  We present calculations of free energy barriers and diffusivities as
functions of temperature for the diffusion of hydrogen in bcc-Fe. This is a
fully quantum mechanical approach since the total energy landscape is computed
using a new self consistent, transferable tight binding model for interstitial
impurities in magnetic iron. Also the hydrogen nucleus is treated quantum
mechanically and we compare here two approaches in the literature both based in
the Feynman path integral formulation of statistical mechanics. We find that
the quantum transition state theory which admits greater freedom for the proton
to explore phase space gives result in better agreement with experiment than
the alternative which is based on fixed centroid calculations of the free
energy barrier. We also find results in better agreement compared to recent
centroid molecular dynamics (CMD) calculations of the diffusivity which
employed a classical interatomic potential rather than our quantum mechanical
tight binding theory. In particular we find first that quantum effects persist
to higher temperatures than previously thought, and conversely that the low
temperature diffusivity is smaller than predicted in CMD calculations and
larger than predicted by classical transition state theory. This will have
impact on future modeling and simulation of hydrogen trapping and diffusion.
