The transition coordinate of the Cj-form... 

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