Valley ridge inflections

B3.5.7.3 BIFURCATION OF THE REACTION PATH AND VALLEY-RIDGE INFLECTION POINTS... 

The geometrical properties of the PES in the vicinity of a transition state mean that the steepest descent path down from the transition state (also generally calculated in mass-weighted coordinates, and called the intrinsic reaction coordinate or IRC) will usually lead only to a single reactant in one direction and a single product (or intermediate) in the other. However, a transition state can sometimes be shared by more than one reactant and/or product. One of these cases arises when the PES possesses a so-called valley-ridge inflection point (VRI). °... 

Figure 1.8 A PES with a bifurcation that takes place after TSl at the valley ridge inflection (VRI). The reaction path splits at the VRI leading in one direction to Product 1 and in the other direction to Product 2.

Let s consider what happens after the molecule passes over TSl. The molecule continues to follow the reaction path downhill until it reaches the valley ridge inflection (VRI) point. At this point, the gradient in one direction perpendicular to the reaction path becomes zero, and after this point, the PES actually falls downhill faster in the direction off of the reaction path than continuing on the reaction path to TS2. It is important to note that the VRI is not a critical point the gradients are not all zero at a VRI point. So at the VRI, the pathway diverges off of the ridge that leads to TS2, in one direction toward Product 1 and in a second direction toward Product 2. What is so unusual about this type of surface is that the reaction paths that lead from the reactant over TSl then proceed onward to two products (Product 1 and Product 2), without crossing any more transition states. In other words, this surface has a TS that leads to two products ... 

Figure 4.19 Bifurcating PES for the dimerization of cyclopentadiene. Heavy black lines are two trajectories that lead from the Diels-Alder bispericyclic TS 77 to the valley-ridge inflection (VRI) point and then diverge, one into the well of product 76a and one into the well of product 76b.

The general shape of the PES for this reaction is given in Figure 8.17. Note that the MEP proceeds from TS 63 to 62, and the MEP does not bifurcate. There is, however, a ridge that some trajectories may cross, allowing for trajectories to go from 63 to 64. This behavior is similar to that of a valley ridge inflection point, but there is no VRl on the MEP in this case. 

Sheppard, A.N., Acevedo, O. Multidimensional exploration of valley-ridge inflection points on potential energy surfaces. J. Am. Chem. Soc. 2009,131, 2530 40. 

VAEs, See Vertical attachment energies (VAEs) Valley-ridge inflection (VRI) points, 39... 

A potential energy surface with a valley-ridge inflection. 

In Figure 4, we give a rough impression of how such a non-physical approach can result in an artificial valley-ridge inflection. In Figure 4 top row, an IRC is the... 

If we set /c(x,y) = 0, we obtain a flank line starting in the valley-ridge-inflection point (VRI) at (0,0) where, per definition, the curvature of the contour line is zero. Its equation is... 

Hirsch, M. Quapp, W. Heidrich, D. The set of valley-ridge inflection points on the potential energy surface of water. Phys. Chem. Chem. Phys. 1999, 1, 5291-5299. 

Figure 1 A model potential energy surface illustrating minima, transition structures, second-order saddle points, a valley-ridge inflection point, and reaction paths (Reproduced from Ref, 2a, with permission from World Scientific Publishing)...

CLQA = corrected local quadratic approximation DDRP = dynamically defined reaction path DRP = dynamic reaction path ES = Euler stabilization method GS = Gonzalez and Schlegel method IMK = Ishida-Morokuma-Kormomicld method LQA = local quadratic approximation MB = Miillar-Brown method MEP = minimum energy path ODE = ordinary differential equations SDRP = steepest descent reaction path VRl = valley-ridge inflection. 

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