The chemical potential of elemental species i is denoted by µ i and n i is the number of atoms of species i added (n i < 0) or removed (n i > 0) from the host supercell to form the defect D. µ i is expressed as µ i = µ 0 + Dµ, where µ 0 is the reference elemental chemical potential and Dµ i is the deviation from the reference value. µ 0 are fitted to a set of experimentally-measured formation enthalpy of compounds in the quaternary Ba-Sr-Cd-P chemical space. The fitted µ 0 values are listed in Table . For a material, the accessible range of Dµ i is constrained by the condition of its thermodynamic phase stability and is calculated through a convex hull analysis. Table lists the Dµ i values at the corners of the polyhedra that defines the region of phase stability of Ba 0.75 Sr 1.25 CdP 2 . The positions of the ions in the defect supercells are relaxed with DFT using a plane-wave energy cutoff of 340 eV. The Brillouin zone of the supercell is sampled with a G-centered 2⇥2⇥2 k-point grid. We consider native vacancies (V Ba , V Sr , V Cd , V P ) and anti-site defects (Cd P , P Cd ). The interstitial defects Ba i and Sr i are not considered in this work because they are expected to have high formation energy due to their large ionic radii. For each defect type, we consider all unique Wyckoff positions in the structure and multiple charge states q = -3, -2, -1, 0, 1, 2, 3. The underestimation of the band gap in DFT is rectified by applying individual valence and conduction band edge shifts (relative to the DFT-computed band edges) as determined from GW quasiparticle energies. Because of the direct band gap at G, the GW quasi-particle energies are calculated for the 40-atom SQS using a G-point only k-point grid.