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Hydrogen compounds are peculiar as the quantum nature of the proton cancrucially affect their structural and physical properties. A remarkable exampleare the high-pressure phases of H$_2$O, where quantum proton fluctuations favorthe symmetrization of the H bond and lower by 30 GPa the boundary between theasymmetric structure and the symmetric one. Here we show that an analogousquantum symmetrization occurs in the recently discovered sulfur hydridesuperconductor with the record superconducting critical temperature $T_c=203$ Kat 155 GPa. In this system, according to classical theory, superconductivityoccurs via formation of a structure of stoichiometry H$_3$S with S atomsarranged on a body-centered-cubic (bcc) lattice. For $P \gtrsim 175$ GPa, the Hatoms are predicted to sit midway between two S atoms, in a structure with$Im\bar3m$ symmetry. At lower pressures the H atoms move to an off-centerposition forming a short H$-$S covalent bond and a longer H$\cdots$S hydrogenbond, in a structure with $R3m$ symmetry. X-ray diffraction experimentsconfirmed the H$_3$S stoichiometry and the S lattice sites, but were unable todiscriminate between the two phases. Our present ab initio density-functionaltheory (DFT) calculations show that the quantum nuclear motion lowers thesymmetrization pressure by 72 GPa. Consequently, we predict that the $Im\bar3m$phase is stable over the whole pressure range within which a high $T_c$ wasmeasured. The observed pressure-dependence of $T_c$ is closely reproduced inour calculations for the $Im\bar3m$ phase, but not for the $R3m$ phase. Thus,the quantum nature of the proton completely rules the superconducting phasediagram of H$_3$S.

Quantum Hydrogen-Bond Symmetrization and High-Temperature Superconductivity in Hydrogen Sulfide