Path integral evaluation of equilibrium isotope effects

Abstract

A general and rigorous methodology to compute the quantum equilibrium isotope effect is described. Unlike standard approaches, ours does not assume separability of rotational and vibrational motions and does not make the harmonic approximation for vibrations or rigid rotor approximation for the rotations. In particular, zero point energy and anharmonicity effects are described correctly quantum mechanically. The approach is based on the thermodynamic integration with respect to the mass of isotopes and on the Feynman path integral representation of the partition function. An efficient estimator for the derivative of free energy is used whose statistical error is independent of the number of imaginary time slices in the path integral, speeding up calculations by a factor of 60 at 500 K. We describe the implementation of the methodology in the molecular dynamics package Amber 10. The method is tested on three [1,5] sigmatropic hydrogen shift reactions. Because of the computational expense, we use ab initio potentials to evaluate the equilibrium isotope effects within the harmonic approximation, and then the path integral method together with semiempirical potentials to evaluate the anharmonicity corrections. Our calculations show that the anharmonicity effects amount up to 30% of the symmetry reduced reaction free energy. The numerical results are compared with recent experiments of Doering and coworkers, confirming the accuracy of the most recent measurement on 2,4,6,7,9-pentamethyl-5-(5,5-${}^2$H${}_2$)methylene-11,11a-dihydro-12H-naphthacene as well as concerns about compromised accuracy, due to side reactions, of another measurement on 2-methyl-10-(10,10-${}^2$H${}_2$)methylenebicyclo[4.4.0]dec-1-ene.

14 pages, 8 figures, 6 tables

Cited by in corpus (8)

• Direct path integral estimators for isotope fractionation ratios
• A generalized class of strongly stable and dimension-free T-RPMD integrators
• Direct evaluation of the temperature dependence of the rate constant based on the quantum instanton approximation
• Monte Carlo evaluation of the equilibrium isotope effects using the Takahashi-Imada factorization of the Feynman path integral
• Accelerating quantum instanton calculations of the kinetic isotope effects
• Three applications of path integrals: equilibrium and kinetic isotope effects, and the temperature dependence of the rate constant of the [1,5] sigmatropic hydrogen shift in (Z)-1,3-pentadiene
• Accelerating equilibrium isotope effect calculations: I. Stochastic thermodynamic integration with respect to mass
• Accelerating equilibrium isotope effect calculations: II. Stochastic implementation of direct estimators