  The transport properties (electrical resistivity, thermal conductivity, and
viscosity) of iron-hydrogen alloy are of great significance in the stability
and evolution of planetary magnetic fields. Here, we investigate the thermal
transport properties of iron doped with varying hydrogen content as functions
of pressure (P) and temperature (T) for the top and bottom of Earth's outer
core and beyond, corresponding to pressures of about 130 to 300 GPa and
temperatures of 4000 to 7000 K. Using first-principles density functional
theory molecular dynamic simulations (FPMD), we verify that crystalline FeH$_x$
is superionic with H diffusing freely. We find a low frequency viscosity of
10-11 mPa$\cdot$s for liquid Fe-H alloys at Earth's outer core conditions by
the linear response Green-Kubo formula. Using the KKR method within density
functional theory (DFT) plus Dynamical mean-field Theory (DMFT), we find
saturation of electrical resistivity with increasing temperatures in liquid
iron at outer core conditions. The effect of H on electrical and thermal
transport we find is small, so that the exact H content of the core is not
needed. The primary effect of H is on the equation of state, decreasing the
density at constant P and T. We find the Lorenz number is smaller than the
ideal value, and obtain for X(H)= 0.20, or 0.45 wt% H , thermal conductivity
$\kappa$ of $\sim$105 and $\sim$190 $Wm^{-1}K^{-1}$, respectively, at
conditions near the core-mantle and inner-outer core boundary.
