$GW$ at the interface: CH$_3$OH and H$_2$O on TiO$_2$(110)

Electronic level alignment at the interface between an adsorbed molecular layer and a semiconducting substrate determines the activity and efficiency of many photocatalytic and photovoltaic materials. However, a quantitative description of the states at the interface remains elusive, due to the computational complexity of quasiparticle $GW$ based algorithms. We compare density functional theory (DFT) calculations and quasiparticle techniques with ultraviolet photoelectron spectra and two photon photoemission spectra to determine the level of theory required to obtain an accurate description of occupied and unoccupied states at the interface. Specifically, we consider GGA DFT, hybrid DFT and $G_0W_0$, $scGW1$, $scGW_0$, and $scGW$ quasiparticle calculations for the interface between rutile TiO$_2$(110) and methanol or water. We find the quasiparticle energy shifts $\Delta$ are linearly dependent on the fraction of the wave function density within the molecular layer $f_{mol}$ and the bulk substrate $f_{bulk}$. For the unoccupied states, the same correlation holds for all the molecular layers studied. This allows one to describe the quasiparticle energy shifts semi-quantitatively for larger molecular layers on TiO$_2$(110) based on more tractable DFT calculations.