^{1,a)}and Michael J. Frisch

^{1}

### Abstract

Exploring potential energy surfaces of large molecular systems can be quite challenging due to the increased number of nuclear degrees of freedom. Many techniques that are well-suited for small and moderate size systems require diagonalization of the energy second-derivative matrix. Since the cost of this step scales as (where *N* _{ atoms } is the number of atomic centers), such methods quickly become infeasible and are eventually rendered cost prohibitive. In this work, the recently developed Euler-based predictor–corrector reaction path integration method [H. P. Hratchian, M. J. Frisch, and H. B. Schlegel, J. Chem. Phys.133, 224101 (2010)]10.1063/1.3514202 is enhanced and proposed as a useful alternative to conventional reaction path following schemes in studies on very large systems. Because this integrator does not require Hessian diagonalization, the bottleneck afflicting other approaches is completely avoided. The effectiveness of the integrator in large system studies is demonstrated with an enzyme-catalyzed reaction employing an ONIOM (QM:MM) model chemistry and involving 5368 atomic centers.

Professor H. B. Schlegel (Wayne State University) is thanked for helpful discussions and encouragement. Dr. M. Caricato and Dr. F. Clemente (Gaussian, Inc.) are also thanked for helpful comments. Dr. M. Lundberg (Stanford University) and Professor K. Morokuma (Kyoto University) are acknowledged for providing initial structures for the IPNS TS.

I. INTRODUCTION

II. METHODS

III. NUMERICAL TESTS

IV. CONCLUSIONS

### Key Topics

- Polynomials
- 12.0
- Chemical reactions
- 9.0
- Enzyme kinetics
- 7.0
- High performance computing
- 6.0
- Nuclear reaction models
- 6.0

## Figures

Reaction scheme of the β-lactam formation step in iso-penicillin synthesis by IPNS. For clarity, only the reactive core of the active site is shown. The atomic centers involved in the β-lactam ring closure are shown in light gray (red online).

Reaction scheme of the β-lactam formation step in iso-penicillin synthesis by IPNS. For clarity, only the reactive core of the active site is shown. The atomic centers involved in the β-lactam ring closure are shown in light gray (red online).

Optimized TS geometry. Atoms included in the ONIOM QM layer are shown in ball and stick form, while atoms included in the MM layer are shown in wire-frame form. (a) Full view of the 5368 atom system. (b) Close-up view of the active site. Atomic centers displayed with a red halo correspond to those shown in light gray (red online) in Fig. 1.

Optimized TS geometry. Atoms included in the ONIOM QM layer are shown in ball and stick form, while atoms included in the MM layer are shown in wire-frame form. (a) Full view of the 5368 atom system. (b) Close-up view of the active site. Atomic centers displayed with a red halo correspond to those shown in light gray (red online) in Fig. 1.

Energy profile of the reaction path for the β-lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

Energy profile of the reaction path for the β-lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

C–N bond distance as a function of the reaction coordinate for the β–lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

C–N bond distance as a function of the reaction coordinate for the β–lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

Fe–S (a) and S–C (b) bond coordinates as functions of the reaction coordinate for the β–lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

Fe–S (a) and S–C (b) bond coordinates as functions of the reaction coordinate for the β–lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

S–C–C angle as a function of the reaction coordinate for the β–lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

S–C–C angle as a function of the reaction coordinate for the β–lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

C–C–C–O dihedral angle (all atoms shown in light gray (red online) in Fig. 1) as a function of the reaction coordinate for the β–lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

C–C–C–O dihedral angle (all atoms shown in light gray (red online) in Fig. 1) as a function of the reaction coordinate for the β–lactam formation reaction using EulerPC integration with a step size of 0.1 amu^{1/2} bohr using Hessian updating at all points (solid curve). Results of EulerPC integration with a step size of 0.2 amu^{1/2} bohr using analytic Hessian re-calculation every 10 points (□), analytic Hessian re-calculation every 20 points (◯), and all updated Hessians (△) are also shown.

## Tables

Performance of EulerPC IRC integration for the β–lactam formation reaction.

Performance of EulerPC IRC integration for the β–lactam formation reaction.

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