Development and application of a hybrid method involving interpolation and ab initio calculations for the determination of transition states

Abstract

Transition state search algorithms, such as the nudged elastic band can fail, if a good initial guess of the transition state structure cannot be provided. The growing string method (GSM) [J. Chem. Phys. 120, 7877 (2004)] eliminates the need for an initial guess of the transition state. While this method only requires knowledge of the reactant and product geometries, it is computationally intensive. To alleviate the bottlenecks in the GSM, several modifications were implemented: Cartesian coordinates were replaced by internal coordinates, the steepest descent method for minimization of orthogonal forces to locate the reaction path was replaced by the conjugate gradient method, and an interpolation scheme was used to estimate the energy and gradient, thereby reducing the calls to the quantum mechanical (QM) code. These modifications were tested to measure the reduction in computational time for four cases of increasing complexity: the Muller-Brown potential energy surface, alanine dipeptide isomerization, H abstraction in methanol oxidation, and C-H bond activation in oxidative carbonylation of toluene to p-toluic acid. These examples show that the modified GSM can achieve two- to threefold speedups (measured in terms of the reduction in actual QM gradients computed) over the original version of the method without compromising accuracy of the geometry and energy of the final transition state. Additional savings in computational effort can be achieved by carrying out the initial search for the minimum energy pathway (MEP) using a lower level of theory (e.g., HF/STO-3G) and then refining the MEP using density functional theory at the B3LYP level with larger basis sets (e.g., 6-31G( *), LANL2DZ). Thus, a general strategy for determining transition state structures is to initiate the modified GSM using a low level of theory with minimal basis sets and then refining the calculation at a higher level of theory with larger basis sets.

Geometry optimization

“…Bounds on the transition structure can be found by approaching it from both the reactant and product side, providing progressively tighter limits 142,154,155. This is closely related to the GSM131–136 (see below). Given two points on opposite sides of the ridge separating reactants from products, the ridge method156,157 can also be used to optimize the transition structure.…”

“…This is known variously as line‐then‐plane,126 reduced gradient following,127–129 and Newton trajectories 130–132. Growing string methods (GSM)131–136 are coordinate driving or reduced gradient following/Newton trajectory methods that start from both the reactant and product side. Because GSMs are double‐ended methods closely related to chain‐of‐states methods like NEB method and SM; they are discussed in the next section.…”

“…The GSM133–136 reduces the total effort by generating the points one at a time starting from the reactants and products. This is similar to the line‐then‐plane method126 and reduced gradient following/Newton trajectories131,132 from both ends of the path.…”

“…If the ends of the growing string are too far apart, a NEB method or SM can be used to optimize of the entire path. To reduce the cost, a low level of theory can be used to grow the string and a higher level of theory can be used to refine the string or to optimize the transition structure 134,135…”

Improved constrained optimization method for reaction-path determination in the generalized hybrid orbital quantum mechanical/molecular mechanical calculations

“…Since then, several variants of the string method have been proposed. [15][16][17][18][19][20][21][22][23][24][25] NEB and string methods work equally well if a suitable optimizer is used. 26,27 There are also several alternatives for calculating RPs, namely, conjugate peak refinement, 28 combined approach, 29 geometric minimum action, 30 holonomic constraints approach, 31 swarm walk approximation, 32 and constrained broyden dimer method.…”