  We address the double hydrogen transfer (DHT) dynamics of the porphycene
molecule: A complex paradigmatic system where the making and breaking of
H-bonds in a highly anharmonic potential energy surface requires a quantum
mechanical treatment not only of the electrons, but also of the nuclei. We
combine density-functional theory calculations, employing hybrid functionals
and van der Waals corrections, with recently proposed and optimized
path-integral ring-polymer methods for the approximation of quantum vibrational
spectra and reaction rates. Our full-dimensional ring-polymer instanton
simulations show that below 100 K the concerted DHT tunneling pathway
dominates, but between 100 K and 300 K there is a competition between concerted
and stepwise pathways when nuclear quantum effects are included. We obtain
ground-state reaction rates of $2.19 \times 10^{11} \mathrm{s}^{-1}$ at 150 K
and $0.63 \times 10^{11} \mathrm{s}^{-1}$ at 100 K, in good agreement with
experiment. We also reproduce the puzzling N-H stretching band of porphycene
with very good accuracy from thermostatted ring-polymer molecular dynamics
simulations. The position and lineshape of this peak, centered at around 2600
cm$^{-1}$ and spanning 750 cm$^{-1}$, stems from a combination of very strong
H-bonds, the coupling to low-frequency modes, and the access to $cis$-like
isomeric conformations, which cannot be appropriately captured with
classical-nuclei dynamics. These results verify the appropriateness of our
general theoretical approach and provide a framework for a deeper physical
understanding of hydrogen transfer dynamics in complex systems.
