## 3 Results and Discussion
### Magnetic Structure of a-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\)
However, this is due to the fact that the energies of substances with very different types of chemical bonding are involved into Equation (1), so that the errors of DFT functionals do not cancel out. However, the experimental result for the enthalpy of formation of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) from the binary oxides at 298 K, \(\Delta E_{f}^{ox}\), \(\approx\) -279.0 \(\pm\) 1.4 kJ/mol [22] while the calculated values (for the ground-state AFM-2 structure) using the PBE, PBE0, and HSE functional translate into \(-\)145.8, \(-\)252.2 and \(-\)252.8 kJ/mol, respectively. One can clearly see that hybrid functionals PBE0 and HSE give formation energy values that are consistent with the experimental result for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\), while the value derived using a semi-local PBE functional is considerably off. Notably, the calculated \(\Delta E_{f}\) and \(\Delta E_{f}^{ox}\) are the lowest for the AFM-2 ordered state as compared with the other three states of magnetic order, when calculated with any of \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**State** & **Functional** & \(\mathbf{\Delta E_{f}}\) & \(\mathbf{\Delta E_{f}^{ox}}\) & \(\mathbf{E_{g}}\) & \(\mathbf{\Delta Q_{Cu}}\) & \(\mathbf{\Delta Q_{P}}\) & \multicolumn{3}{c}{\(\mathbf{\Delta Q_{Q}}\)} \\ \hline FM & PBE & \(-\)22.616 & \(-\)1.416 & 0 & \(-\)1.19 & \(-\)3.59 & 1.36 & 1.35 & 1.51 & 1.32 \\ \(\uparrow\uparrow\uparrow\uparrow\) & PBE0 & \(-\)24.453 & \(-\)2.534 & 3.784 & \(-\)1.35 & \(-\)3.74 & 1.45 & 1.44 & 1.56 & 1.42 \\  & HSE & \(-\)26.741 & \(-\)2.562 & 3.084 & \(-\)1.35 & \(-\)3.74 & 1.45 & 1.44 & 1.56 & 1.42 \\ \hline AFM-1 & PBE & \(-\)22.671 & \(-\)1.475 & 0.390 & \(-\)1.19 & \(-\)3.64 & 1.39 & 1.36 & 1.52 & 1.32 \\ \(\downarrow\downarrow\uparrow\uparrow\) & PBE0 & \(-\)24.519 & \(-\)2.604 & 4.344 & \(-\)1.35 & \(-\)3.79 & 1.46 & 1.46 & 1.58 & 1.42 \\  & HSE & \(-\)26.785 & \(-\)2.610 & 3.566 & \(-\)1.35 & \(-\)3.79 & 1.46 & 1.46 & 1.58 & 1.42 \\ \hline AFM-2 & PBE & \(-\)22.704 & \(-\)1.511 & 0.730 & \(-\)1.19 & \(-\)3.64 & 1.39 & 1.36 & 1.52 & 1.32 \\ \(\downarrow\uparrow\uparrow\downarrow\) & PBE0 & \(-\)24.530 & \(-\)2.614 & 4.670 & \(-\)1.35 & \(-\)3.79 & 1.46 & 1.46 & 1.58 & 1.42 \\  & HSE & \(-\)26.796 & \(-\)2.620 & 3.966 & \(-\)1.35 & \(-\)3.79 & 1.46 & 1.46 & 1.58 & 1.42 \\ \hline AFM-3 & PBE & \(-\)22.649 & \(-\)1.447 & 0.362 & \(-\)1.20 & \(-\)3.65 & 1.39 & 1.36 & 1.52 & 1.33 \\ \(\uparrow\downarrow\uparrow\downarrow\) & PBE0 & \(-\)24.530 & \(-\)2.605 & 4.395 & \(-\)1.35 & \(-\)3.79 & 1.46 & 1.46 & 1.58 & 1.42 \\  & HSE & \(-\)26.785 & \(-\)2.611 & 3.683 & \(-\)1.35 & \(-\)3.79 & 1.46 & 1.46 & 1.58 & 1.43 \\ \hline  & Previous & \(-\)21.725 \({}^{1}\) & \(-\)2.892 \({}^{1}\) & 2.85 \({}^{2}\), 3.17 \({}^{3}\), & & & & & & \\  & studies & & & 3.36 \({}^{4}\), 3.79 \({}^{5}\) & & & & & & & \\ \hline \end{tabular} \({}^{1}\) Experiment (calorimetry) Ref. [22]. \({}^{2}\) Theory (GGA+_l_) Ref. [26]. \({}^{3}\) Theory (GGA+_l_) Ref. [23]. \({}^{4}\) Theory (LDA+_l_) Ref. [12]. \({}^{5}\) Experiment (UV-spectrometry) Ref. [17].  
\end{table} 
Table 3: Calculated formation energy \(\Delta E_{f}\) (eV/f.u. ), formation energy from oxides \(E_{f}^{ox}\) (eV/f.u. ), band gap \(E_{g}\) (eV), and effective atomic charges \(\Delta Q_{Cu}\), \(\Delta Q_{P}\), and \(\Delta Q_{O}\), in \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\). Elementary charge units (_e_) are used; negative charge values denote a lack of electrons. Previous experimental and theoretical values are quoted for comparison. the PBE, PBE0, or HSE functionals. This demonstrates that AFM-2 is indeed the most stable ordered structure, which supports the findings of previous studies that \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) exhibits an antiferromagnetic ordering bellow 22-27 K [12; 23; 24; 25; 26].