  Understanding the reactivity of ubiquitous molecules on complex oxides has
broad impacts in energy applications and catalysis. The garnet-type
Li$_7$La$_3$Zr$_2$O$_{12}$ is a promising solid-state electrolyte for
lithium(Li)-ion batteries, and it readily reacts with H$_2$O and CO$_2$ when
exposed to ambient air. Such reactions form a contamination layer on
Li$_7$La$_3$Zr$_2$O$_{12}$ that is detrimental to the battery operations. The
strong interactions of Li$_7$La$_3$Zr$_2$O$_{12}$ with H$_2$O and CO$_2$,
however, make Li$_7$La$_3$Zr$_2$O$_{12}$ a promising support to catalyze H$_2$O
dissociation and CO$_2$ adsorption. Here, using first-principles calculations,
we investigate the adsorption and reactions of H$_2$O and CO$_2$ on a
Li$_7$La$_3$Zr$_2$O$_{12}$ surface. We show that H$_2$O reacts through the
exchange of proton and Li$^{+}$ and produces metal hydroxide species. At high
H$_2$O coverage, half of the H$_2$O molecules dissociate while the other half
remain intact. CO$_2$ reacts with the Li$_7$La$_3$Zr$_2$O$_{12}$ surface
directly to produce carbonate species. We clarify that the individual reactions
of H$_2$O and CO$_2$ with Li$_7$La$_3$Zr$_2$O$_{12}$ are more thermodynamically
favorable than the co-adsorption of H$_2$O and CO$_2$. Finally, we demonstrate
that low temperature and high partial pressure promote the reactions of H$_2$O
and CO$_2$ with Li$_7$La$_3$Zr$_2$O$_{12}$. For energy storage application of
Li$_7$La$_3$Zr$_2$O$_{12}$, our study guides processing conditions to minimize
surface contamination. From a catalysis point of view, our findings reveal the
potential of using complex oxides, such as Li$_7$La$_3$Zr$_2$O$_{12}$ as a
support for reactions requiring H$_2$O dissociation and strong CO$_2$
adsorption.
