  For a long time, performing converged path integral simulations at ultra-low,
but finite temperatures of a few Kelvin has been a nearly impossible task.
However, recent developments in advanced colored noise thermostatting schemes
for path integral simulations, namely the Path Integral Generalized Langevin
Equation Thermostat (PIGLET) and the Path Integral Quantum Thermal Bath
(PIQTB), have been able to greatly reduce the computational cost of these
simulations, thus making the ultra-low temperature regime accessible in
practice. In this work, we investigate the influence of these two
thermostatting schemes on the description of hydrogen-bonded systems at
temperatures down to a few Kelvin as encountered, for example, in helium
nanodroplet isolation or tagging photodissociation spectroscopy experiments.
For this purpose, we analyze the prototypical hydrogen bond in the Zundel
cation (H$_5$O$_{2}^{+}$) as a function of both, oxygen-oxygen distance and
temperature in order to elucidate how the anisotropic quantum delocalization
and, thus, the shape of the shared proton adapts depending on the
donor-acceptor distance. The underlying electronic structure of the Zundel
cation is described in terms of Behler's Neural Network Potentials of
essentially converged Coupled Cluster accuracy, CCSD(T*)-F12a/AVTZ. In
addition, the performances of the PIQTB and PIGLET methods for energetic,
structural, and quantum delocalization properties are assessed and directly
compared. Overall, our results emphasize the validity and practical usefulness
of these two modern thermostatting approaches for path integral simulations of
hydrogen-bonded systems even at ultra-low temperatures.
