  We investigate the quantum-mechanical localization of protonated and
deterated isotopes in the symmetric low-barrier hydrogen-bonds of potassium
dihydrogen phosphate (KDP) crystals in the paraelectric phase. The spatial
density distributions of these hydrogen atoms are suspected to be responsible
for the surprisingly large isotope effect observed for the ferroelectric phase
transition in KDP. We employ ab initio path integral molecular dynamics
simulations to obtain the nuclear real-space and momentum-space densities n(R)
and n(k) of protons and deuterons, which are compared to experimental Neutron
Compton Scattering data. Our results suggest a qualitative difference in the
nature of the paraelectic phase in KDP between the two isotopes. We are able to
discriminate between real quantum delocalization and vibration-assisted hopping
and thus provide evidence for two distinct mechanisms of the ferroelectric
phase transition in this class of materials.
