We use first principles calculations to study structural, vibrational and superconducting properties of H$_2$S at pressures $Pge 200$ GPa. The inclusion of zero point energy leads to two different possible dissociations of H$_2$S, namely 3H$_2$S $to$ 2H$_3$S + S and 5H$_2$S $to$ 3H$_3$S + HS$_2$, where both H$_3$S and HS$_2$ are metallic. For H$_3$S, we perform non-perturbative calculations of anharmonic effects within the self-consistent harmonic approximation and show that the harmonic approximation strongly overestimates the electron-phonon interaction ($lambdaapprox 2.64$ at 200 GPa) and T$_c$. Anharmonicity hardens HS bond-stretching modes and softens H--S bond-bending modes. As a result, the electron-phonon coupling is suppressed by $30%$ ($lambdaapprox 1.84$ at 200 GPa). Moreover, while at the harmonic level T$_c$ decreases with increasing pressure, the inclusion of anharmonicity leads to a T$_c$ that is almost independent of pressure. High pressure hydrogen sulfide is a strongly anharmonic superconductor.
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