Intrinsic Reaction Coordinate Methods

The optimization methods described in Sections 12.2-12.4 concentrate on locating stationary points on an energy surface. The important points for discussing chemical reactions are minima, corresponding to reactant(s) and product(s), and saddle points. 

The simplest method for integrating eq. (12.33) is the Eukr method. A series of steps are taken in the opposite direction of the normalized gradient at the current geometry x . 

This corresponds to a steepest descent minimization with a fixed step size As. As discussed in Section 12.2.1, such an approach tends to oscillate around the true path and consequently requires a small step size to follow the IRC accurately. 

The fourth-order Runge-Kutta (RK4) method generates four intermediate gradients, 

Illustration of the Gonzales-Schlegel constrained optimization method for following 

A more advanced method is the Runge-Kutte (RK). The idea here is to generate some intermediate steps which allow a better and more stable estimate of the next geometry -Tor a given step size. The second-order Runge-Kutte (RK2) method first calculates the 

Although it is clear that RK4 is more stable and accurate than the Euler method for a given step size, this does not necessarily mean that it is the most efficient method. Since the RK4 method requires four gradient calculations for each step, the simple Euler can employ a step size four times as small for the same computational cost. Similarly, 

Schlick, Rev. Comput. Chem., 3(1992), 1 H. B. Schlegel, Adv. Chem. Phys., 67 (1987), 249 H. B. Schlegel, Modern Electronic Structure Theory, Parti, ed. D. Yarkony, World Scientific, 1995, p. 459-500 R. Fletcher Practical Methods of Optimizations, Wiley, 1980 M. L. McKee and M. Page, Rev. Comput. Chem., 4 (1993), 35. 

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Deng, L., Ziegler, T., 1997, Theoretical Study of the Oxidation of Alcohols to Aldehyde by d° Transition-Metal-Oxo Complexes Combined Approach Based on Density Functional Theory and die Intrinsic Reaction Coordinate Method , Organometallics, 16, 716. 

An alternative and more costly approach is to actually follow the reaction from transition state to both the reactants and (independently) the products. In practice, this involves optimization subject to a fixed position along the reaction coordinate. A number of schemes for doing this have been proposed, and these are collectively termed Intrinsic Reaction Coordinate methods. Note, that no scheme is unique while the reactants, products and transition state are well defined points on the overall potential energy surface, there are an infinite number of pathways linking them together, just like there are an infinite number of pathways leading over a mountain pass. 

Deng L, Exploring Potential Energy Surfaces using Density Functional Theory and Intrinsic Reaction Coordinate Methods, PhD Thesis (University of Calgary, Calgary, 1996)... 

Crehuet R (2005) The reaction path intrinsic reaction coordinate method and the Hamilton Jacobi theory. J Chem Phys 122 234105... 

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