## 3 Results and Discussion  
To model the \(\rm YSr_{2}FeCu_{2}O_{7.5}\) phase, the four possible configurations in the \(\rm 2a_{p}\times 2a_{p}\times 3a_{p}\) supercell were generated (Figure 2a). Note that, despite the experimental suggestions from the neutron powder diffraction (NPD) results for the \(\rm YSr_{2}Cu_{2}FeO_{7.56}\) phase [21], none of the four modelled configurations have Fe in solely tetrahedral [FeO4] and octahedral [FeO6] environments. The configuration where Fe forms tetrahedra (Th) [FeO4], octahedra (Oh) [FeO6], and square pyramids (SQP) [FeO5] (Figure 2a) is the most stable, with a maximum energy difference obtained from geometrical optimization with the SCAN functional of 300 meV/f.u. with respect to the less stable configuration (all Fe ions in SQP). For a \(\rm YSr_{2}FeCu_{2}O_{7.5}\) composition, the \(\rm 2a_{p}\times 2a_{p}\times 3a_{p}\) cell is not compatible with Fe ions occupying exclusively Th and Oh. Hence, a \(\rm 2a_{p}\times 4a_{p}\times 3a_{p}\) supercell was used to generate additional \(\rm YSr_{2}Cu_{2}FeO_{7.5}\) configurations. In this case, 91 configurations were generated using the Pymatgen package, some of them presenting Fe in solely tetrahedral [FeO4] and octahedral [FeO6] coordination. Among those configurations, the Ewald method allows the selection of the most stable. The SCAN geometry optimization shows that this configuration in the \(\rm 2a_{p}\times 4a_{p}\times 3a_{p}\) supercell is 194 meV/f.u. less stable (more positive) than the most stable configuration in the \(\rm 2a_{p}\times 2a_{p}\times 3a_{p}\) supercell (Fe in Oh, Th, and SQP). These findings advocate for detailed structural characterizations to precisely identify the local ordering of vacancies in the synthetized \(\rm YSr_{2}Cu_{2}FeO_{7+\delta}\) materials with \(\delta\sim 0.5\). Interestingly, from the calculated total energies, it can be inferred that the most stable configuration for \(\rm YSr_{2}Cu_{2}FeO_{7.5}\) (in the \(\rm 2a_{p}\times 2a_{p}\times 3a_{p}\) unit cell, where Fe is found in Th, Oh, and SQP) is energetically favored respective to the end terms in 157 meV/f.u., according to the reaction \(\rm 1\%\)\(\rm YSr_{2}Cu_{2}FeO_{8}+\rm\gamma\)\(\rm YSr_{2}Cu_{2}FeO_{7}\rightarrow\rm YSr_{2}Cu_{2}FeO_{7.5}\). This is in line with the feasibility of synthesizing YSCFO terms with \(\delta\sim 0.5\), as was previously reported [15; 21]. Figure 2: (**a**) Schematic representation of FeO\({}_{1+\delta}\) planes in the four possible configurations for \(\rm YSr_{2}Cu_{2}FeO_{7.5}\) considering the \(\rm 2a_{p}\times 2a_{p}\times 3a_{p}\) supercell. Bader charges and magnetic moments are shown for the most stable configuration. (**b**) View of the crystal structure of the most stable model for \(\rm YSr_{2}Cu_{2}FeO_{7.5}\) (\(\rm 2a_{p}\times 2a_{p}\times 3a_{p}\) supercell). Color code: Fe in brown, Cu in blue, Sr in green, Y in grey, and O in red. For this configuration of YSr\({}_{2}\)FeCu\({}_{2}\)O\({}_{7.5}\), AIMD calculations were performed at 1000, 1200, 1400, 1600, and 2000 K. The MSD slopes can be used to calculate the self-diffusion coefficient at each simulated temperature; for instance, Figure 3a shows the results at 1400 K that yield a diffusivity of 3.67 \(\times\) 10\({}^{-6}\) cm\({}^{2}\)/s. An activation energy of 0.37 eV is extracted from the Arrhenius plot (Figure 3b). It is found that a high oxygen ion mobility occurs in the FeO\({}_{1+8}\) plane (Figure 3c), with a predicted diffusivity of 1.28 \(\times\) 10\({}^{-7}\) cm\({}^{2}\)/s at 500 \({}^{\circ}\)C. Calculated AIMD oxygen diffusivity values of 3.1 \(\times\) 10\({}^{-9}\) cm\({}^{2}\)/s (E\({}_{\text{a}}\) = 0.50 eV) were previously reported for GdBaCo\({}_{2}\)O\({}_{5.5}\) (GBCO) [36], whose diffusion mechanism has been extensively studied from AIMD simulations [37; 38]. The Na\({}_{0.5}\)Bi\({}_{0.5}\)Ti\({}_{0.96}\)Mg\({}_{0.04}\)O\({}_{2.96}\) fast oxide ion conductor presents a calculated diffusivity from AIMD of 2.1 \(\times\) 10\({}^{-8}\) cm\({}^{2}\)/s and E\({}_{\text{a}}\) = 0.61 eV [39].