  The step-by-step hydrogen-terminated (SSHT) model is proposed as a model for
the surfaces of metal oxides. Using this model, it is found that the vibronic
coupling density (VCD) can be employed as a reactivity index for surface
reactions. As an example, the regioselectivity of CO$_2$ adsorption on the
Ag-loaded Ga$_2$O$_3$ photocatalyst surface is investigated based on VCD
analysis. The cluster model constructed by the SSHT approach reasonably
reflects the electronic structures of the Ga$_2$O$_3$ surface. The geometry of
CO$_2$ adsorbed on the Ag-loaded Ga$_2$O$_3$ cluster has a bent structure,
which is favorable for its photocatalytic reduction to CO.
