## 4 Conclusions  
To summarize, we have carried out a comparative hybrid-density functional study (using PBE, PBE0, and HSE functionals) of electronic structure, magnetic ordering, phonon  
Figure 7: Temperature-dependent (**a**) total (internal) energy \(E\), (**b**) Helmholtz free energy \(F\), (**c**) entropy \(S\) and (**d**) heat capacity \(C_{V}\) of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) calculated per mole (of formula units) at a molecular volume of 130.268 Å\({}^{3}\). dispersion relations, and thermodynamic properties of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\). First of all, the calculated formation energy, cohesive energy, elastic constants, and phonon dispersion relations indicate that \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) is mechanically and thermodynamically stable. The mechanical property indicators based on elastic moduli reveal that \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) should exhibit a ductile behavior with a relatively weak volume deformation resistance and shear deformation resistance. Secondly, the spin-polarized calculations indicate that the FM, AFM-1, AFM-2, and AFM-3 magnetic configurations have rather similar electronic structures, except for some slight differences at the bottom of the conduction band where the hybridization between the electron states of Cu and O atoms in the AFM-2 structure is so strong that the conduction band is shaped almost like a flat level. Thus AFM-2 magnetic ordering produces the largest band gap of 3.966 eV and the smallest local magnetic moments on Cu atoms within HSE functional calculations. Correspondingly, the states at the valence band maximum are mainly dominated by O states with a small contribution due to Cu electron states, particularly from Cu-\(d_{x^{2}-y^{2}}\) orbital. Thirdly, by comparing the energies for the different types of magnetic ordering in \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\), we find that AFM-2 ordering is the most stable structure having the strongest antiferromagnetic interactions. Specifically, the proposed one-dimensional spin system in AFM-2 ordering originates from \(-\)Cu\({}_{1}\)-Cu\({}_{2}\)-Cu\({}_{3}\)-Cu\({}_{4}\)- chains along the \(b\) axis, in which there are two identical Cu\(-\)O\(-\)Cu paths, each forming a bond angle of 100.301\({}^{\circ}\). Within HSE functional, the partial magnetic moments on Cu atoms in FM, AFM-1, AFM-2 and AFM-3 ordered states are 0.791 \(\mu_{\rm B}\), \(\pm\)0.790 \(\mu_{\rm B}\), \(\pm\)785 \(\mu_{\rm B}\) and \(\pm\)0.788 \(\mu_{\rm B}\), respectively. Fourthly, phonon dispersion relations indicate that there are four ranges of vibrational frequencies: (1) 0-12 THz contributed by acoustic and optical modes involving Cu, P, and O atoms, (2) 13-18 THz mainly dominated by vibrations corresponding to bending of pyrophosphate groups; (3) a narrow band at 20 THz due to the stretching modes of a P-O-P bridge; (4) two bands in the range 25-34 THz due to O-P stretching modes. Our systematic study on structural stability, electronic and magnetic properties, and phonon dispersion relations of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) opens a way for further computational studies of related materials such as transition metal pyro compounds, M\({}_{2}\)X\({}_{2}\)O\({}_{7}\).