## 3 Results and Discussion
### Electronic and Magnetic Structure of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\)
But as a whole, HSE results seem to be in better agreement with the experiment than PBE0 results. Intriguingly, the total energy calculations show that the most favorable magnetic structure of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) is AFM-2, for which the largest band gap value is obtained no matter which functional (PBE, PBE0 or HSE) is used. Besides, the fat band structures calculated within HSE functional are depicted in Figure 2. They are resolved into the atoms to gain more physical insights into the electronic structure variations, in which blue-, cyan- and red-colored balls denote the contribution by Cu, P, and  
Figure 2: Fat band structures for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in (**a**) FM spin-up channel, (**b**) FM spin-down channel, (**c**) AFM-1, (**d**) AFM-2, and (**e**) AFM-3 magnetic states calculated within HSE density functional. The zero energy is chosen at the highest occupied level at 0 K. The inset in panel (**a**) shows the BZ shape and paths used in the electronic and phonon spectrum calculations. O atoms, respectively. Similar to the four magnetic states, one finds that the valence bands are mainly dominated by O atoms with little contribution from Cu atoms. In contrast, the conduction bands are dominated by hybridized states of Cu and O atoms. What makes FM ordering special is that the conduction band states, in that case, are completely in the spin-down channel and belong to both Cu and O atoms, as shown in Figure 2b. It is worth mentioning that the hybridization of Cu and O atoms at the bottom of the conduction band of AFM-2 configuration is so strong that it produces an almost flat level, which leads to the smallest magnetic moments on Cu atoms, as shown in Figure 2d. Relative to that case, the exchange coupling of Cu and O atoms is weak and split into wider band energy regions in the cases of FM, AFM-1, and AFM-3 ordering. Based on above discussion one can conclude that \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) is an antiferromagnetic semiconductor with a band gap of \(\sim\)3.96 eV. To get a deeper insight into the band structure, the spin-polarized total density of states (TDOS) and partial density of states (PDOS), computed using HSE functional, have been plotted together in Figure 3. In all the four cases of magnetic ordering, the broad oxygen bands are located between \(-\)9 and 0 eV, with an admixture of Cu 3\(d\) band states above \(-\)5.5 eV corresponding to the [CuO\({}_{5}\)] pyramid. Specifically, the top of the valence band (VBM) is primarily composed of O-2\(p\) electron states with a slight admixture of Cu-3\(d\) states, particularly Cu \(d_{x^{2}-y^{2}}\) orbital states. Oppositely, the CBM is mainly composed of Cu-3\(d\) electron states hybridized with O-2\(p\) states, especially in the AFM-2 case, which is resolved as the magnetically active orbital [11]. This relevant hybridization was identified using Wannier functions and LDA for Cu \(d_{x^{2}-y^{2}}\) states [11; 58]. Figure S5 zooms into the magnetic layers of [Cu\({}_{2}\)O\({}_{6}\)] structural dimers, containing two edge-sharing [CuO\({}_{4}\)] plaquettes, which are separated from each other by nonmagnetic [PO\({}_{4}\)] tetrahedra in \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\). More exactly, the spacer layers consist of [P\({}_{2}\)O\({}_{7}\)] pyrophosphate groups formed by joining two [PO\({}_{4}\)] tetrahedra into a corner-sharing structural unit. Thus, the magnetic orbital lies in the plane of [CuO\({}_{4}\)] plaquette which for two possible magnetic coupling pathways, i.e., Cu\({}_{1}\)-O\({}_{1}\)-Cu\({}_{2}\) and Cu\({}_{1}\)-O\({}_{2}\)-Cu\({}_{2}\) pathways. The charge density cross-sections along the (\(-\)1.5, 1, \(-\)2.5) plane for [Cu\({}_{2}\)O\({}_{6}\)] structural dimers in \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) at 0.3 \(e\)/A\({}^{3}\) are presented in Figure S6 for the four different magnetic states considered in this work.