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Quantum effects in proton-conducting oxides: an exhaustive study in barium stannate

Density-functional theory calculations are performed to investigate hydrogen transport in the proton conductor BaSnO$_3$. Structural optimizations in the stable and saddle point configurations for transfer and reorientation allow description of the high-temperature classical and semi-classical regimes, in which diffusion occurs by over-barrier motion. At lower temperature (typically below 300 K), we describe a thermally-assisted quantum regime. In this regime, transfer and reorientation occur when the surrounding matrix adopts particular "coincidence" configurations in which quantum tunneling is favored. Both the non-adiabatic and the adiabatic cases are examined. In the adiabatic case, the energy landscape of hydrogen in the coincidence configuration is very flat, with very low coincidence energy barriers. Path-integral molecular dynamics simulations of the H atom in the coincidence potential reveal, in the transfer case, highly quantum behavior up to T=300 K (the density of probability of H in the coincidence configuration, has its maximum at the saddle point, due to the fact that the zero-point energy exceeds the coincidence energy barrier). Arguments are given that support the adiabatic picture for the transfer mechanism. This suggests existence of this state of hydrogen during the very short lifetime of the coincidence configurations ($\sim$ 10$^{-13}$ s), as a transition state for the transfer mechanism. Remarkably, such state is identical to that of ice X, a highly quantum phase of ice observed at high pressures $\approx$ 100 GPa. In the case of reorientation, typical times for existence of the coincidence configuration and for protonic motion are roughly equal, suggesting that the adiabatic picture is not valid. Protonic transfer and reorientation in oxides are therefore governed by radically different mechanisms below room temperature.