Minimal energy path

Fig. 15. Minimal energy path for hydrogen motion in the A2A (3s) electronic state of C2H5. (From Zyubin et al.123)...

As a conclusion, the simple analytical form of the potential energy surface allows to calculate the minimal energy path, step by step from HS to the LS energy minimum. It is obvious that along the path the contributions of the different modes will change. At HS only JT active modes contribute. After the first step the symmetry is lowered and the other modes as mentioned will mix in. This allows getting very detailed picture on the interaction between the deformation of the electron distribution and the displacements of the nuclei. 

The transition state exists only in the vicinity of the saddle point in a small distance d of the minimal energy path between reactants and products. If is the average velocity of the transition state toward the products, defined as direction x, r is given by... 

Figure il. Minimal energy path of the ground 2 displacement of the (OH)3 plane from the... 

Fig. 5.7. Minimal energy path of attack upon the carbon atom of the formaldehyde molecule by the hydride-anion according to ab initio [93] (dashed line) and MINDO/3 [95] (solid line) calculations. The circles a, b, c, d, e, f denote the positions of the atoms H, C, O of the molecule being attacked and H", OH, with the distances between the nucleophile and the carbon atom being, respectively, 4.0, 3.0, 2.5, 1.5, and 1.12 A. In the initial state the formaldehyde molecule lies in the XOZ plane...

Fig. 5.10. Minimal energy path of attack upon a nitrogen atom of the HNO molecule by the fluoride ion and the trajectory of the hydride ion abstracted from the intermediate HNOF , calculated by the MINDO/3 method [163]. The points a, b, c denote the positions of the atoms H, N, O, F" with the distance between the nucleophile F and the nitrogen atom being, respectively, 3.0, 2.0 and 1.5 A (relative energies are 30.6 and 27.0kcal/mol). The points a-e denote the positions of the atoms F, N, O, H with the distance between the abstracted H " and the nitrogen atom equalling 2.0 and 3.0 A, respectively, (relative energies are 13.6 and 87.5 kcal/mol). In the initial state the HNO molecule lies in the XOY plane, while the FNO molecule lies in the plane VOX. The energy of the intermediate HNOF is taken as the zero point (point b, structure LIV)...

Figure 11.22 A simple unified model for free energy change of a chemical reaction in solution. Each parabola is a function of the solvent displacement coordinate rand the solute reaction coordinate q. The minimal energy path followed by the system, as shown, involves both coordinates, just as in the toy model of Section 11.2.2.1. Here we do not couple the two motions, although we could. To apply transition state theory we need to know where is the lowest energy barrier and what is its height.

This dissociation was verified by minimal energy path quantum calculations during the adsorption of hydrogen on metals [SOU 67]. 

Henkelman G, Uberuaga BP, Jonsson H. A climbing image nudged elastic band method for finding minimal energy paths and saddle points. J Chem Phys 2000 113 9901. 

---