A hydrated electron in water at different densities and temperatures is studied via a set of density functional based molecular dynamics simulations, showing that a localization of an excess electron is still present even at very low densities. Space variations of the molecular dipole moments are analyzed, proposing a simple algorithm to identify the region of localization of the wavefunction relative to the solvated electron in terms of orientation of the H2O molecular dipole moments. Finally, the effects of the self-interaction corrections on the optical absorption spectra are analyzed and compared with both available experimental data and path integral molecular dynamics calculations, showing that a weighted subtraction of the self-interaction yields a systematic improvement in the position of the absorption peak.