Diffusive barrier crossing rates from variationally determined eigenvalues

Area of Science:

• Chemical Kinetics
• Statistical Mechanics
• Theoretical Chemistry

Background:

• Kramers' procedure calculates activated process rates using dividing surfaces and steady-state flux.
• An alternative method estimates the rate constant from the first non-zero eigenvalue of the dynamics operator.

Purpose of the Study:

• Establish the relationship between Kramers' procedure and eigenvalue-based methods for diffusive dynamics.
• Derive Kramers' flux-over-population expression from a variational eigenvalue approach.

Main Methods:

• Utilized a variational principle for the eigenvalue of interest.
• Employed a trial function with two adjustable dividing surfaces.
• Leveraged the modern theory of activated rate processes, incorporating the committor probability.

Main Results:

• Demonstrated that Kramers' flux-over-population rate constant can be obtained from the variationally determined eigenvalue for high barriers.
• Showcased that the upper bound for the eigenvalue can be expressed using mean first-passage times and mean transition-path times.

Conclusions:

• The study provides a theoretical link between two prominent methods for calculating activated process rates.
• Highlights the utility of variational principles and the committor probability in understanding rate processes.