*ab initio*calculations for the determination of transition states

^{1}, Alexis T. Bell

^{2,a)}and Martin Head-Gordon

^{3,a)}

### 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 Müller–Brown potential energy surface, alanine dipeptideisomerization, H abstraction in methanol oxidation, and C–H bond activation in oxidative carbonylation of toluene to -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., , 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.

I. INTRODUCTION

II. THEORY

A. Coordinate system

B. Conjugate gradient

C. Potential energy surfaceinterpolation

D. Modified-GSM algorithm

E. Hybrid low-level/high-level strategy

III. RESULTS AND DISCUSSION

A. Müller–Brown potential energy surface

B. Alanine dipeptideisomerization

C. H-abstraction in isolated vanadate sites supported on silica

D. C–H bond activation in the oxidative carbonylation of toluene to -toluic acid

IV. CONCLUSIONS

### Key Topics

- Interpolation
- 49.0
- String theory
- 24.0
- Potential energy surfaces
- 14.0
- Basis sets
- 10.0
- Eigenvalues
- 10.0

## Figures

A flow sheet for the evolution step of the modified GSM. The orthogonal force , refers to the negative of the component of the gradient that is orthogonal to the reaction path. The criterion for using the interpolation scheme is if there are points within a confidence length .

A flow sheet for the evolution step of the modified GSM. The orthogonal force , refers to the negative of the component of the gradient that is orthogonal to the reaction path. The criterion for using the interpolation scheme is if there are points within a confidence length .

A flow sheet for a TS-finding strategy that uses a combined low level/high level of theory approach. A TS search using a high level of theory can be initiated directly from the converged low-level string. Otherwise, if the topology of the PES is correct, the low-level string can be further refined at a high level of theory.

A flow sheet for a TS-finding strategy that uses a combined low level/high level of theory approach. A TS search using a high level of theory can be initiated directly from the converged low-level string. Otherwise, if the topology of the PES is correct, the low-level string can be further refined at a high level of theory.

Contour plot of the Müller–Brown potential energy surface showing the three minima (A, B, C) and two saddle points, or TSs (TS1, TS2).

Contour plot of the Müller–Brown potential energy surface showing the three minima (A, B, C) and two saddle points, or TSs (TS1, TS2).

Error in the estimate of the highest saddle point from the true saddle point (TS1) on the MB PES. Results are shown at 18 nodes for the GSM and modified GSM. The open circles represent points where an interpolation step was performed in the modified GSM.

Error in the estimate of the highest saddle point from the true saddle point (TS1) on the MB PES. Results are shown at 18 nodes for the GSM and modified GSM. The open circles represent points where an interpolation step was performed in the modified GSM.

Final energy profile of the MEP for the MB PES using the GSM and the modified GSM for 18 nodes. The points in each curve are joined with a spline.

Final energy profile of the MEP for the MB PES using the GSM and the modified GSM for 18 nodes. The points in each curve are joined with a spline.

Alanine dipeptide isomerization from species (reactant) to (product).

Alanine dipeptide isomerization from species (reactant) to (product).

Error in the estimate of the highest saddle point from the true saddle point for alanine dipeptide isomerization. Results are shown at 11 nodes for the GSM and modified-GSM. The open circles represent points where an interpolation step was performed in the modified GSM.

Error in the estimate of the highest saddle point from the true saddle point for alanine dipeptide isomerization. Results are shown at 11 nodes for the GSM and modified-GSM. The open circles represent points where an interpolation step was performed in the modified GSM.

Final energy profile of the MEP for alanine dipeptide isomerization using the GSM and the modified GSM for 11 nodes. The reaction path shown for the modified GSM is after single-point QM calculations have been performed on interpolated points that exist in the final string. The points in each curve are joined with a spline.

Final energy profile of the MEP for alanine dipeptide isomerization using the GSM and the modified GSM for 11 nodes. The reaction path shown for the modified GSM is after single-point QM calculations have been performed on interpolated points that exist in the final string. The points in each curve are joined with a spline.

H abstraction in methanol oxidation on .

H abstraction in methanol oxidation on .

Error in the estimate of the highest saddle point from the true saddle point for H abstraction in methanol oxidation on . Results are shown at 11 nodes for the GSM and modified GSM. The open circles represent points where an interpolation step was performed in the modified GSM.

Error in the estimate of the highest saddle point from the true saddle point for H abstraction in methanol oxidation on . Results are shown at 11 nodes for the GSM and modified GSM. The open circles represent points where an interpolation step was performed in the modified GSM.

Final energy profile of the MEP for H abstraction in methanol oxidation on using the GSM and the modified GSM for 11 nodes. The reaction path shown for the modified GSM is after single-point QM calculations have been performed on interpolated points that exist in the final string.

Final energy profile of the MEP for H abstraction in methanol oxidation on using the GSM and the modified GSM for 11 nodes. The reaction path shown for the modified GSM is after single-point QM calculations have been performed on interpolated points that exist in the final string.

C–H bond activation in toluene on a Rh complex coordinated with two CO species and three TFA solvent ligands.

C–H bond activation in toluene on a Rh complex coordinated with two CO species and three TFA solvent ligands.

Error in the estimate of the highest saddle point from the true saddle point for C–H bond activation in toluene over Rh/TFA complex. The result shown is at 11 nodes for the modified GSM. The GSM is not shown because it was unable to converge. The open circles represent points where an interpolation step was performed in the modified GSM.

Error in the estimate of the highest saddle point from the true saddle point for C–H bond activation in toluene over Rh/TFA complex. The result shown is at 11 nodes for the modified GSM. The GSM is not shown because it was unable to converge. The open circles represent points where an interpolation step was performed in the modified GSM.

Final energy profile of the MEP for C–H bond activation in toluene over Rh/TFA complex using the modified GSM for 11 nodes. The modified GSM was first used using HF/STO-3G, and then refined using B3LYP/6-31G/LANL2DZ. The points in each curve are joined with a spline.

Final energy profile of the MEP for C–H bond activation in toluene over Rh/TFA complex using the modified GSM for 11 nodes. The modified GSM was first used using HF/STO-3G, and then refined using B3LYP/6-31G/LANL2DZ. The points in each curve are joined with a spline.

## Tables

Comparison of TS geometries from final string calculations for alanine dipeptide isomerization. The energies are relative to the reactant, .

Comparison of TS geometries from final string calculations for alanine dipeptide isomerization. The energies are relative to the reactant, .

Time required to determine the TS for alanine dipeptide isomerization.

Time required to determine the TS for alanine dipeptide isomerization.

Comparison of TS geometries from final string calculations for H abstraction in methanol oxidation on . The strings were grown using the functional/basis sets listed. All TS estimates were refined at .

Comparison of TS geometries from final string calculations for H abstraction in methanol oxidation on . The strings were grown using the functional/basis sets listed. All TS estimates were refined at .

Time required to determine the TS for H transfer in .

Time required to determine the TS for H transfer in .

Comparison of TS geometry using different functionals/basis sets for C–H bond activation in toluene on .

Comparison of TS geometry using different functionals/basis sets for C–H bond activation in toluene on .

Time required to determine the TS for C–H activation in toluene over .

Time required to determine the TS for C–H activation in toluene over .

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