From Free-Energy Profiles to Activation Free Energies

TL;DR

This paper critically examines the use of free-energy profiles (FEPs) for estimating activation free energies in chemical reactions, deriving an exact expression that clarifies the limitations and dependencies on the choice of collective variables.

Contribution

The authors derive an exact formula for activation free energies that removes ambiguities caused by the choice of collective variables, improving accuracy over traditional FEP-based methods.

Findings

FEP-based approximation is valid only at low temperatures and with small effective mass CVs.

The activation free energy depends on reactant probability, transition state density, and thermal wavelength.

Choice of CV significantly affects estimated activation free energies, but not the overall free energy change.

Abstract

Given a chemical reaction going from reactant (R) to the product (P) on a potential energy surface (PES) and a collective variable (CV) that discriminates between R and P, one can define a free-energy profile (FEP) as the logarithm of the marginal Boltzmann distribution of the CV. The FEP is not a true free energy, however, it is common to treat the FEP as the free-energy analog of the minimum energy path on the PES and to take the activation free energy, $\Delta F^\ddagger_\mathrm{RP}$, as the difference between the maximum of the FEP at the transition state and the minimum at R. We show that this approximation can result in large errors. Since the FEP depends on the CV, it is therefore not unique, and different, discriminating CVs can yield different activation free energies for the same reaction. We derive an exact expression for the activation free energy that avoids this ambiguity with respect to the choice of CV. We find $\Delta F^\ddagger_\mathrm{RP}$ to be a combination of the probability of the system being in the reactant state, the probability density at the transition state surface, and the thermal de~Broglie wavelength associated with the transition from R to P. We then evaluate the activation free energies based on our formalism for simple analytic models and realistic chemical systems. The analytic models show that the widespread FEP-based approximation applies only at low temperatures for CVs for which the effective mass of the associated pseudo-particle is small. Most chemical reactions of practical interest involve polyatomic molecules with complex, high-dimensional PES that cannot be treated analytically and pose the added challenge of choosing a good CV, typically through heuristics. We study the influence of the choice of CV and find that, while the reaction free energy is largely unaffected, $\Delta F^\ddagger_\mathrm{RP}$ is quite sensitive.