We present a first-principles computational investigation on the adsorption mode and electronic structure of the highly efficient heteroleptic ruthenium dye C101, [NaRu(4,4'-bis(5-hexylthiophene-2-yl)-2,2'-bipyridine)(4-carboxylic acid-4'-carboxylate-2,2'-bipyridine) (NCS)(2)], on anatase TiO2 models exposing the (001) and (101) surfaces. The electronic structure of the TiO2 models shows a conduction band energy upshift for the (001)-surface ranging between similar to 50 and similar to 110 meV compared with the (101) surface, in agreement with previous interfacial impedance and recent spectro-electrochemical data. TDDFT excited-state calculations provided the same optical band gap, within 0.01 eV, for the (001)- and (101) models. Two dominant adsorption modes for C101 dye Adsorption on the (001) and (101) surfaces were found, which differ by the binding of the dye carboxylic groups to the TiO2 surfaces (bridged bidentate vs monodentate), leading to sizably different tilting of the anchoring bipyridine plane with respect to the TiO2 surface. The different adsorption mode leads to a smaller dye coverage on the (001) surface, as experimentally found, due to partial contact of the thiophene and alkyl bipyridine substituents with the TiO2 surface. For the energetically favored adsorption modes, we calculate a larger average spatial separation, by 1.3 angstrom, between the dye-based HOMO and the semiconductor surface in (001) and (101) TiO2 models. In terms of simple nonadiabatic electron-transfer considerations, our model predicts a retardation of the charge recombination kinetics, in agreement with the experimental observations.