Fractionation of isotopes among distinct molecules or phases is a quantum effect which is often exploited to obtain insights on reaction mechanisms, biochemical, geochemical, and atmospheric phenomena. Accurate evaluation of isotope ratios in atomistic simulations is challenging, because one needs to perform a thermodynamic integration with respect to the isotope mass, along with time-consuming path integral calculations. By re-formulating the problem as a particle exchange in the ring polymer partition function, we derive new estimators giving direct access to the differential partitioning of isotopes, which can simplify the calculations by avoiding thermodynamic integration. We demonstrate the efficiency of these estimators by applying them to investigate the isotope fractionation ratios in the gas-phase Zundel cation, and in a few simple hydrocarbons.

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Note that in cases where the A and B systems contain a different number of equivalent positions that can be occupied by the minority isotope (e.g., consider HOD versus CH3D), the equilibrium (1) also involves a component of configurational entropy, that means that also classicallyαA-B≠0. Here, unless otherwise specified, we will consider the equilibrium between two specified positions, i.e., we will write αA-B to indicate the sole quantum mechanical component to fractionation αA-Btot/αA-Bclass.

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