## 2 Methodology  
### Computational Details  
In this work, density functional theory (DFT) [27] based calculations for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) have been performed using the projector-augmented-wave (PAW) type pseudopotentials with the _Vienna Ab initio Simulation Package_ (VASP) code [28; 29]. Four possible magnetic configurations are considered, FM, AFM-1, AFM-2, and AFM-3, see Figure 1. Three different approximations to treat the exchange and correlation contributions have been applied: (i) the PBE approach utilizing a standard generalized gradient approximation scheme [30], (ii) the hybrid PEB0 approach [31; 32] and (iii) the hybrid HSE method [33; 34] as implemented in VASP. In the PBE0 functional, 25% of PBE exchange has been replaced by the exact non-local exchange showing the normal \(1/r\) kernel. When the degree of exact exchange \(\alpha\) is allowed to vary, \(E_{x}^{\text{PBE0}}(\alpha)=\alpha E_{x}^{\text{exact}}+(1-\alpha)E_{x}^{\text {PBE}}\); such a case is referred to as the generalized class of functionals [31; 32]. Both PBE0 and HSE functionals used in this work mix 25% of the exact Hartree-Fock (HF) exchange with 75% of the PBE exchange functional in their short-range part. However, in the HSE functional, the long-range part of the exact exchange is smoothly replaced with the local expression of the PBE exchange. The valence basis set included \(3d\) and \(4s\) states on Cu, \(3s\) and \(3p\) states on P, and \(2s\) and \(2p\) states on O. A cutoff energy of 520 eV was chosen for the plane-wave basis set. For each considered functional and magnetic configuration, full structural optimization was performed including the shape and volume of the unit cell. Internal atomic positions were fully relaxed until the Hellmann-Feynman forces on them were less than 0.01 eV/A. A \(3\times 3\times 3\) Monkhorst-Pack \(k\)-point mesh [35] was used to integrate over the Brillouin zone (BZ) for each configuration of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\), whose unit cell contained 22 atoms. A dense \(5\times 5\times 3\)\(k\)-point mesh was selected to compute the density of states (DOS) and the charge density distributions. Spin polarization was turned on in all the calculations. To have a quantitative description of the charge distribution and transfer, the Bader charge analysis was employed [36]. Phonon frequencies were calculated based on the density functional perturbation theory (DFPT) as implemented in the VASP and PHONOPY [37] packages for the lowest-energy AFM-2 configuration using PBE functional at the PBE-optimized volume. In the interpolation of the force constants for the phonon dispersion curve calculations, \(3\times 3\times 3\)\(k\)-point mesh in the first BZ was used. Vibrational contributions to  
Figure 1: Optimized crystal structures for four collinear configurations of magnetic moments on Cu atoms in \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\), one ferromagnetic (FM) and three antiferromagnetic (AFM). (**a**) FM; (**b**) AFM-1; (**c**) AFM-2; (**d**) AFM-3. Large blue, grey, and red balls denote copper, phosphorus, and oxygen atoms, respectively. thermodynamic properties such as (free) energy, entropy, and heat capacity were evaluated, as functions of temperature, from the computed phonon dispersion relationships of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) using the methodology detailed in Section 3.5. The relative chemical stability of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) was calculated according to the definition of formation energy (per formula unit),  
\[\Delta E_{f}=\frac{1}{2}\{E(\mathrm{Cu}_{4}\mathrm{P}_{4}\mathrm{O}_{14})-4 \mu(\mathrm{Cu})-4\mu(\mathrm{P})-14\mu(\mathrm{O})\}, \tag{1}\]  
where \(E(\mathrm{Cu}_{4}\mathrm{P}_{4}\mathrm{O}_{14})\) is the total ground-state energy of the primitive cell containing two Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) formula units and \(\mu(\mathrm{Cu})\), \(\mu(\mathrm{P})\) and \(\mu(\mathrm{O})\) are the chemical potentials associated with the pure elements in their standard states [38].