## 3 Results and Discussion
### Magnetic Structure of a-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\)
In particular, the ground-state energy (\(E_{tot}\)) of AFM-2 ordered state is smaller than that of FM state by 0.189, 0.158, or 0.116 eV (per primitive cell) when calculated using PBE, PBE0, or HSE functionals, respectively, see Table 4 for further details. Elastic Properties and Mechanical Stability of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\)  
Before discussing the calculated elastic and other properties of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) in detail, it is instructive to demonstrate its mechanical stability. For a monoclinic structure, the Born criteria of mechanical stability [49; 50] require that the following equations should be satisfied [51],  
\[ \begin{split}& C_{ii}>0\ (i=1,2,3,4,5,6),\ \ C_{11}+C_{12}+C_{33}+2(C_{12}+C_{13}+C_{23})>0,\\ & C_{33}C_{55}-C_{35}^{2}>0,\ \ C_{44}C_{66}-C_{46}^{2}>0,\ \ (C_{22}+C_{33}-2C_{23})>0,\\ &[C_{22}(C_{33}C_{55}-C_{35}^{2})+2C_{23}C_{25}C_{35}-C_{23}^{2}C_ {55}-C_{25}^{2}C_{33}]>0,\\ &\{2[a_{1}C_{15}C_{25}+b_{1}C_{15}C_{35}+c_{1}C_{25}C_{35}]-[d_{ 1}C_{15}^{2}+e_{1}C_{25}^{2}+f_{1}C_{35}^{2}]\}+g_{1}C_{35}>0.\end{split}  \tag{6}\]  
Here \(a_{1}=C_{33}C_{12}-C_{13}C_{23}\), \(b_{1}=C_{22}C_{13}-C_{12}C_{23}\), \(c_{1}=C_{11}C_{23}-C_{12}C_{13}\), \(d_{1}=C_{22}C_{33}-C_{23}^{2}\), \(e_{1}=C_{11}C_{33}-C_{13}^{2}\), \(f_{1}=C_{11}C_{22}-C_{12}^{2}\), and \(g_{1}=C_{11}C_{22}C_{33}-C_{11}C_{23}^{2}-C_{22}C_{13}^{2}-C_{33}C_{12}^{2}+2C _{12}C_{13}C_{23}\). The calculated elastic constants for \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) are collected in Table 5. According to Equation (6) and the obtained values, we can definitely conclude that \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) is mechanically stable since the calculated elastic constants meet the Born stability criteria. The calculated elastic constants \(C_{11}\), \(C_{22}\) and \(C_{33}\) are lower than 200 GPa, indicating that \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) has a relatively weak deformation resistance along the \(a\)-, \(b\)- and \(c\)-axes. Also, the elastic constants \(C_{44}\), \(C_{55}\) and \(C_{66}\) are calculated to be small. It can therefore be concluded that \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) has a weak shear deformation resistance. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**State** & **Functional** & **\(E_{tot}\) (eV)** & **\(M_{tot}\) (\(\mu_{\text{B}}\)/Cell)** & **\(M_{\text{Cu}}\) (\(\mu_{\text{B}}\))** & **\(M_{\text{P}}\) (\(\mu_{\text{B}}\))** & **\(M_{\text{O}}\) (\(\mu_{\text{B}}\))** \\ \hline FM & PBE & \(-140.547\) & 3.539 & 0.606 & 0.006 & 0.078 \\ \(\uparrow\uparrow\uparrow\uparrow\) & PBE0 & \(-189.559\) & 3.944 & 0.793 & 0.008 & 0.062 \\  & HSE & \(-179.043\) & 3.944 & 0.791 & 0.008 & 0.063 \\ \hline AFM-1 & PBE & \(-140.664\) & 0.000 & \(\pm 0.599\) & \(\pm 0.002\) & \(\pm 0.020\) \\ \(\downarrow\downarrow\uparrow\uparrow\) & PBE0 & \(-189.699\) & 0.000 & \(\pm 0.791\) & \(\pm 0.002\) & \(\pm 0.033\) \\  & HSE & \(-179.140\) & 0.000 & \(\pm 0.790\) & \(\pm 0.002\) & \(\pm 0.029\) \\ \hline AFM-2 & PBE & \(-140.736\) & 0.000 & \(\pm 0.579\) & \(\pm 0.003\) & \(\pm 0.034\) \\ \(\downarrow\uparrow\uparrow\downarrow\) & PBE0 & \(-189.717\) & 0.000 & \(\pm 0.787\) & \(\pm 0.002\) & \(\pm 0.028\) \\  & HSE & \(-179.159\) & 0.000 & \(\pm 0.785\) & \(\pm 0.002\) & \(\pm 0.028\) \\ \hline AFM-3 & PBE & \(-140.609\) & 0.000 & \(\pm 0.612\) & \(\pm 0.004\) & \(\pm 0.067\) \\ \(\uparrow\downarrow\uparrow\downarrow\) & PBE0 & \(-189.701\) & 0.000 & \(\pm 0.789\) & \(\pm 00.004\) & \(\pm 0.052\) \\  & HSE & \(-179.140\) & 0.000 & \(\pm 0.788\) & \(\pm 0.004\) & \(\pm 0.052\) \\ \hline FM & GGA [23] & \(-140.567\) & & 0.554 & & \\  & GGA+\(U\)[23] & \(-99.822\) & & 0.840 & & \\ AFM & GGA [23] & \(-140.598\) & & \(\pm 0.610\) & & \\  & GGA+\(U\)[23] & \(-99.830\) & & \(\pm 0.838\) & & \\ \hline \end{tabular}
\end{table}