A theoretical prediction of ion conductivity for solid state HfO2 is carried out in analogy to ZrO2 based on the density functional calculation. Geometric and electronic structures of pure bulks exhibit similarity for the two materials. Negative formation enthalpy and negative vacancy formation energy are found for YSH (yttria-stabilized hafnia) and YSZ (yttria- stabilized zirconia), suggesting the stability of both materials. Low activation energies (below 0.7 eV) of diffusion are found in both materials, and YSH’s is a little higher than that of YSZ. In addition, for both HfO2 and ZrO2 , the supercells with native oxygen vacancies are also studied. The so-called defect states are observed in the supercells with neutral and +1 charge native vacancy but not in the +2 charge one. It can give an explanation to the relatively lower activation energies of yttria-doped oxides and +2 charge vacancy supercells. A brief discussion is presented to explain the different YSH ion conductivities in the experiment and obtained by us, and we attribute this to the different ion vibrations at different temperatures. A theoretical prediction of ion conductivity for solid state HfO2 is carried out in analogy to ZrO2 based on the density functional calculation. Geometric and electronic structures of pure bulks exhibit similarity for the two materials. Negative formation enthalpy and negative vacancy formation energy are found for YSH (YTtria-stabilized hafnia) and YSZ (Yttria-stabilized zirconia), suggesting the stability of both materials. Low activation energies (below 0.7 eV) of diffusion are found in both materials, and YSH’s is a little higher than that of YSZ. In addition, for both HfO2 and ZrO2, the supercells with native oxygen vacancies are also studied. The so-called defect states are also in the supercells with neutral and +1 charge native vacancy but not in the +2 charge one. explanation to the relatively lower activation energies of yttria-doped oxides and +2 charge vacancy supercells. A brief discussion is presented to explain the different YSH ion conductivities in the experiment and obtained by us, and we attribute this to the different ion vibrations at different temperatures.