We perform high-level, quantum molecular dynamics simulations of proton transfer in the protonated water dimer, H5O2+. The electronic structure of the system is calculated concurrently with the nuclear motion using Born-Oppenheimer molecular dynamics plus density functional theory. Performing the calculations at finite (thermal) temperatures allows us to observe and quantify such effects as the broadening of the electronic density of states, the thermal splitting of degenerate states, the shift of the highest occupied molecular orbital, the thermal expansion of the dipole moment, and the thermal shift, coupling and broadening of the vibrational density of states. At two of the temperatures considered (225 K and 360 K), we find that H5O2+ exists in a dynamical equilibrium state in which the proton oscillates between two water molecules. The characteristic frequencies of the proton motion are very sensitive to temperature. At 40 K and 225 K, strong peaks are identified in the vibrational spectrum corresponding to the motion of the proton between the two oxygen atoms. At 360 K, the frequencies of this motion are distributed among a series of peaks between 1100 and 1800 cm(-1). At all temperatures investigated, the proton motion is coupled strongly to other degrees of freedom in the water molecules and the dimer. Statistically, the proton is localized near one of the two H2O molecules to form H3O+ with a probability of 0.09 at 225 K, and 0.18 at 360 K. At low temperature (40 K), the proton remains localized near the midpoint of the two oxygens, and has almost zero probability to exist as H3O+. (C) 1997 American Institute of Physics. [S0021-9606(97)51144-5].