## 3 Results and Discussion
### Electronic and Magnetic Structure of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\)  
The spin-resolved band structure of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) has been calculated to evaluate the exchange coupling using PBE, PBE0, and HSE functionals. As Figure S3a shows, the band structure calculated within PBE functional for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in FM state is metallic. Specifically, the states at the Fermi level (chosen to be the zero of energy) belong to both spin-up and spin-down channels, while the states from just above the Fermi level to the top of the valence band have pure spin-down character (exhibit 100% spin polarization). It is well known that PBE functional tends to underestimate band gaps [57]; therefore, additional calculations using hybrid functionals PBE0 and HSE have been performed to get better estimates of the band gap, as depicted in Figure S3 and Figure 2, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & \multicolumn{3}{c}{**Averaging Method:**} \\  & **[PERSON]** & **[PERSON]** & **[PERSON]** \\ \hline \(B\) (GPa) & 93.298 & 80.624 & 86.961 \\ \(G\) (GPa) & 46.882 & 30.018 & 38.450 \\ \(Y\) (GPa) & 120.468 & 80.112 & 100.533 \\ \((B/G)\) & 1.990 & 2.686 & 2.262 \\ \(\
u\) & 0.285 & 0.334 & 0.300 \\ \(H\) (GPa) & 6.413 & 3.326 & 4.819 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Direction-averaged bulk modulus \(B\), shear modulus \(G\), [PERSON]’s modulus \(Y\), [PERSON]’s ratio \(\
u\), bulk-to-shear modulus ratio \((B/G)\), and hardness \(H\) of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) calculated according to [PERSON], [PERSON], and [PERSON] approximations. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**State** & \(C_{\bf 11}\) & \(C_{\bf 22}\) & \(C_{\bf 33}\) & \(C_{\bf 44}\) & \(C_{\bf 55}\) & \(C_{\bf 66}\) & \(C_{\bf 12}\) & \(C_{\bf 13}\) & \(C_{\bf 23}\) & \(C_{\bf 46}\) & \(C_{\bf 15}\) & \(C_{\bf 25}\) & \(C_{\bf 35}\) \\ \hline AFM-2 & 120.8 & 97.7 & 174.1 & 59.9 & 61.7 & 56.6 & 71.5 & 79.9 & 72.1 & 8.4 & \(-34.1\) & \(-1.2\) & \(-43.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Calculated elastic constants \(C_{ij}\) (GPa) of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\). Surprisingly, a band gap as large as 3.784 eV and 3.084 eV opens up in the electronic structure of FM \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) upon going from PBE to, respectively, PBE0 and HSE functionals. In particular, we observe a sizeable enhancement of the band gap value in the spin-down channel. According to a previous study, the local Coulomb interaction \(U\) in GGA+\(U\) or LDA+\(U\) calculations appears to produce a similar band gap opening in the electronic structure of FM \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\), indicating that the insulating ground state of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) is of the Mott type [12; 23]. In the case of AFM-1 magnetic ordering, due to the shifts of the valence band maximum (VBM) and conduction band minimum (CBM) positions, the band gap values have been calculated to be 0.390 eV, 4.344 eV, and 3.566 eV within PBE, PBE0, and HSE functionals, respectively. The calculated band gap values for AFM-2 state of magnetic order are 0.730 eV, 4.670 eV, and 3.966 eV within PBE, PBE0, and HSE functional, respectively. Finally, the obtained band gap values for the AMF-3 state are 0.362 eV, 4.395 eV, and 3.683 eV within PBE, PBE0, and HSE functionals, respectively. All calculations that consider an antiferromagnetic order of Cu magnetic moments predict \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) to be a semiconductor. Additionally, the same variation trend of the band structures is observed under PBE0 and HSE hybrid functional calculations. All these results are collected in Table 3. The band gap values calculated in this work using hybrid functionals agree reasonably well with the band gap values of 3.17 eV, 2.85 eV, 3.36 eV, and 3.80 eV calculated previously using LDA+\(U\) and GGA+\(U\) approaches [12; 16; 17; 26].