Path integral evaluation of equilibrium isotope effects

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 and more at room temperature. 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 et al., [J. Am. Chem. Soc. 128, 9080 (2006); J. Am. Chem. Soc.129, 2488 (2007)] confirming the accuracy of the most recent measurement on 2,4,6,7,9-pentamethyl-5-(5,5-2H2) methylene- 11,11a-dihydro- 12H-naphthacene as well as concerns about compromised accuracy, due to side reactions, of another measurement on 2-methyl-10-(10,10-2H2) methylenebicyclo [4.4.0]dec-1- ene.

Accelerating path integral evaluation of equilibrium and kinetic isotope effects

Konstantin Karandashev

Investigating the effect of isotope substitution on equilibrium and kinetic properties of molecules has become an important tool for estimating the importance of nuclear quantum effects. In this work, we discuss calculating both equilibrium and kinetic isotope effects, i.e., the isotope effects on a system's partition function and a reaction's rate constant. With the help of Feynman's path integral formalism, both quantities can be estimated using standard Monte Carlo methods that scale favorably with system's dimensionality; improving efficiency of such approaches is the main focus of this work. First of all, we developed a novel procedure for changing mass stochastically during an equilibrium isotope effect calculation, and evaluated the numerical benefits of combining it with two popular approaches for calculating isotope effects, using either direct estimators or thermodynamic integration. We demonstrate that the modification improves statistical convergence of both methods, and that it additionally allows to eliminate integration error of thermodynamic integration. The improved methods are tested on equilibrium isotope effects in a model harmonic system and in methane. Then we turn our attention to kinetic isotope effect calculations with the quantum instanton approximation, a method whose path integral implementation belongs among the most accurate approaches for evaluating reaction rate constants in polyatomic systems. To accelerate quantum instanton calculations of kinetic isotope effects, we combine higher-order Boltzmann operator factorization with virial estimators, allowing us to speed up both the convergence to the quantum limit and statistical convergence of the calculation. We estimate the overall resulting acceleration using H+H2/D+D2 as a benchmark system, and then apply the accelerated method to several kinetic isotope effects associated with the H+CH4=H2+CH3 exchange. Last but not least, we explored ways to improve on the quantum instanton approximation for reaction rate constants. To that end, we review quantum instanton and Hansen-Andersen approximations, and propose a combined method, which, as the Hansen-Andersen approximation, has the correct high-temperature behavior, and at the same time, as the quantum instanton approximation, has more flexibility by allowing the dividing surface for the reaction to split into two surfaces at low temperatures. The properties of the combined method are tested on symmetric and asymmetric Eckart barrier.

EPFL2018

Accelerating path integral evaluation of equilibrium and kinetic isotope effects

Konstantin Karandashev

EPFL2018