First-principles real-space study of electronic and optical excitations in rutileTiO2nanocrystals

We model rutile titanium dioxide nanocrystals (NCs) up to $\ensuremath{\sim}1.5$ nm in size to study the effects of quantum confinement on their electronic and optical properties. Ionization potentials (IPs) and electron affinities (EAs) are obtained via the perturbative $GW$ approximation (${G}_{0}{W}_{0}$) and $\ensuremath{\Delta}\mathrm{SCF}$ method for NCs up to 24 and 64 ${\mathrm{TiO}}_{2}$ formula units, respectively. These demanding $GW$ computations are made feasible by using a real-space framework that exploits quantum confinement to reduce the number of empty states needed in $GW$ summations. Time-dependent density functional theory (TDDFT) is used to predict the optical properties of NCs up to 64 ${\mathrm{TiO}}_{2}$ units. For a NC containing only 2 ${\mathrm{TiO}}_{2}$ units, the offsets of the IP and the EA from the corresponding bulk limits are of similar magnitude. However, as NC size increases, the EA is found to converge more slowly to the bulk limit than the IP. The EA values computed at the ${G}_{0}{W}_{0}$ and $\ensuremath{\Delta}\mathrm{SCF}$ levels of theory are found to agree fairly well with each other, while the IPs computed with $\ensuremath{\Delta}\mathrm{SCF}$ are consistently smaller than those computed with ${G}_{0}{W}_{0}$ by a roughly constant amount. TDDFT optical gaps exhibit weaker size dependence than $GW$ quasiparticle gaps, and result in exciton binding energies on the order of eV. Altering the dimensions of a fixed-size NC can change electronic and optical excitations up to several tenths of an eV. The largest NCs modeled are still quantum confined and do not yet have quasiparticle levels or optical gaps at bulk values. Nevertheless, we find that classical Mie-Gans theory can quite accurately reproduce the line shape of TDDFT absorption spectra, even for (anisotropic) ${\mathrm{TiO}}_{2}$ NCs of subnanometer size.