  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.
