Automatic integration of the reaction path using diagonally implicit Runge-Kutta methods.

Area of Science:

• Numerical analysis
• Computational chemistry
• Chemical kinetics

Background:

• Steepest descent reaction path following is crucial for understanding chemical reactions.
• Existing numerical methods may lack efficiency or stability for complex systems.
• Diagonally implicit Runge-Kutta methods offer a promising avenue for improved integration.

Purpose of the Study:

• To establish the diagonally implicit Runge-Kutta framework as a general approach for stable and efficient steepest descent reaction path integrators.
• To develop and review embedded and non-embedded methods for adaptive step-size control without additional computational cost.
• To evaluate the performance of these integrators for chemical systems.

Main Methods:

• Review of existing embedded and non-embedded diagonally implicit Runge-Kutta methods of orders two, three, and four.
• Development of new embedded methods specifically for reaction path following.
• Application and comparison of integrators to the Muller-Brown potential and two gas-phase chemical reactions.

Main Results:

• The diagonally implicit Runge-Kutta framework provides a general and stable approach for reaction path integration.
• Embedded methods enable tolerance-driven, adaptive step-size control with no extra computational expense.
• Tested integrators efficiently and reliably followed reaction paths within desired error tolerances for all systems.

Conclusions:

• Diagonally implicit Runge-Kutta methods offer a robust and efficient solution for automated reaction path following.
• The developed framework allows for accurate determination of reaction paths by specifying only desired accuracy and the transition state.
• This approach simplifies the computational study of chemical reaction dynamics.