## 3 Results and Discussion
### Phonon Dispersion Curves and Lattice Thermodynamic Properties  
In addition to thermodynamic stability and mechanical stability, dynamical stability can also decide the structural stability of a material. Thus the phonon dispersion curves along the Y\(-\Gamma-\)A\(-\)M\(-\Gamma-\)L\(-\)N directions in the BZ have been calculated for the lowest-energy AFM-2 configuration using PBE functional at the PBE-optimized volume and are displayed in Figure 6a; the corresponding phonon density of state (PhDOS) is plotted in Figure 6b, see also Figure 2a for BZ path notations. It can be seen that there are no imaginary phonon frequencies, implying the dynamical stability of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\) which depends on the vibration frequencies of all atoms in the material. The entire phonon spectrum consists of five separate frequency bands. The low-frequency band below 12 THz includes the acoustic and optical branches of vibrations involving all ions, Cu, P, and O. The broad optical band between 13 and 18 THz may be associated with bending modes of pyrophosphate [P\({}_{2}\)O\({}_{7}\)]\({}^{4-}\) groups consisting in changing the P\(-\)O\(-\)P bridge angle. Then there occurs a distinct gap in the phonon dispersion relations with a narrow vibrational band at about 20 THz associated with stretching modes of the P\(-\)O\(-\)P bridge. The two high-frequency bands in the range from 25 to 34 THz above the gap are mainly due to O\(-\)P bond stretching modes. Temperature dependencies of the total energy, Helmholtz free energy, entropy, and specific heat contributions from the lattice vibrations have been evaluated by integrating the phonon density of states obtained from the interaction force constants calculated in the quasi-harmonic approximation (QHA) [56; 63]. In particular, the Helmholtz free energy (\(F\)) as a function of temperature (\(T\)) and volume (\(V\)) is expressed as following equation,  
\[F(V,T)=E(V)+F_{\rm ph}(V,T)+F_{\rm ele}(V,T), \tag{7}\]  
where \(E(V)\) is the electronic ground-state total energy of the system, \(F_{\rm ph}(V,T)\) denotes the phonon-free energy, and \(F_{\rm ele}(V,T)\) is the electronic excitation energy at a given unit cell volume (\(V\)) and temperature (\(T\)). For wide-band-gap semiconductors at moderate temperatures, the electronic free energy contribution may safely be neglected. Under the QHA, the phonon contributions to Helmholtz free energy \(F(V,T)\), the internal energy (\(E\)), and the vibrational entropy (\(S\)) can be obtained from PhDOS g(\(\omega\)) using the following formulas [64],  
\[F(V,T)=3 nNk_{\rm B}T\int_{0}^{\infty}g(\omega)\ln[2\sinh(\frac{\hbar\omega}{2k _{\rm B}T})]d\omega, \tag{8}\]  
Figure 6: Phonon dispersions (**a**) and phonon density of states (**b**) of \(\alpha\)-Cu\({}_{2}\)P\({}_{2}\)O\({}_{7}\), as calculated within PBE density functional. For BZ path notations, see Figure 2.  
\[E(V,T)=3 nN\frac{\hbar}{2}\int_{0}^{\infty}\omega\coth\ (\frac{\hbar\omega}{2k_{\rm B }T})g(\omega)d\omega, \tag{9}\]  
\[S(V,T)=3 nNk_{\rm B}\int_{0}^{\infty}[\frac{\hbar\omega}{2k_{\rm B }T}\coth\ (\frac{\hbar\omega}{2k_{\rm B}T})-\ln\{2\sinh(\frac{\hbar\omega}{2k_{\rm B }T})\}]g(\omega)d\omega, \tag{10}\]  
where \(\omega=\omega(V)\) stands for the frequency of the phonon mode calculated at a primitive cell volume \(V\), \(g(\omega)\) is the phonon density of state with the norm \(\int_{0}^{\infty}g(\omega)d\omega=1\). \(F(V,T)\) and \(E(V,T)\) at zero temperature are both equal to the zero-point energy, which is given by \(F_{0}=E_{0}=3 nN\int_{0}^{\infty}\frac{\hbar\omega}{2}g(\omega)d\omega\), where \(n\) is the number of atoms per primitive cell, \(N\) is the number of primitive cells. Here \(\hbar\) and \(k_{\rm B}\) are the Plank constant and the Boltzmann constant, respectively.