## 4 Results and Discussion
### Interatomic Bonding  
The best way to describe the interatomic interaction is to show the BO vs BL distribution in each crystal and then use the data to obtain the TBOD and PBOD for each structural unit. The BO vs BL distribution for the five pyrophosphate crystals turns out to be quite similar, except for C3 and C4 containing Cu. We present the distribution for crystal C2 ((NH\({}_{4}\))\({}_{2}\)Zn(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\):2H\({}_{2}\)O) and C4 (4[K\({}_{2}\)Cu (H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\):2H\({}_{2}\)O) as examples in Figure 5a,b, respectively. More detailed discussion for C1 has been described in [21]. Essentially, there are seven types of different interatomic bonds for each crystal. The five common bonds in each crystal are covalent bonds (P-O, O-H), hydrogen bond (HB) \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **K\({}_{2}\)** & **(NH\({}_{4}\))\({}_{2}\)** & **Mg** & **Zn** & **Cu** & **(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\)** & **2**-**H\({}_{2}\)O** \\ \hline
**C1** & 1.68 & - & 1.26 & - & - & -2.97 & 0.03 \\ \hline
**C2** & - & 1.55 & - & 0.91 & - & -2.57 & 0.11 \\ \hline
**C3** & 1.73 & - & - & - & 0.67 & -2.62 & 0.22 \\ \hline
**C4** & 1.74 & - & - & - & 0.59 & -2.36 & 0.03 \\ \hline
**C5** & 1.73 & - & - & 0.85 & - & -2.71 & 0.13 \\ \hline \end{tabular}
\end{table} 
Table 2: Partial charge values from three different structural units in five crystals. The first five columns are the elements in the first structural unit. Figure 4: Calculated partial charge distribution in five pyrophosphate crystals. (**a**) C1, (**b**) C2, (**c**) C3, (**d**) C4, (**e**) C5. (O\(\cdots\)H), bridging bonds (O-H-O), and negligibly weak (O-H) bonds when far apart. The covalent P-O bonds within the structural unit H\({}_{2}\)P\({}_{2}\)O\({}_{7}\) is the strongest bonds, and they are only slightly different in each crystal, which implies the strong tetrahedral unit PO\({}_{4}\) as is true in all phosphates. Mg-O and Zn-O bonds are stronger in comparison to K-O bonds, and they form octahedral units in the respective crystal. In C2, the unique covalent N-H bonds are very strong since they are part of the intramolecular bonds in NH\({}_{4}\). The BO for Cu-O in C3 and C4 with same chemical composition are quite different (Not shown). In C3, some Cu-O bonds have a larger BL of 2.50 A with a lower BO of 0.054 e. This shows difference in bonding due to crystal structure. The sum of total bond order values in the crystal when normalized by volume gives the TBOD, a very useful parameter to identify internal cohesion in pyrophosphate crystals, and they are listed in Table 3. It turns out that C2 has the highest cohesion with TBOD of 0.02743 e./A\({}^{3}\), much larger than the other four crystals with very similar TBOD. The PBOD from different bond types is shown in Figure 5c in the form of histogram. P-O bonds have highest contribution in all crystals and is responsible for its internal cohesion. Apart from the strong N-H bonds in C2, O-H is second most significant bonds in these crystals. However, all other bonds also play its part in crystal cohesion of pyrophosphates including the O\(\cdots\)H hydrogen bonds. Figure 5: (**a**) Bond order (BO) vs. BL for all pairs of atoms in C2 (NH\({}_{4}\))\({}_{2}\)Zn(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\)\(\cdot\)2H\({}_{2}\)O, (**b**) C4 [K\({}_{2}\)Cu(H\({}_{2}\)P\({}_{2}\)O\({}_{7}\))\({}_{2}\)\(\cdot\)2H\({}_{2}\)O], (**c**) bar graph of contributions from different bonds types in the five pyrophosphate crystals.