## 3 Results and Discussion
### Magnetic Properties of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\)  
The alignment of magnetic moments arises from a competition between two physical mechanisms, the intraatomic exchange splitting and the interatomic interactions involving Cu 3\(d\) states. The magnitude of the magnetic moment increases with the strength of the exchange interaction. As shown in Table 2 and Figure 4, the shortest Cu-Cu separation distance of 3.056 A, occurs between Cu\({}_{1}\) and Cu\({}_{2}\) ionic positions, as well as between Cu\({}_{3}\) and Cu\({}_{4}\) positions in the structure. For comparison, the bond length between Cu\({}_{2}\) and Cu\({}_{3}\) positions is 3.261 A. The short interatomic distance is expected to result in a strong direct interaction between the open 3\(d\) shells of these Cu\({}^{2+}\) ions. The unoccupied Cu 3\(d\) states contribute to the narrow conduction band lying at about 4 eV in Figure 2d that forms a sharp DOS peak at the same energy in Figure 3c. All calculated information about the total and partial magnetic moments is collected in Table 4. Previously calculated values of magnetic moments are also given for comparison. With regards to FM ordering, the conduction bands come totally from spin-down states, with the dominant contribution due to Cu 3\(d\) states with a slight exchange splitting, as shown in Figure 2b. The splitting signifies a higher magnetic moment of the Cu atom. Within HSE functional the partial magnetic moments of Cu, P, and O in the FM case of orders are calculated to be 0.791 \(\mu_{B}\), 0.008 \(\mu_{B}\) and 0.063 \(\mu_{B}\), respectively. The spin density in the interstitial region is rather small: it adds up to about 0.12 electrons per primitive cell in the calculations employing HSE functional. The total magnetic moment of the FM state is 3.944 \(\mu_{B}\) per primitive cell, which is close to the integer value corresponding to one \(d\)-orbital hole per each of the four Cu\({}^{2+}\) ions in the cell. For comparison, the local atomic moments are 0.785 \(\mu_{B}\) for Cu, 0.002 \(\mu_{B}\) for P, and 0.028 \(\mu_{B}\) for O in AFM-2 ordered state, in which the net magnetic moment is 0 \(\mu_{B}\) since the total DOS is symmetrical in the spin up and spin down channels. As Table 4 shows, the magnetic moments obtained in this work are in good agreement with the results of previous GGA+\(U\) calculations [23]. The local magnetic moments computed in this work are characteristic of Cu oxidation state 2+, which is in line with the paramagnetic moment of 0.15 emu/g [60] or 1.89 \(\mu_{B}\)[12] experimentally reported for Cu\({}_{2}\)P\({}_{2}\)O\({}_{5}\) in the rage from \(-\)123 to \(+\)127 \({}^{cit}\)C. Some nanocrystalline specimens at room temperature have been found to exhibit a weak ferromagnetic or superparamagnetic response to an applied strong magnetic field [17; 60]. Unfortunately, no experimental data on local moments are available for the ordered antiferromagnetic state. Theoretically, \(\beta\)-Cu\({}_{2}\)V\({}_{2}\)O\({}_{7}\) has been proposed to be a quasi-one-dimensional spin system composed of magnetic Cu\({}^{2+}\) ions and a local magnetic moment of 0.73 \(\mu_{B}\) per Cu atom was obtained from GGA+\(U\) calculations [61]. It is consistent with the values calculated in this work for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\), due to structural similarities between the two phases where the Cu\({}^{2+}\) ions are arranged into the [CuO\({}_{4}\)] plaquettes linked via two corner-sharing [PO\({}_{4}\)] tetrahedra. For comparison, the shortest Cu-Cu distance is of 2.9974 A in \(\alpha\)-Cu\({}_{2}\)V\({}_{2}\)O\({}_{7}\), which is slightly shorter than the Cu\({}_{1}\)-Cu\({}_{2}\) (or Cu\({}_{3}\)-Cu\({}_{4}\)) bond length of 3.056 A calculated here for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in AFM-2 ordered state, see Figure 4. The length of the next-nearest neighbor bond, Cu\({}_{2}\)-Cu\({}_{3}\), is 3.266 A as obtained here from HSE functional calculations for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in AFM-2 state.