  The final step of the water formation network on interstellar grain surfaces
starting from the H + O$_2$ route is the reaction between H and H$_2$O$_2$.
This reaction is known to have a high activation energy and therefore at low
temperatures it can only proceed via tunneling. To date, however, no rate
constants are available at temperatures below 200 K. In this work, we use
instanton theory to compute rate constants for the title reaction with and
without isotopic substitutions down to temperatures of 50 K. The calculations
are based on density functional theory, with additional benchmarks for the
activation energy using unrestricted single-reference and multireference
coupled-cluster single-point energies. Gas-phase bimolecular rate constants are
calculated and compared with available experimental data not only for H +
H$_2$O$_2$ $\rightarrow$ H$_2$O + OH, but also for H + H$_2$O$_2$ $\rightarrow$
H$_2$ + HO$_2$. We find a branching ratio where the title reaction is favored
by at least two orders of magnitude at 114 K. In the interstellar medium this
reaction predominantly occurs on water surfaces, which increases the
probability that the two reactants meet. To mimic this one, two, or three
spectator H2O molecules are added to the system. Eley-Rideal bimolecular and
Langmuir-Hinshelwood unimolecular rate constants are presented here. The
kinetic isotope effects for the various cases are compared to experimental data
as well as to expressions commonly used in astrochemical models. Both the
rectangular barrier and the Eckart approximations lead to errors of about an
order of magnitude. Finally, fits of the rate constants are provided as input
for astrochemical models.
