Hydrogen interstitials inside bulk reduced-CeO$_{\mathrm{2}}$: charge state, defect chemistry and dynamics by \textit{ab-initio} calculations*

CeO$_{\mathrm{2}}$ is a well-known and widely-used solid oxide fuel cells (SOFC) electrolyte and catalyst anode support, due to its facile oxygen vacancy formation and diffusion within its symmetric and capacious fluorite lattice. In real SOFC working conditions, hydrogen fuels will dissociate on anode surface and possibly permeate inside CeO$_{\mathrm{2}}$-based anode support and electrolyte. Studying hydrogen defect inside CeO$_{\mathrm{2}}$ lattice thus has two significant impacts: To see how hydrogen alters the ``oxygen buffering'' inside CeO$_{\mathrm{2}}$ as an anode support, and to see how it affects oxygen vacancy's diffusion and clustering inside electrolyte CeO$_{\mathrm{2}}$. Hereby \textit{ab-initio} calculations in Kohn-Sham Density Functional Theory with Hubbard model of self-interaction correction is carried out to investigate the electron polaron and vacancy formation in CeO$_{\mathrm{2}}$ affected by hydrogen, the hydrogen defect state within the band gap of CeO$_{\mathrm{2}}$, the chemistry of defect interactions and its effect on oxygen vacancy mobility.