## This paper reports the hybrid density functional determination

We have performed first principles density functional calculations of SnO6-containing oxides with two different methods. The first approach is density functional calculations with the PBEsol generalized gradient approximation (GGA) [36] and the projector-augmented-wave method as implemented in VASP [37] and [38]. We considered the following valence CEP-37440 configuration: 5s26s25p65d1 for Ba, 4s25s24p64d1 for Sr, 3s23p64s23d0 for Ca, 4d105s25p2 for Sn, and 2s22p4 for O. The electronic wave functions were expanded with plane waves up to a kinetic-energy cutoff of 400 eV except for structural optimization, where a kinetic energy cutoff of 520 eV has been applied to reduce the effects of Pulay stress. The momentum space integrations were performed using a 4×5×5 Γ-centered Monkhorst–Pack k-mesh [39]. For the various symmetries examined, the lattice constants and internal coordinates were fully optimized until the residual Hellmann–Feynman forces became smaller than 10−1 meV/Å. In the PBEsol calculation, the correct band gap could not be obtained, as noted by previous researchers [11], [12], [13] and [14], because of ignoring the Hartree–Fock exact exchange part. In addition, the plane wave approach for the hybrid functional calculation incurs considerable computational cost for low symmetry structural optimization. Therefore, we have applied the hybrid density functional calculation of CRYSTAL09 [40] to study the electronic structure of SnO6-containing oxides for a systematic band gap prediction with B3LYP hybrid functional [41]. For the oxygen atom, we have used all electrons Gaussian Type Function (GTF) and the Hay–Wadt pseudo-potential for the core electron has been used for the Ba, Sr, and Ca atoms [42]. We have applied the Durand 21G type pseudo-potential for Sn atoms [42]. For each given system, we have performed the independent structural optimization using the B3LYP hybrid functional [32] and [41] and also LDA-PZ [26] and PBE-GGA [27] functional. The total energy calculations have been performed in the momentum space integrations using an 8×8×8 Γ-centered Monkhorst–Pack k-mesh [39] for the CRYSTAL09 calculation. After each structural optimization starting from experimental values, structural parameters have been summarized based on the optimized structure and the band diagrams have been obtained to check the band gap obtained from the density of state calculation. We would like to emphasize that our B3LYP hybrid density functional calculation is done with independent optimized structure. Details of the structural parameters have been converted to CIF file format for GGA and B3LYP hybrid density functional and can be downloaded from the Supplementary material for all 9 compounds [43].