Symmetry, crystal, and electronic structure properties for the second-order Ruddlesden-Popper phase of ${\mathrm{Sr}}_3{\mathrm{Fe}}_2{\mathrm{O}}_7$ are comprehensively analyzed with the help of group-theoretical approaches and hybrid DFT calculations. This study is motivated by recent findings on cooperative Jahn-Teller (JT) distortion in ${\mathrm{Sr}}_2{\mathrm{FeO}}_4$ [G. Zvejnieks et al., Sci. Rep. 13, 16446 (2023)] and the charge order (CO) transition in ${\mathrm{Sr}}_3{\mathrm{Fe}}_2{\mathrm{O}}_7$ [J.-H. Kim et al., Phys. Rev. Lett. 127, 097203 (2021)]. We employed the amplimodes method to identify irreducible representations (irreps) of symmetry modes responsible for phase transitions in ${\mathrm{Sr}}_3{\mathrm{Fe}}_2{\mathrm{O}}_7$. The total electronic energies were calculated using the linear combination of atomic orbitals formalism and were compared for the two distinct orthorhombic symmetries and the tetragonal phase of ${\mathrm{Sr}}_3{\mathrm{Fe}}_2{\mathrm{O}}_7$. The hybrid DFT functionals used in this analysis, PBE0, WC3PW, HSE06, ${\mathrm{r}}^2\mathrm{SCANh}$, and ${\mathrm{r}}^2\mathrm{SCAN}0$, represent the most important types of hybrid DFT functionals. Depending on the choice of hybrid DFT functional, we identify as the ground state either the orthorhombic space group $Cmcm$ (CO-related symmetry mode $\mathrm{X}2-$, No. 63) or SG $Cmce$ (JT-related symmetry mode $\mathrm{X}2+$, No. 64) in the relaxed atomic structures. The interplay between the cooperative effects, such as JT distortion and charge ordering, highlights the importance of selecting the appropriate hybrid DFT functional to capture relevant exchange effects accurately. The ${\mathrm{r}}^2\mathrm{SCANh}$ functional with 10% exact Hartree-Fock (HF) exchange and ${\mathrm{r}}^2\mathrm{SCAN}0$ (with 25% exact HF exchange) yield the SG $Cmcm$ (No.63) ground state in agreement with the recent experimental observations. We thoroughly analyze and compare the crystal structure properties calculated for the chosen hybrid DFT functionals. Furthermore, only test calculations are performed to evaluate the effect of noncollinear magnetic structures. Finally, the electronic density of states for both the orthorhombic and tetragonal structures, calculated with the ${\mathrm{r}}^2\mathrm{SCANh}$ functional, emphasizes the changes induced by the cooperative effects on the electronic properties.

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• Charge density waves
• Charge order
• Jahn-Teller effect
• Charge-transfer insulators
• Band structure methods
• Density functional theory
• Hubbard model
• Quantum chemistry methods
• Symmetries