As a model system for proton configuration optimization, scandium-doped barium zirconate was employed according to the crystal structure shown in Figure . The mother compound was the BaZrO3 perovskite, and fast proton conduction has previously been observed in its doped materials, wherein a high proton conductivity was achieved by substituting zirconium with scandium. Under these conditions, oxygen vacancies are generated, and hydroxy groups can be introduced into the system in the presence of water vapor to compensate for these vacancies. In the current study, eight initial proton positions are considered at each oxygen atom, as determined by DFT structural optimizations for several configurations, including those of the protons. Each proton position was directed to the second nearest oxygen atom to form a hydrogen bond (see Figure ). Thus, these proton positions can be used as the initial proton configurations for the relaxation calculations. A 2 × 2 × 2 supercell model was adopted to search for a stable configuration. Based on the number of scandium ions (𝑥) in the supercell, the composition of the model was expressed as Ba8Zr8-xScxO24Hx, corresponding to the condition that all oxygen vacancies are compensated for by hydroxy groups; therefore, the number of introduced protons is the same as the number of scandium ions. When the doped positions of the scandium ions are fixed, the number of proton configurations is 24Cx × 8 x , without considering symmetry. Thus, when the value of 𝑥 increases, many proton configurations must be considered. Of course, when symmetry is considered, the number of independent initial proton configurations is reduced; however, a combinatorial explosion of the initial configurations occurs when larger systems are considered.