## 4 Conclusions
**Supplementary Materials:** The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13010498/s1](https://www.mdpi.com/article/10.3390/app13010498/s1)]([https://www.mdpi.com/article/10.3390/app13010498/s1](https://www.mdpi.com/article/10.3390/app13010498/s1)), Formalism based on Voigt-Reuss-Hill (VRH) approximation for the evaluation of direction-averaged bulk modulus (B) and shear modulus (G) [52; 53; 54]; Figure S1: Optimized crystal structures of (a) CuO and (b) P\({}_{2}\)O\({}_{5}\); Figure S2: The local structure of a pyrophosphate group [P\({}_{2}\)O\({}_{7}\)]\({}^{4-}\) in \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\); Figure S3: Electronic energy band structures calculated using PBE functional for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in (a) FM, (b) AFM-1, (c) AFM-2, and (d) AFM-3 magnetic states; Figure S4: Electronic energy band structures calculated using PBE0 functional for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in (a) FM, (b) AFM-1, (c) AFM-2, and (d) AFM-3 magnetic states; Figure S5: Different representations of the Cu local environment: (a) the [CuO\({}_{5}\)] pyramid and (b) the [CuO\({}_{4}\)] distorted plaquette; Figure S6: Charge density in the (\(-\)1.5, 1, \(-\)2.5) plane with [Cu\({}_{2}\)O\({}_{6}\)] layer in \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in four different magnetic states calculated using HSE functional. The isosurface level of the charge density is set at 0.3 \(e\)/A\({}^{3}\). **Author Contributions:** Conceptualization, P.K. ; methodology, X.Y. ; software, P.K. ; validation, X.Y. and P.K. ; formal analysis, X.Y. and P.Z. ; investigation, X.Y. and P.K. ; resources, P.K. ; data curation, X.Y. ; writing--original draft preparation, X.Y. ; writing--review and editing, P.K. ; visualization, P.K. and P.Z. ; supervision, P.K. ; project administration, P.K. ; funding acquisition, X.Y. and P.K. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was performed at the Vinnova Competence Centre \"Hero-m 2i\", funded jointly by the Swedish Governmental Agency for Innovation Systems (Vinnova, grant number 2016-00668), Swedish industry, the Swedish steel producers' association Jernkontoret, and KTH Royal Institute of Technology. P.K. acknowledges support by Svensk Karnbar'shelantering AB (SKB). X.Y. sincerely appreciates the support of National Natural Science Foundation of China (Grant No. 11705152 and 11974055) and by Sfitelsen for Tillampad Termodynamik, Sweden. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. Data Availability Statement: The data presented in this study are available in the article and supplementary material. The computational resources are provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre (NSC) in Linkoping, Sweden, and at the Center for High-Performance Computing (PDC) in Stockholm, Sweden, partially funded by the Swedish Research Council through grant agreement No. 2018-05973. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: \begin{tabular}{l l} AFM & Antiferromagnetic \\ BZ & Brillouin zone \\ DFT & Density functional theory \\ DOS & Density of states \\ FM & Ferromagnetic \\ GGA & Generalized gradient approximation \\ HSE & Heyd-Scuseria-Erzzerhof hybrid functional \\ LDA & Local density approximation \\ PAW & Projector-augmented-wave \\ PBE & Perdew-Burke-Erzzerhof semi-local functional \\ PBE0 & Perdew-Burke-Erzzerhof hybrid functional \\ PDOS & Partial density of states \\ PhDOS & Phonon density of states \\ PM & Paramagnetic \\ QHA & Quasi-harmonic approximation \\ TDOS & Total density of states \\ VASP & Vienna Ab initio Simulation Package \\ XRD & X-ray diffraction \\ \end{tabular}