## 2 Crystal Structures  
The five pyrophosphates in this paper are K\({}_{2}\)Mg(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O, (NH\({}_{4}\))\({}_{2}\)Zn(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O, K\({}_{2}\)Cu(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O, 4[K\({}_{2}\)Cu(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O], and 4[K\({}_{2}\)Zn(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O]. These crystals were synthesized using the wet method. This method is generally applied by the strategy followed by our scientific team ([PERSON], KB, and MH) to obtain multicomponent compounds such as acidic pyrophosphates studied in this work. This method is based on the preparation of a solution of the various precursors, which are generally sodium or potassium pyrophosphates (Na\({}_{4}\)P\({}_{2}\)O\({}_{7}\)(H\({}_{2}\)O)10, K\({}_{4}\)P\({}_{2}\)O\({}_{7}\)), NH\({}_{4}\)Cl, MgCl\({}_{2}\) (H\({}_{2}\)O)\({}_{6}\), ZnCl\({}_{2}\)(H\({}_{2}\)O)\({}_{4}\), and CuCl\({}_{2}\)(H\({}_{2}\)O)\({}_{2}\). We then resort to the solubilization of all these precursors by the addition of a strong acid of the HCl type (pH of solute must be controlled), sometimes with slow stirring. Afterward, slow evaporation at room temperature for all the solute leads to the formation of single crystals. These pyrophosphates have well-characterized crystal structure as well as vibrational analysis, Raman, and infrared spectra reported such as K\({}_{2}\)Mg(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O by Harcharras et al. Harcharras et al. (2012), (NH\({}_{4}\))\({}_{2}\)Zn(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O by Capitelli et al. Capitelli et al. (2013), and 4[K\({}_{2}\)Zn(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O] by [PERSON] et al. [PERSON] et al. (2014). K\({}_{2}\)Cu(H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O and 4[K\({}_{2}\)Cu (H\({}_{2}\)P\({}_{2}\)O)\({}_{7}\)\(\cdot\)2H\({}_{2}\)O] are same crystal structure with different space group. i.e., _P_-1 and _Pnma_ respectively. Their Raman and infrared vibrational spectroscopic analysis is done by [PERSON] et al. [PERSON] et al. (2014). The structure of these five crystals after optimization are shown in Figure 1. As already pointed out in ref [PERSON] et al. (2019), crystal containing light H atoms have difficulties in accurately locating the positions of H atoms because of the weak intensity signal in the measurement. This results in the unrealistically short O-H bonds. This deficiency has been rectified by accurate ab initio geometry optimization implemented in the Vienna Ab initio Simulation Package (VASP) to be explained in section below. The optimized crystal parameters for all five crystals are shown in Table 1. In the later discussion on the structural characteristics and their physical properties of these five crystals, we will focus on the difference and interplay between the three structural units. The first unit is the cationic group consisting of metallic elements. The only difference in the crystal composition is in the first structural unit K or NH\({}_{4}\) and metallic element Mg, Zn, or Cu. The composition from the other two structural units,pyrophosphate group H\({}_{2}\)P\({}_{2}\)O\({}_{7}\) and water are the same. To save space, we will name these five crystals as C1 to C5 with difference only in the first two components of unit 1: C1 for K\({}_{2}\)Mg(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\)\(\cdot\)2H\({}_{2}\)O, C2 for (NH\({}_{4}\))\({}_{2}\)Zn(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\)\(\cdot\)2H\({}_{2}\)O, C3 for K\({}_{2}\)Cu(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\)\(\cdot\)2H\({}_{2}\)O, C4 for 4[K\({}_{2}\)Cu(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\)\(\cdot\)2H\({}_{2}\)O], and C5 for 4[K\({}_{2}\)Zn(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\)\(\cdot\)2H\({}_{2}\)O]. It should be pointed out that C3 and C4 have the same chemical composition and formula unit but with different crystal symmetry and volumes.