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Path integral calculations of equilibrium isotope effects and isotopicfractionation are expensive due to the presence of path integral discretizationerrors, statistical errors, and thermodynamic integration errors. Whereas thediscretization errors can be reduced by high-order factorization of the pathintegral and statistical errors by using centroid virial estimators, two recentpapers proposed alternative ways to completely remove the thermodynamicintegration errors: Cheng and Ceriotti [J. Chem. Phys. 141, 244112 (2015)]employed a variant of free-energy perturbation called "direct estimators,"while Karandashev and Van\'{\i}\v{c}ek [J. Chem. Phys. 143, 194104 (2017)]combined the thermodynamic integration with a stochastic change of mass andpiecewise-linear umbrella biasing potential. Here we combine the formerapproach with the stochastic change of mass in order to decrease itsstatistical errors when applied to larger isotope effects, and perform athorough comparison of different methods by computing isotope effects first ona harmonic model, and then on methane and methanium, where we evaluate allisotope effects of the form$\mathrm{CH}_{\mathrm{4-x}}\mathrm{D}_{\mathrm{x}}/\mathrm{CH}_{4}$ and$\mathrm{CH}_{\mathrm{5-x}}\mathrm{D}^{+}_{\mathrm{x}}/\mathrm{CH}^{+}_{5}$,respectively. We discuss thoroughly the reasons for a surprising behavior ofthe original method of direct estimators, which performed well for a muchlarger range of isotope effects than what had been expected previously.

Accelerating equilibrium isotope effect calculations: II. Stochastic implementation of direct estimators