In turn, the vibrational contribution F vib of small and supported metal-oxide clusters were found to depend on the low frequency contributions of the metal atoms only. The difference between clusters with the same number of metal atoms is therefore small. The differences in zero-point energy contributions are more significant, but proportional to the number of O atoms and accountable in an approximate way (~ 0.1 eV per O atom in the particle). The chemical potential of oxygen in the gas phase is evaluated in terms of the temperature and partial pressure of the molecular oxygen reservoir and the ZPE-corrected DFT energy of molecular oxygen. Hence, for any value of the temperature, the chemical potential can be expressed as shown in equation 3: