## 3 Results and Discussion
### Magnetic Structure of a-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\)
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**State** & **Functional** & \(a\) **(Å)** & \(b\) **(Å)** & \(c\) **(Å)** & _\(\beta\)_ **(\({}^{\circ}\))** & **V (Å\({}^{3}\))** \\ \hline FM & PBE & 7.245 & 8.237 & 9.411 & 111.922 & 521.060 \\ \(\uparrow\uparrow\uparrow\uparrow\) & PBE0 & 7.072 & 8.115 & 9.291 & 111.003 & 497.660 \\  & HSE & 7.076 & 8.118 & 9.299 & 111.028 & 498.58 \\ \hline AFM-1 & PBE & 7.244 & 8.236 & 9.405 & 111.907 & 520.52 \\ \(\downarrow\downarrow\uparrow\uparrow\) & PBE0 & 7.066 & 8.112 & 9.251 & 110.914 & 495.36 \\  & HSE & 7.068 & 8.121 & 9.256 & 110.910 & 496.34 \\ \hline AFM-2 & PBE & 7.246 & 8.231 & 9.405 & 111.925 & 520.440 \\ \(\downarrow\uparrow\uparrow\downarrow\) & PBE0 & 7.063 & 8.112 & 9.252 & 110.876 & 495.260 \\  & HSE & 7.073 & 8.116 & 9.262 & 110.976 & 496.48 \\ \hline AFM-3 & PBE & 7.232 & 8.214 & 9.422 & 111.778 & 519.80 \\ \(\uparrow\downarrow\uparrow\downarrow\) & PBE0 & 7.062 & 8.108 & 9.257 & 110.886 & 495.18 \\  & HSE & 7.063 & 8.112 & 9.262 & 110.897 & 495.82 \\ \hline AFM & GGA \({}^{1}\) & 7.217 & 8.212 & 9.429 & 111.14 & 521.20 \\  & GGA+_U_\({}^{1}\) & 7.095 & 8.095 & 9.355 & 111.16 & 501.03 \\  & GGA+_U_+D2 \({}^{1}\) & 6.890 & 8.104 & 9.156 & 111.09 & 476.99 \\ \hline PM & XRD \({}^{2}\) & 6.876 & 8.113 & 9.162 & 109.54 & 481.67 \\  & XRD \({}^{3}\) & 6.895 & 8.113 & 9.164 & 109.62 & 482.86 \\  & XRD \({}^{4}\) & 6.887 & 8.108 & 9.156 & 109.54 & 481.59 \\  & XRD \({}^{5}\) & 6.901 & 8.108 & 9.176 & 109.65 & 483.53 \\ \hline \end{tabular} \({}^{1}\) Theory: PBE, also including Hubbard-_U_ and Van der Waals D2 corrections, Ref. [23]. \({}^{2}\) Experiment: X-ray diffraction (XRD), powder, Cu Ka, Ref. [15]. \({}^{3}\) Experiment: XRD, single-crystal, Mo K\(\alpha\), Ref. [14]. \({}^{4}\) Experiment: XRD, powder, Cu Ka, Ref. [22]. \({}^{5}\) Experiment: XRD, nanocrystals, Cu Ka, Ref. [23].  
\end{table} 
Table 1: Unit cell parameters of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in comparison with previous theoretical [23] and room-temperature experimental [14; 15; 22; 23] results. Up and down arrows denote a possible order of magnetic moments on sites Cu1–Cu4, see Figure 1.  
paramagnetic (PM) at room temperature, with an effective paramagnetic moment of 1.89 \(\mu_{B}\) corresponding to spin-1/2 moment on each Cu\({}^{2+}\) ion [12]. Another structural degree of freedom is related to the [P\({}_{2}\)O\({}_{7}\)]\({}^{4-}\) polyhedron, consisting of two corner-sharing [PO\({}_{4}\)]\({}^{3-}\) tetrahedra, as shown in Figure S2. The structure of [PO\({}_{4}\)]\({}^{3-}\) polyhedron may be perturbed by changing the angle between these two [PO\({}_{4}\)]\({}^{3-}\) tetrahedral sub-units and the values of P-O bond lengths. The P-O(1)-P bond angle is calculated to be about 154.736\({}^{\circ}\), which is consistent with the experimental value of 157\({}^{\circ}\)[15]. The obtained inner and outer P-O bond lengths for [P\({}_{2}\)O\({}_{7}\)]\({}^{4-}\) polyhedron are 1.585 A and 1.539 A, respectively, in agreement with the corresponding values of 1.58 A and 1.53 A measured in Ref. [15]. The calculated volume of [PO\({}_{4}\)]\({}^{3-}\) tetrahedron is about 1.9 A\({}^{3}\). Intriguingly, the lattice parameters in Tables 1 and 2 show very slight differences among the four studied magnetic configurations. It is worth mentioning that, compared with the PBE functional, all of these values shrink when a hybrid density functional, PBE0 or HSE, is used. The calculated formation energies (\(\Delta E_{f}\) and \(\Delta E_{f}^{ox}\)) of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in the four considered states of magnetic order are collected in Table 3. The obtained negative values indicate an exothermic formation process and the thermodynamic stability of the compound. By comparing the experimental and calculated formation energies \(\Delta E_{f}\) listed in Table 3, one may get an impression that PBE works slightly better than the hybrid functionals for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\).