## 4 Results and Discussion
### Optical Properties  
The optical properties for the five pyrophosphate crystals are calculated in the form of complex dielectric function based on the one-electron theory of interband optical transition. The calculated real (\(\epsilon_{1}\)) and imaginary (\(\epsilon_{2}\)) parts of frequency-dependent dielectric function are shown in black and red color respectively in Figure 7 (center panel (a) to (e)). Optical absorption spectra in all five crystals above the absorption threshold of 5.0 eV show five peaks--A, B, C, D, and E--roughly in the ultraviolet region. This feature is related to the similarity in the TDOS of Figure 3 dominated by the PDOS of the H\({}_{2}\)P\({}_{2}\)O\({}_{7}\) unit. However, for the absorption spectra for the Cu containing crystals C3 and C4 in Figure 7c,d, additional absorptions occur in the 0.0 eV to 5.0 eV region, which are in the infrared, visible, ultraviolet spectral region (see the far left column in Figure 7). This is due to the presence of unoccupied part of the metallic band discussed in Section 4.1. This made the optical properties in C4 resemble a metallic material with huge peaks at energy close to 0.0 eV. On the other hand, the optical properties for C3 are quite reasonable, despite the same chemical composition and formula with C4. The optical absorption (\(\epsilon_{2}\)) in C3 occurs at a higher photon energy than in C4. After Kramers-Kronig transformation of \(\epsilon_{2}\) to \(\epsilon_{1}\), \(\epsilon_{1}\) at zero is only slightly larger than those in C1, C2, and C5 but are still reasonable. A plausible explanation on the difference between C3 and C4 is that they have different crystal symmetry (space group \(P\)-1(Ci) and \(Pnma\)(D\({}_{2n}\)[16]), respectively) despite these two Cu-containing pyrophosphates have the same chemical formula. This is one of the very rare examples that the crystal symmetry can play a critical role in the optical properties in the infrared frequency region.  
The refractive index for the five pyrophosphate crystals can be obtained from the square root of zero-frequency limit of real part of the dielectric function (\(\epsilon_{1}\)(0)). They are listed in Table 3 and are in the range of 1.44 to 1.53. The value of n = 17.06 for C4 is questionable due to large absorption in the infrared region. These refractive index values can be compared with 60% glucose solution in water (1.44) [61], BaF2 (1.47) [62], CaF2 (1.43) [62], SrF2 (1.44) [62], CsF (1.48) [62], RbBr (1.55) [62], and RbCl (1.49) [62] etc.  
In the right panel of Figure 7a'-e', we display the calculated energy loss function (ELF) for the five crystals whose optical absorption spectra are shown in the center panel. The peak of ELF is known as plasma frequency (\(\omega_{\rm p}\)), \(\omega_{\rm p}\) is the frequency of collective excitation of electrons in solid at high energy  
Figure 6: Shear modulus G vs bulk modulus K for the 5 pyrophosphates and other selected crystals such as chalcogenides [56], oxide glasses [57], organic molecules [58], alkali metal halides [59; 60], thallium halides [60] within the same range of G and K. The slope of the dashed lines gives the G/K values 0.611 and 0.457.  
and can be easily measured experimentally. The \(\omega_{\rm p}\) for all five crystals are listed in Table 3. They range from 20.85 eV in C2 to 23.14 eV in C3.