## 3 Results and Discussion
### Magnetic Properties of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\)
From Figure S6, one can see that the covalent component of bonding between Cu and O ions is very pronounced; the effect facilitates itself in the form of electron density streaks connecting the neighboring atoms and may be mainly attributed to the hybridization between Cu-3\(d\) and O-2\(p\) atomic orbitals upon the formation of the chemical bond. Also, Figure 4 shows that electron density accumulates near the short Cu\({}_{1}\)-Cu\({}_{2}\) bonds; somewhat weaker density accumulation may be seen near the longer Cu\({}_{2}\)-Cu\({}_{3}\) bonds. Thus, chemical bonding produces zig-zag \(-\)Cu\({}_{1}\)\(-\)Cu\({}_{2}\)\(-\)Cu\({}_{3}\)\(-\)Cu\({}_{4}\)- chains along the \(b\)-axis in the structure of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) with AFM-2 ordering. The occurrence of these chains provides valuable theoretical support and a qualitative explanation to the hypothesis that transition metal pyro compounds (M\({}_{2}\)X\({}_{2}\)O\({}_{7}\)) are quasi-one-dimensional spin systems composed of magnetic Cu\({}^{2+}\) ions, as proposed by [PERSON] et al. [12] for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) and by [PERSON] et al. [11] for \(\alpha\)-Cu\({}_{2}\)As\({}_{2}\)O\({}_{7}\). Contour plots (isosurfaces) of the spin density are depicted in Figure 5 for the studied magnetic configurations of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\). The Figure clearly shows that the magnetic moments of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) arise from Cu atoms; the O and P atoms are essentially nonmagnetic and the spin density in the interstitial region is negligible. This is in accordance with the above-discussed results on the calculated magnetic moments. Spin densities for these four cases look similar: the colored clouds, indicating loci of accumulated spin-up and spin-down density, are almost identical for the different Cu positions. Particularly, the spin density along the Cu chains is localized in one orbital, i.e., \(d_{x^{2}-y^{2}}\). Besides, one can see small clouds of spin density accumulated on O atoms, which is of importance for opening the superexchange channels like Cu\({}_{1}\)\(-\)O\({}_{1}\)\(-\)Cu\({}_{2}\) and Cu\({}_{1}\)\(-\)O\({}_{2}\)\(-\)Cu\({}_{2}\). Based on the above discussion, we can generally conclude that all four studied cases of magnetic ordering are reasonable with the consideration of electron interactions of each Cu atom with its first, second, and third nearest-neighbor atoms. After a more comprehensive analysis, we can conclude that the energetically preferable magnetic state in \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) is AFM-2 ordering with quasi-one-dimensional antiferromagnetic \(-\)Cu\({}_{1}\)\(-\)Cu\({}_{2}\)\(-\)Cu\({}_{3}\)\(-\)Cu\({}_{4}\)\(-\) chains along the \(b\) axis, in which two identical Cu\(-\)O\(-\)Cu paths make an angle of 100.301\({}^{\circ}\) to each other. Figure 4: **(a)** Atomic structure of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) for AFM-2 state in the (0, 1, 29.5) cross-section plane chosen for charge density calculation. **(b)** Contour plot of the charge density calculated in the (0, 1, 29.5) cross-section plane within HSE density functional. The isosurface level of the charge density is set at 0.3 \(e/\)Å\({}^{3}\); the image is produced using VESTA [62]. The white solid and dotted lines indicate, respectively, the distances of 3.056 Å for \(d_{\text{Cu}_{1}-\text{Cu}_{2}}\) bond and 3.266 Å for \(d_{\text{Cu}_{2}-\text{Cu}_{3}}\) bond. Figure 5: Spin density of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in **(a)** FM, **(b)** AFM-1, **(c)** AFM-2 and **(d)** AFM-3 states of magnetic order, as calculated within HSE density functional. The isosurface level of the charge density is set at 0.2 \(e/\)Å\({}^{3}\); the image is produced using VESTA [62].