## 5 Summary and Conclusions  
We have presented detailed results on the quantum mechanical calculation on electronic structure, partial charge, interatomic bonding, mechanical and optical properties of five pyrophosphates with very complex structures. To our knowledge, these are the first time such calculations have been attempted. This enable us to make detailed comparisons on the structure and properties among them.  
Figure 7: Optical properties for the five pyrophosphate crystals: Center panel: real (\(\epsilon_{1}\)) and imaginary (\(\epsilon_{2}\)) parts of the dielectric function. Far left panel: expanded illustration of \(\epsilon_{1}\) and \(\epsilon_{2}\)) from 0.0 to 5 eV. Right panel, the electron energy loss function (ELF). (**a**,**a**) for C1; (**b**,**b**) for C2; (**c**,**c**) for C3; (**d**,**d**) for C4; and (**e**,**e**) for C5.  
The fundamental understanding of pyrophosphate crystals can open the door for many applications. The general conclusions of this study can be succinctly summarized as follows:  
1. The electronic structure and bonding in these crystals appear to be quite similar, but there are subtle differences in bonding in these crystals due to differences in their compositions in different metallic elements and crystal geometry.  
2. The presence of very narrow metallic state of Cu in the crystals can induce empty states above the Fermi level, which make its optical properties more complicated and interesting.  
3. NH\({}_{4}\) is a unique group of atoms to replace the metallic ion. It is the only organic group in the five crystals and has the largest TBOD.  
4. Crystal symmetry plays an important role, triclinic vs orthorhombic. This implies short-range intermolecular action is more critical than longer-range crystal symmetry.  
5. H\({}_{2}\)P\({}_{2}\)O\({}_{7}\) is the most important and dominating unit in pyrophosphates. The water molecule provides the unique H\(\cdots\)O bonds, and metallic elements can influence the structure bonding and reactivity to a certain extent.  
6. The mechanical properties of these five crystals vary the most, but detailed correlation to structure and electronic properties remains clear.  
The work presented represent the first step in understanding the structure and properties of pyrophosphates. Much work remains to be done, especially in the direction of having different transition metal elements replacing Cu or Zn and invoke spin-polarized calculations for magnetic properties.  
W.-Y.C. and R.K. initiated the project. P.A. and W.-Y.C. did the calculations. R.K., M.H., and K.B. developed the synthesis method of phosphate materials including acidic pyrophosphates. All the authors participated in the discussion and interpretation of the results. W.-Y.C., P.A., and R.K. wrote the paper. All authors edited and proof-read the final manuscript.  
This research received no external funding.  
This research used the resources of the National Energy Research Scientific Computing Center supported by the DOE under Contract No. DE-AC03-76 SF00098 and the Research Computing Support Services (RCSS) of the University of Missouri System. PA is supported by a research grant from the School of Graduate Studies at UMKC.  
The author declares no conflict of interest.