The structure and energetics of excitons and individual electron and hole polarons in a model anatase TiO2 nanoparticle (NP) are investigated by means of Density Functional Theory (DFT) and Time Dependent (TD)-DFT calculations. The effect of the Hartree-Fock exchange (HF-exc) contribution in the description of TiO2 NPs with unpaired electrons is examined by comparing the results from semilocal and hybrid DFT functionals with different HF-exc percentages, including a long-range corrected hybrid functional. The performances of TD-DFT and ground state (SCF) DFT approaches in the description of the photoexcited polaron states in TiO2 NPs are also analyzed. Our results confirm that the HF-exc contribution is essential to properly describe the self-trapping of the charge carriers. They also suggest that long-range corrected functionals are needed to properly describe excited state relaxation in TiO2 NPs. TD-DFT geometry optimization of the lowest excited singlet and triplet states deliver photoluminescence values in close agreement with the experimental data.