  The mass-dependent equilibrium stable isotope fractionation between different
materials is an important geochemical process. Here we present an efficient
method to compute the isotope fractionation between complex minerals and fluids
at high pressure, P, and temperature, T, representative for the Earth's crust
and mantle. The method is tested by computation of the equilibrium
fractionation of lithium isotopes between aqueous fluids and various Li bearing
minerals such as staurolite, spodumene and mica. We are able to correctly
predict the direction of the isotope fractionation as observed in the
experiments. On the quantitative level the computed fractionation factors agree
within 1.0 permil with the experimental values indicating predictive power of
ab initio methods. We show that with ab initio methods we are able to
investigate the underlying mechanisms driving the equilibrium isotope
fractionation process, such as coordination of the fractionating elements,
their bond strengths to the neighboring atoms, compression of fluids and
thermal expansion of solids. This gives valuable insight into the processes
governing the isotope fractionation mechanisms on the atomic scale. The method
is applicable to any state and does not require different treatment of crystals
and fluids.
