The formation of polarons is a pervasive phenomenon in transition metal oxide compounds, with a strong impact on the physical properties and functionalities of the hosting materials. In its original formulation, the polaron problem considers a single charge carrier in a polar crystal interacting with its surrounding lattice. Depending on the spatial extension of the polaron quasiparticle, originating from the coupling between the excess charge and the phonon field, one speaks of small or large polarons. This chapter discusses the modeling of small polarons in real materials, with a particular focus on the archetypal polaron material TiO2. After an introductory part, surveying the fundamental theoretical and experimental aspects of the physics of polarons, the chapter examines how to model small polarons using first-principles schemes in order to predict, understand, and interpret a variety of polaron properties in bulk phases and surfaces. Following the spirit of this handbook, different types of computational procedures and prescriptions are presented with specific instructions on the setup required to model polaron effects.
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