Proton Quantum Effects in H$_3$S Electronic Structure: A Multicomponent DFT study via Nuclear-Electronic Orbital Method

We investigate the impact of the quantum effects of protons on the electronic structure of high-pressure H$_3$S, a benchmark hydrogen-rich superconductor with a critical temperature ($T_c$) exceeding 200 K. Using Nuclear-Electronic Orbital Density Functional Theory (NEO-DFT), we treat hydrogen nuclei quantum mechanically on the same footing as electrons within a first-principles framework. Our calculations reveal that nuclear quantum effects (NQEs) induce subtle modifications to the electronic band structure and density of states (DOS) near the Fermi energy, including features associated with van Hove singularities. However, the resulting changes in the DOS would increase $T_c$ by only a few percent. On the other hand, calculations of the phonon dispersion with the NEO-DFT method show large changes in the hydrogen-dominated phonons that arise from a stiffening of the S-H bonds due to NQEs. These findings imply that the experimentally observed reduction in $T_c$ upon deuteration arises predominantly from changes in the phonon properties, while NQEs-induced modifications to the electronic structure itself are minimal.