Abstract
Path integral calculations of equilibrium isotope effects and isotopic fractionation are expensive due to the presence of path integral discretization errors, statistical errors, and thermodynamic integration errors. Whereas the discretization errors can be reduced by high-order factorization of the path integral and statistical errors by using centroid virial estimators, two recent papers proposed alternative ways to completely remove the thermodynamic integration errors: Cheng and Ceriotti [J. Chem. Phys. 141, 244112 (2015)] employed a variant of free-energy perturbation called "direct estimators," while Karandashev and Vaníček [J. Chem. Phys. 143, 194104 (2017)] combined the thermodynamic integration with a stochastic change of mass and piecewise-linear umbrella biasing potential. Here, we combine the former approach with the stochastic change in mass in order to decrease its statistical errors when applied to larger isotope effects and perform a thorough comparison of different methods by computing isotope effects first on a harmonic model and then on methane and methanium, where we evaluate all isotope effects of the form CH4-xDx/CH4 and CH5-xDx +/CH5 +, respectively. We discuss the reasons for a surprising behavior of the original method of direct estimators, which performed well for a much larger range of isotope effects than what had been expected previously, as well as some implications of our work for the more general problem of free energy difference calculations.
Highlights
The equilibrium isotope effect and a closely related concept of isotope fractionation1–3 belong among the most useful experimental tools for uncovering the influence of nuclear quantum effects on molecular properties.4,5 The equilibrium isotope effect measures the effect of isotopic substitution on the equilibrium constant of a chemical reaction and is defined as the ratio of equilibrium constants, EIE ∶= K(B)/K(A), (1)where A and B are two isotopologues of the reactant
In Ref. 16, we introduced a method following the first philosophy for eliminating the integration error—in particular, we showed that an effective Monte Carlo procedure for changing the mass reduces the thermodynamic integration error and that a special mass-dependent biasing potential renders the integration error exactly zero
While in path integral molecular dynamics gradients of V are available and the centroid virial estimator is “free,” in path integral Monte Carlo implementations only the potential energy is required for sampling, and in order to avoid an unnecessary evaluation of forces, one may evaluate the centroid virial estimators by a single finite difference differentiation with respect to λ
Summary
The equilibrium isotope effect and a closely related concept of isotope fractionation belong among the most useful experimental tools for uncovering the influence of nuclear quantum effects on molecular properties. The equilibrium (or thermodynamic) isotope effect measures the effect of isotopic substitution on the equilibrium constant of a chemical reaction and is defined as the ratio of equilibrium constants, EIE ∶= K(B)/K(A),. Since the statistical convergence of thermodynamic integration can be improved by changing the mass stochastically during the simulation, it seems natural to combine this stepwise approach with the procedure for changing mass introduced in Ref. 16; testing the performance of the resulting combined method on large isotope effects was the main goal of this paper Such a combination with direct estimators is only possible for a mass sampling procedure that allows finite steps with respect to mass, making the Monte Carlo procedure of Ref. 16 suitable for the task but disqualifying standard λ-dynamics algorithms based on molecular dynamics.
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