First-Principles Calculations of the Structural, Electronic, Optical, and Mechanical Properties of 21 Pyrophosphate Crystals

Abstract

Pyrophosphate crystals have a wide array of applications in industrial and biomedical fields. However, fundamental understanding of their electronic structure, optical, and mechanical properties is still scattered and incomplete. In the present research, we report a comprehensive theoretical investigation of 21 pyrophosphates A₂M (H₂P₂O₇)₂•2H₂O with either triclinic or orthorhombic crystal structure. The molecule H₂P₂O₇ is the dominant molecular unit, whereas A = (K, Rb, NH₄, Tl), M = (Zn, Cu, Mg, Ni, Co, Mn), and H₂O stand for the cation elements, transition metals, and the water molecules, respectively. The electronic structure, interatomic bonding, partial charge distribution, optical properties, and mechanical properties are investigated by first-principles calculations based on density functional theory (DFT). Most of these 21 crystals are theoretically investigated for the first time. The calculated results show a complex interplay between A, M, H₂P₂O₇, and H₂O, resulting in either metallic, half-metallic, or semi-conducting characteristics. The novel concept of total bond order density (TBOD) is used as a single quantum mechanical metric to characterize the internal cohesion of these crystals to correlate with the calculated properties, especially the mechanical properties. This work provides a large database for pyrophosphate crystals and a road map for potential applications of a wider variety of phosphates.

1. Introduction

Despite longstanding research interest, phosphate crystals have attracted a great deal of attention in recent years because of their application in novel devices. Generally speaking, phosphate crystals can be classified into three categories depending on the bonding nature of the P-O groups: orthophosphates, pyrophosphates, and polyphosphates. Among them, the pyrophosphate structure contains a P₂O₇ or P₂H₂O₇ dimer. Phosphates have rich and unique crystal structure types, therefore they participate in a wide range of applications of phosphate materials, such as sodium ion metal batteries [1], nonlinear optical (NLO) crystals with short cutoff edges [2,3,4,5,6,7,8], near ultraviolet light-emitting diode (NUV LED) applications [9], and birefringent materials [10]. For example, potassium dihydrogen phosphate (KH₂PO₄/KDP), as a soft-brittle optical crystal, is extensively used in the construction of inertial confinement fusion (ICF) facilities [11,12]. Other types of phosphate crystals are used to reduce pollution and climate change effects due to their capacity for CO₂ and CH₄ sorption [13].

Inorganic acidic pyrophosphate (PPi) (or inorganic diphosphate) materials are omnipresent in nature [14,15]. These compounds belong to the family of inorganic acidic diphosphates with the very general formula A₂M B₂•nH₂O, where A is a monovalent cation such as ammonium (NH₄) or an alkaline earth cation (Na, K, and Rb), and M can be a divalent transition metal cation (Cu, Ni, Co, and Mn) or Zn²⁺ and Mg²⁺. With the acidic diphosphates, several forms of the acidic anions B can exist in the molecular structure, such as H₂P₂O₇²⁻, HP₂O₇³⁻, and H₃P₂O₇⁻, causing these acidic diphosphates to exhibit important biological properties such as antitumor, antibacterial, and antifungal activity against Salmonella typhimurium, Enterococcus faecium, and Candida albicans [16]. Inorganic acidic pyrophosphates basically have two forms: crystalline and glassy forms. Both forms have attracted considerable attention as promising inorganic materials. They possess many applications in industry due to their uses as solid-state laser hosts [17], dielectric materials [18], magnetic materials [19], rechargeable lithium batteries [20], in ceramic applications [21], and in the area of gas separation [22]. Inorganic acidic pyrophosphates also have biological importance such as their use in some catalytic processes [23], or as ion exchangers [22]. Of special interest are the crystals: K₂M (H₂P₂O₇)·2H₂O and (NH₄)₂M(H₂P₂O₇)·2H₂O where (M = Mg, Zn, Cu, Ni, Co, and Mn). These materials are polymorphic structures with three different space groups, namely triclinic P-1, orthorhombic Pnma, and monoclinic P2/m [13,24,25,26,27,28,29,30,31,32].

PPi, which can be produced through a variety of biosynthetic reactions, such as the biosynthesis of nucleotides, lipids, urea, and oxidative phosphorylation [33], is also a prevalent reactant in some metabolic pathways including fat metabolism, protein synthesis, nucleoside diphosphate sugar synthesis, and DNA and RNA synthesis [34]. For instance, orthophosphates (with orthophosphate group: PO₄³⁻), metaphosphate (with metaphosphate group: PO₃⁻), and pyrophosphate (with pyrophosphate group: P₂O₇⁴⁻) have many biological uses [35], especially their role in some enzyme-catalyzed reaction processes [36]. The P₂O₇⁴⁻ anion is unstable in hydrous solutions and rapidly hydrolyzes to inorganic phosphate in the presence of divalent metal ions (P₂O₇⁴⁻ + H₂O→2HPO₄²⁻), which promotes the splitting of both inorganic and biological pyrophosphates, especially in acidic conditions [37]. In the case of pyrophosphates, it has been found that they are very effective in certain catalytic processes such as the conversion of butane to maleic anhydride [38,39,40]. They also have a crucial role in some bioenergetic processes [41,42], such as the energy circulating in functional cells [43,44]. Moreover, they can be used as inhibitors in the in vitro formation and dissolution of apatite crystals [45].

Biological pyrophosphates hydrolyze to a family of enzymes known as pyrophosphatases (PPases) [46]. Soluble pyrophosphatases (sPPases) are very common and are found in all organisms in two forms: (i) organic and (ii) inorganic [47]. These enzymes are fundamental for the regulation of intra-cellular levels of inorganic phosphate and for the removal of pyrophosphate products of nucleotide coupling reactions [48]. There are two unrelated families of inorganic pyrophosphatases, the ubiquitous Mg-dependent family, and the Mn-dependent family, and both of them can be found in bacteria and archaea [49,50]. The soluble pyrophosphatases have several functions, including some steroid biosynthetic pathways (such as juvenile hormone synthesis), regulation of cell motility, nucleotide metabolism, and they may have a nuclear role in regulating transcription of some genes [51,52,53,54]. PPases need divalent metal cations to be more active. The efficiency of such cations as an activator decreases in the order: Mg > Zn > Co >Mn > Cd [55,56]. These enzymes can have as many as four functional divalent metal ions in the active site, with two metal ions bound as essential cofactors, whereas the third and fourth are ligated to pyrophosphate/phosphate-forming motifs [55,56].

Several theoretical and experimental investigations of some important pyrophosphate crystals have appeared in the literature in recent years. Using experimental and computational methods, some uses of pyrophosphates as water oxidation catalysts were investigated by Kim et al. [57]. Other pyrophosphate materials were investigated separately via density functional theory (DFT) [58,59,60]. Recently, another pair of complex acidic metal pyrophosphate crystals have been experimentally investigated by F. Elhafiane et al. [61,62]. The crystal structure of the orthorhombic phase of K₂Zn (H₂P₂O₇)₂·2H₂O, K₂Cu (H₂P₂O₇)₂·2H₂O, and K₂Ni (H₂P₂O₇)₂·2H₂O crystals was investigated experimentally for the first time by R. Essehli et al. [24]. Vibrational analysis, Raman, and infrared spectra studies of K₂Zn (H₂P₂O₇)₂·2H₂O and K₂Cu (H₂P₂O₇)₂·2H₂O crystals were reported [31,63]. The electronic structure, optical properties, and mechanical properties of the orthorhombic phase of K₂Zn (H₂P₂O₇)₂·2H₂O and K₂Cu (H₂P₂O₇)₂·2H₂O crystals and the triclinic phase of K₂Mg (H₂P₂O₇)₂·2H₂O, (NH₄)₂Mg (H₂P₂O₇)₂·2H₂O, and (NH₄)₂Zn (H₂P₂O₇)₂·2H₂O crystals were theoretically investigated recently via DFT [64,65], but they did not include the spin-polarized effect in their calculations. The ammonium diphosphates crystals: (NH₄)₂Ni (H₂P₂O₇)₂·2H₂O, (NH₄)₂Co (H₂P₂O₇)₂·2H₂O, and (NH₄)₂Mn (H₂P₂O₇)₂·2H₂O were synthesized and prepared by R. Essehli et al. [26,28] and F. Capitelli et al. [27]; however, to our knowledge, no theoretical investigations have been attempted for the electronic structure, optical properties, and mechanical properties of these crystals. Even though the crystal structure and vibrational analysis of Rb₂Zn (H₂P₂O₇)₂·2H₂O, Rb₂Mg (H₂P₂O₇)₂·2H₂O, and K₂Co (H₂P₂O₇)₂·2H₂O crystals were experimentally investigated by R. Essehli et al. [18] and A. Lamhamdi et al. [66], their electronic structure, optical properties, and mechanical properties have not yet been theoretically investigated. The crystal structure of thallium diphosphates: Tl₂M (H₂P₂O₇)₂•2H₂O (M = Mg, Mn, Co, Ni, and Zn) crystals was synthesized for the first time by B. El Bali [67], but no theoretical investigations have been reported for their electronic structure, optical properties, and mechanical properties.

With the complex structures of pyrophosphate crystals and their widespread presence, the primary understanding of their structures and other properties has been a subject of many studies. However, these studies are still not as common as for other inorganic materials such as silicates and chalcogenides. With high-accuracy calculations that can be provided by quantum mechanical principles, more information on the electronic structure, molecular interaction, optical properties, and mechanical properties can be revealed. In this work, we present the results of DFT calculations of 21 pyrophosphate crystals as listed in

Table 1. Experimental and calculated lattice parameters for the 21 crystals.

#	Crystal	Abbreviation	Space Group	a, b, c(Å), α, β, γ (Degree) (Experimental)	a, b, c(Å), α, β, γ (Degree) (Our Calculations)
1	K2Zn(H2P2O7)2·2H2O	K2Zn-1	Ortho, Pnma(62)	9.770, 10.980, 13.420, 90.0, 90.0, 90.0 [24]	10.385, 11.229, 13.566, 90.0, 90.0, 90.0
2	K2Zn(H2P2O7)2·2H2O	K2Zn-2	Tric, P-1(2)	6.827, 7.333, 7.570, 80.753, 72.547, 83.442 [30]	6.972, 7.466, 7.630, 80.754, 73.419, 84.039
3	K2Cu(H2P2O7)2·2H2O	K2Cu-3	Ortho, Pnma(62)	9.899, 10.781, 13.401, 90.0, 90.0, 90.0 [24]	9.835, 11.662, 14.083, 90.0, 90.0, 90.0
4	K2Cu(H2P2O7)2·2H2O	K2Cu-4	Tric, P-1(2)	6.856, 7.314, 7.557, 81.028, 72.327, 83.697 [31]	7.074, 7.424, 7.604, 78.940, 71.785, 83.500
5	K2Ni(H2P2O7)2·2H2O	K2Ni-5	Ortho, Pnma(62)	9.912, 10.774, 13.422, 90.0, 90.0, 90.0 [24]	10.815, 11.132, 14.568, 90.0, 90.0, 90.0
6	K2Ni(H2P2O7)2·2H2O	K2Ni-6	Tric, P-1(2)	6.855, 7.312, 7.561, 81.012, 72.301, 83.691 [32]	7.153, 7.382, 7.658, 77.933, 70.398, 83.217
7	K2Co(H2P2O7)2·2H2O	K2Co-7	Tric, P-1(2)	6.874, 7.357, 7.614, 80.740, 72.397, 83.484 [66]	7.087, 7.647, 7.548, 77.860, 67.386, 87.327
8	K2Mg(H2P2O7)2·2H2O	K2Mg-8	Tric, P-1(2)	6.857, 7.362, 7.620, 81.044, 72.248, 83.314 [29]	6.968, 7.478, 7.662, 81.170, 73.303, 83.516
9	Rb2Zn(H2P2O7)2·2H2O	Rb2Zn-9	Tric, P-1(2)	6.957, 7.362, 7.794, 81.851, 70.622, 86.263 [18]	7.068, 7.486, 7.813, 81.840, 71.764, 86.863
10	Rb2Mg(H2P2O7)2·2H2O	Rb2Mg-10	Tric, P-1(2)	6.955, 7.375, 7.812, 81.986, 70.275, 85.988 [18]	7.054, 7.488, 7.835, 82.119, 71.265, 86.504
11	Rb2Co(H2P2O7)2·2H2O	Rb2Co-11	Tric, P-1(2)	6.980, 7.370, 7.816, 81.740, 70.350, 86.340 [68]	7.071, 7.359, 7.709, 82.405, 71.758, 87.859
12	(NH4)2Zn(H2P2O7)2·2H2O	(NH4)2Zn-12	Tric, P-1(2)	7.003, 7.330, 7.789, 81.229, 71.064, 88.172 [25]	7.330, 6.930, 7.732, 84.653, 71.551, 88.116
13	(NH4)2Mg(H2P2O7)2·2H2O	(NH4)2Mg-13	Tric, P-1(2)	7.076, 7.431, 7.894, 81.420, 70.900, 87.780 [13]	7.138, 7.432, 7.818, 81.516, 71.444, 89.924
14	(NH4)2Ni(H2P2O7)2·2H2O	(NH4)2Ni-14	Tric, P-1(2)	7.034, 7.321, 7.792, 81.530, 70.910, 88.210 [26]	7.266, 7.338, 7.917, 79.058, 68.138, 88.466
15	(NH4)2Co(H2P2O7)2·2H2O	(NH4)2Co-15	Tric, P-1(2)	7.067, 7.368, 7.852, 81.230, 70.680, 88.540 [28]	7.220, 7.286, 7.804, 80.824, 69.610, 89.739
16	(NH4)2Mn(H2P2O7)2·2H2O	(NH4)2Mn-16	Tric, P-1(2)	7.003, 7.440, 7.877, 80.444, 71.359, 87.408 [27]	7.144, 7.322, 7.739, 81.831, 71.537, 90.547
17	Tl2Zn(H2P2O7)2·2H2O	Tl2Zn-17	Tric, P-1(2)	6.963, 7.363, 7.740, 81.590, 71.441, 86.323 [67]	7.078, 7.523, 7.707, 81.487, 73.016, 86.740
18	Tl2Ni(H2P2O7)2·2H2O	Tl2Ni-18	Tric, P-1(2)	6.966, 7.308, 7.702, 81.801, 71.185, 86.528 [67]	7.089, 7.432, 7.617, 81.500, 73.213, 87.502
19	Tl2Mg(H2P2O7)2·2H2O	Tl2Mg-19	Tric, P-1(2)	6.962, 7.364, 7.771, 81.801, 70.860, 86.137 [67]	7.066, 7.519, 7.723, 81.825, 72.726, 86.339
20	Tl2Mn(H2P2O7)2·2H2O	Tl2Mn-20	Tric, P-1(2)	6.958, 7.475, 7.825, 80.723, 71.912, 85.661 [67]	7.095, 7.439, 7.625, 81.901, 73.026, 86.945
21	Tl2Co(H2P2O7)2·2H2O	Tl2Co-21	Tric, P-1(2)	6.977, 7.363, 7.769, 81.421, 71.114, 86.424 [67]	7.096, 7.399, 7.597, 81.827, 73.038, 87.597

with more information about their experimental and calculated lattice parameters and symmetry. These 21 crystals have the same structural formula: A₂M (H₂P₂O₇)₂•2H₂O. They have the same pyrophosphate group, H₂P₂O₇, and the same number of H₂O molecules. The only difference between these 21 crystals is the type of the cation A (A = K, Rb, NH₄, and Tl) and metallic element M (M = Mg, Zn, Cu, Ni, Co, and Mn). Some crystals under study include ferromagnetic elements (Ni, Mn, Co) in their structure. To explore the electronic and optical characteristics of these crystals, the spin-polarized DFT calculations were performed and their results are presented in a separate section. With the detailed and accurate calculations of the electronic structure, bonding, optical properties, and mechanical properties, we offer perceptible insights that give better understanding of these pyrophosphate crystals. The 21 pyrophosphate crystals are listed in Table 1 in specific order, from 1 to 21. For consistency, we follow this order in all tables, figures, and discussions. In Table 1, we give each crystal an abbreviation and this abbreviation will be used in all other tables to save space. Figure 1 depicts the structure of the crystals 1 and 2 in ball-and-stick form. Studying the set of 21 pyrophosphate crystals enables us to develop important correlations among the properties of the different crystals. For example, we use a single quantum mechanical metric—the total bond order density (TBOD), to describe the strength and internal cohesion of the crystal. Our focus will be on the bonding, mechanical properties, and optical properties, and their correlations with the TBOD. In the next section, we briefly describe the crystal structures and computational methods used in the calculation. The main results are presented and discussed in Section 3. The paper ends with a summary and some conclusions.

2. Materials and Methods

In this work, we used two density functional theory (DFT)-based methods to perform our calculations: the Vienna ab initio simulation package (VASP) [69] and the orthogonal linear combination of atomic orbitals method (OLCAO) [70]. VASP was used to optimize the crystal structures and calculate the mechanical properties. The one-electron orbitals and the electronic charge density were expanded in a plane wave basis set with an energy cut-off of 500 eV. The energy cut-off value was found to be the best choice, balancing the fast convergence and accurate ground state energy. We used the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE) [71] as the exchange and correlation potential for solving the Kohn–Sham equation. In the VASP calculation, the electronic and ionic force convergence criteria were set at 10⁻⁶ eV and 10⁻⁴ eV/Å, respectively. We used the Monkhorst scheme [72] with different k-point meshes ranging from 4 × 4 × 4 for larger crystals such as the orthorhombic phase of crystals K₂Zn (H₂P₂O₇)₂·2H₂O, K₂Cu (H₂P₂O₇)₂·2H₂O, and K₂Ni (H₂P₂O₇)₂·2H₂O (124 atoms) to 8 × 8 × 8 for the other smaller crystals which have triclinic space group such as (NH₄)₂Mg (H₂P₂O₇)₂·2H₂O (31 atoms). The VASP-optimized crystal structures were used as inputs to calculate the electronic structure, interatomic bonding, and optical properties, using the in-house developed OLCAO package.

OLCAO is an all-electron method based on the local density approximation (LDA). Atomic orbitals expanded as Gaussian-type orbitals (GTO) are used in OLCAO as the basis set for expanding the solid-state wave function. The use of localized atomic orbitals in the basis expansion, which contrasts with the plane-wave expansion used in VASP, is successful for the DFT calculations of both crystalline [73,74,75,76,77,78] and non-crystalline materials [79,80], especially those with complex structures typical in biomolecular systems [81,82]. A suitably large number of Monkhorst k-points were used, (10 × 10 × 10) for all crystals, for density of states (DOS), band structure, and optical properties calculations. The Mulliken scheme [83] was used to calculate the partial charge (PC) and interatomic bonding. We define the PC of an atom as the difference between the effective charge Q* and the electronic charge of a neutral atom (Q₀) in units of electron. Mathematically, ∆Q = Q₀ − Q*. Negative ∆Q implies a gain of electrons (i.e., an electronegative ion), and positive ∆Q implies a loss of electrons (i.e., electropositivity). The formulae of effective charge (Q_α*) and bond order (BO) values are shown in Equations (1) and (2), also called the overlap population, ρ_αβ between any pair of atoms (α, β).

$$Q_{\alpha }^{*}={\sum }_i{\sum }_{m,occ}{\sum }_{j,\beta }{C}_{i\alpha }^{*m}{C}_{j\beta }^m{S}_{i\alpha ,j\beta }$$

(1)

$${\rho }_{\alpha \beta }={\sum }_{m, occ}{\sum }_{i,j}{C}_{i\alpha }^{*m}{C}_{j\beta }^m{S}_{i\alpha ,j\beta }$$

(2)

In the above equations, $S_{i\alpha .j\beta }$ are the overlap integrals between the ith orbital of the αth atom and the jth orbital of the βth atom. $C_{j\beta }^m$ is the eigenvector coefficients of the mth occupied band. The BO (Equation (2)) defines the relative strength of the bond between two atoms. The summation of all BO values in the crystal gives the total bond order (TBO). The total bond order density (TBOD) is obtained when TBO is normalized by the cell volume. TBOD is a single quantum mechanical metric to describe the internal cohesion of the crystal [84] and can be decomposed into partial components (PBOD) for any structural units or groups of bonded atomic pairs. The combination of using VASP and OLCAO packages with two different basis expansions is highly effective in revealing the subtle features in the material properties, especially regarding atomic scale details. More details about the methods used in the optical and mechanical properties calculations are given in the Supplementary Material.

3. Results and Discussion

3.1. Electronic Structure

Understanding the electronic structure is important for identifying the optoelectronic behaviors of the materials and their multiple uses in technological fields. For this purpose, we calculated the total density of states (TDOS) of the 21 pyrophosphate crystals. To achieve more deep and detailed information about the interaction between different atoms and the differences among the 21 crystals, we resolved the TDOS into partial density of states (PDOS) for each crystal structural unit. The TDOS and PDOS from −15 to 15 eV are displayed for all 21 crystals in Supplemental Figure S1 in the Supplementary Information (SI). The energy gaps of the 21 crystals are presented in

Table 2. Energy band gap (Eg), total bond order density (TBOD), refractive index (n), and plasma frequency (ω_p) for the 21 crystals.

#	Crystal	# of Atoms	Eg (eV)	TBOD (e−/Å3)	n	ωp (eV)
1	K2Zn-1	124	4.7	0.0194	1.483	23.10
2	K2Zn-2	31	5.0	0.0217	1.466	22.10
3	K2Cu-3	124	-	0.0188	-	-
4	K2Cu-4	31	3.8	0.0122	1.533	23.00
5	K2Ni-5	124	-	0.0086	-	-
6	K2Ni-6	31	-	0.0220	1.500	23.10
7	K2Co-7	31	-	0.0222	-	-
8	K2Mg-8	31	5.4	0.0217	1.449	22.50
9	Rb2Zn-9	31	4.9	0.0207	1.450	22.00
10	Rb2Mg-10	31	5.1	0.0209	1.456	21.50
11	Rb2Co-11	31	0.6	0.0215	1.718	22.05
12	(NH4)2Zn-12	39	4.6	0.0288	1.481	21.90
13	(NH4)2Mg-13	39	5.1	0.0276	1.446	21.80
14	(NH4)2Ni-14	39	1.2	0.0280	1.483	21.00
15	(NH4)2Co-15	39	-	0.0284	1.500	21.90
16	(NH4)2Mn-16	39	0.6	0.0286	1.503	21.80
17	Tl2Zn-17	31	3.05	0.0202	1.703	21.10
18	Tl2Ni-18	31	2.05	0.0202	-	-
19	Tl2Mg-19	31	3.2	0.0203	1.688	21.10
20	Tl2Mn-20	31	0.1	0.0210	1.844	20.50
21	Tl2Co-21	31	0.5	0.0209	2.098	21.30

. Most of these 21 crystals are insulators with different values of E_g. However, a closer inspection of the TDOS and PDOS shows the need for more accurate analysis. We selectively discuss some of the crystals in more detail, focusing on those crystals that have an unclear energy band gap and those that we deem to be particularly interesting. In Supplemental Figure S2, we plotted TDOS and PDOS for crystals numbered from 1 to 9, and crystals 15, 16, 18, and 21 with an energy range of −6 to 6 eV. Supplemental Figure S2 shows that there is a slight overlap of the valence bands above the Fermi level in all crystals. These bands mainly consist of 2p states of the O atom in crystals 1, 2, 8, and 9, while they mainly consist of 2p states of the O atom and 3d states of Cu and Ni atoms in crystals 3, 4, 5, and 6. In crystals 15 and 16, the valence bands cross the Fermi level and these bands mainly consist of 4s and 3d states of the Co and Mn atoms. In crystal 18, valence bands cross the Fermi level and these bands mainly consist of 2p states of O atom and 4s and 3d states of Ni atom, while in crystal 21, the valence bands that cross the Fermi level consist mainly of 2p states of O atom and 4s and 3d states of Co atom. In conclusion, the accurate analysis of the electronic structure of the discussed crystals indicates semi-metallic behavior.

3.2. Interatomic Bonding

The ability to calculate the interatomic bonding properties is a key strength of this work. Revealing and understanding the bonding properties of phosphates is crucial, especially for biological processes. For example, protein phosphorylation, which can be described in a very general picture, is a common mechanism for modifying protein function [85]; it partially depends on the bonding nature between the P atom and other atoms in DNA and RNA. Much of its use depends on the reversibility of the process, which is often mediated by protein phosphatases. Protein phosphorylation is an important post-translational modulation that is an integral part of cellular function [86,87]. The phosphorylation of any amino acid residue results in a change in charge and thus in the protein surface potential. For instance, phosphorylation of the amino acids: serine, threonine, and tyrosine, results in a change from neutral to negative (−2). Hence, there is no surprise that phosphorylation can affect protein conformation, protein–protein interactions, and biochemical pathways, and its dysregulation is connected to many disease states [88,89,90]. The interatomic interaction between atoms can be explained more clearly by showing the BO vs. BL (bond length) distribution for each crystal. The BO data can also be used to obtain the TBOD for each structural unit. BO vs. BL distribution for each crystal is shown in Supplemental Figure S3 (1–21) in the SI. In crystals 1 and 3 (the orthorhombic phase), there are six different types of bonds, while in crystal 5 (the orthorhombic phase), there are seven different types of bonds. All the remaining triclinic crystals contain five different types of bonds, except for crystal 12 ((NH₄)₂Zn(H₂P₂O₇)₂·2H₂O), which contains six different types of bonds. The four most common bonds in all crystals are the P-O and H-O bonds, hydrogen bonds (H-H), and negligibly weak (H-O) bonds when far apart. P-O and H-O bonds can be described as polar covalent bonds. We should mention that considering these bonds as polar covalent bonds is not final, because one should keep in mind that the difference in the electronegativity value between the P atom (with electronegativity = 2.19) and the O atom (with electronegativity = 3.44), or between the H atom (with electronegativity = 2.2) and the O atom (with electronegativity = 3.44) is not small (higher than 1.2).

These 21 crystals also have some bonds that are considered to be ionic bonds, such as the Zn-O, K-O, Mg-O, Rb-O, and Tl-O bonds. The polar covalent P-O bonds are the strongest bonds, and they are only slightly different in each crystal, which implies a strong tetrahedral PO₄ unit as is true in all phosphates. All P-O bonds occur at short BLs, between 1.5 Å and 1.8 Å. Uniquely, crystals 1 and 5 have very strong P-H bonds which occur at a BL of 1.5 Å. The H-O bonds in all crystals are distributed along long BLs, between 1.0 Å and 3.5 Å. The H-O bonds that occur with short BLs (about 1.0 Å) are strong bonds with a high BO value. The Zn-O, Cu-O, Ni-O, Mg-O, Co-O, and Mn-O bonds in all crystals have comparable values of BO and BL, while K-O, Rb-O, and Tl-O bonds are weak bonds with small values of BO. The unique covalent N-H bonds in crystals from 12 to 16 are very strong bonds because they are part of the intramolecular bonds in NH₄. In Figure 2 and Figure 3, we plotted the BO versus BL distribution for the crystals from 1 to 6. The pie charts in Figure 4 to be shown later give the proportion contributions from diverse bond types to the total bond order. P-O bonds have the highest contribution in all crystals and are responsible for their internal cohesion. All other bonds can play their part in the crystal cohesion of pyrophosphates, including the H-O hydrogen bonds. 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 2. In Figure 5, we display the TBOD for the 21 crystals in sequential order as designated earlier in Table 1. The 21 crystals were divided into four groups with four different colors: black for A₂M (H₂P₂O₇)₂•2H₂O(A = K), red for A₂M (H₂P₂O₇)₂•2H₂O(A = Rb), green for A₂M (H₂P₂O₇)₂•2H₂O(A = NH₄), and blue for A₂M (H₂P₂O₇)₂•2H₂O(A = Tl). It turns out that the crystals from 12 to 16 have the highest cohesion, with the highest values of TBOD. This feature for these crystals (12–16) can be explained by the existence of unique, very strong covalent N-H bonds. In Supplemental Figure S4, we plotted the crystal structure of crystal 2 in ball-and-stick format. As can be seen, H-O bonds occur at different BLs: short BLs (about 1.0 Å), medium BLs (about 1.8 Å), and long BLs (about 3.5 Å). H-O bonds with short BLs belong to H₂O molecule, while other H-O bonds with medium and long BLs probably belong to H₂P₂O₇ molecule. There are some very weak H-O bonds with long BLs that link H₂O molecule with H₂P₂O₇ molecule.

3.3. Partial Charge

The partial charge (PC) or the charge transfer for an atom in a crystal is the deviation of the effective charge Q* from the neutral atom charge Q₀. It is a crucial part of the electronic structure. The calculated Q* on each atom in the pyrophosphate crystals is shown in

Table 3. Effective charge (Q*) for each atom in the 21 crystals.

#	Crystal	Q*(in e−)
1	K2Zn-1	11.092(Zn), 6.127(K), 2.992(P), 6.836(O), 0.665(H)
2	K2Zn-2	11.132 (Zn), 6.197(K), 2.807(P), 6.918(O), 0.569(H)
3	K2Cu-3	10.269(Cu), 6.114(K), 2.792(P), 6.873(O), 0.670(H)
4	K2Cu-4	10.349(Cu), 6.202(K), 2.809(P), 6.904(O), 0.569(H)
5	K2Ni-5	9.378(Ni), 6.142(K), 3.044(P), 6.814(O), 0.643(H)
6	K2Ni-6	9.344(Ni), 6.214(K), 2.804(P), 6.905(O), 0.567(H)
7	K2Co-7	8.348(Co), 6.232(K), 2.799(P), 6.906(O), 0.562(H)
8	K2Mg-8	0.826(Mg), 6.199(K), 2.816(P), 6.934(O), 0.571(H)
9	Rb2Zn-9	11.125(Zn), 6.191(Rb), 2.812(P), 6.920(O), 0.565(H)
10	Rb2Mg-10	0.825(Mg), 6.190(Rb), 2.821(P), 6.936(O), 0.568(H)
11	Rb2Co-11	8.358(Co), 6.196(Rb), 2.817(P), 6.904(O), 0.566(H)
12	(NH4)2Zn-12	11.125(Zn), 5.803(N), 2.801(P), 6.921(O), 0.583(H)
13	(NH4)2Mg-13	0.794(Mg), 5.807(N), 2.805(P), 6.942(O), 0.581(H)
14	(NH4)2Ni-14	9.318(Ni), 5.816(N), 2.789(P), 6.911(O), 0.582(H)
15	(NH4)2Co-15	8.332(Co), 5.809(N), 2.795(P), 6.910(O), 0.583(H)
16	(NH4)2Mn-16	6.349(Mn), 5.801(N), 2.797(P), 6.910(O), 0.581(H)
17	Tl2Zn-17	11.124(Zn), 12.296(Tl), 2.801(P), 6.911(O), 0.563(H)
18	Tl2Ni-18	9.322(Ni), 12.298(Tl), 2.806(P), 6.899(O), 0.559(H)
19	Tl2Mg-19	0.814(Mg), 12.301(Tl), 2.809(P), 6.927(O), 0.565(H)
20	Tl2Mn-20	6.346(Mn), 12.306(Tl), 2.804(P), 6.895(O), 0.562(H)
21	Tl2Co-21	8.348(Co), 12.303(Tl), 2.807(P), 6.895(O), 0.562(H)

. The distribution of the calculated PC for each element in each crystal is shown in Supplemental Figure S5 (1–21) in the SI. As can be seen in Supplemental Figure S5, K, Cu, Zn, Ni, Mg, Co, Rb, Mn, P, and H atoms have positive PC whereas O, Tl, and N atoms have negative PC. Mg atom has the largest positive PC among other elements. Small variations of the PC for P, H, or O are due to their locations in the structural units of H₂P₂O₇ or H₂O. P and H elements in the orthorhombic phase of K₂Zn (H₂P₂O₇)₂·2H₂O (crystal 1) can lose or gain charge, whereas they only lose charge in the triclinic phase (crystal 2). P element in the orthorhombic phase of K₂Ni (H₂P₂O₇)₂·2H₂O (crystal 5) can lose or gain charge, while it only loses charge in the triclinic phase (crystal 6).

Figure 4. The pie charts show the percentages of different bonding types for crystals from 1 to 6.

Figure 4. The pie charts show the percentages of different bonding types for crystals from 1 to 6.

Figure 5. Distribution of TBOD for the 20 crystals. The black color is for the crystals from 1 to 8, red color is for crystals from 9 to 11, green color is for crystals from 12 to 16, and blue color is for crystals from 17 to 21.

Figure 5. Distribution of TBOD for the 20 crystals. The black color is for the crystals from 1 to 8, red color is for crystals from 9 to 11, green color is for crystals from 12 to 16, and blue color is for crystals from 17 to 21.

3.4. Optical Properties

We have explored the optical properties of the 21 pyrophosphate crystals through the frequency-dependent dielectric function which is mainly connected with the electronic structures. The frequency-dependent dielectric function can be used to calculate many optical parameters such as refractive index, absorption, energy loss, reflectivity, and optical conductivity [91]. The complex dielectric function, which has real and imaginary parts, is given by [92]:

$$\epsilon \left(\omega \right)={\epsilon }_1\left(\omega \right)+i{\epsilon }_2\left(\omega \right)$$

(3)

The first part of the equation above gives the dispersive part, while the second part represents the absorption part of complex dielectric function. The imaginary part can be estimated by the means of density of states and momentum matrix elements associated with occupied and unoccupied states. The imaginary part can be calculated relatively easily within the one-electron random phase approximation using the OLCAO method. More information about the mathematical formulas involved in these calculations is given in Section 1 in the Supplementary Material. To investigate the linear response of a system based on the inter-band transitions, the real part of dielectric function (ε₁(ω)), which is directly related to polarizability in the electromagnetic spectra, must be discussed in some detail. An intra-band transition is a transition between two electronic energy states in the same band, while an inter-band transition is a transition between two electronic energy states in two different bands. Optical dielectric functions of the 21 crystals are shown in Supplemental Figure S6 (1–21) in the SI. As can be seen in Supplemental Figure S6 (1–21), each crystal has its unique absorption features; we selectively discuss the crystals 1, 6, 8, 12, 13, 17, and 19 in some detail. The maximum value of dispersion is observed in the energy range from 6.5 eV to 8.0 eV for crystals 1, 6, 8, 12, and 13, while the maximum value of dispersion is observed in the energy range from 3.0 eV to 3.5 eV for crystals 17 and 19. After attaining its maximum value, ε₁(ω) starts decreasing after a few vibrations. The dispersive part along the energy range (0–40 eV) shows a positive value due to semi-conducting behavior of crystals 1, 6, 8, 12, 13, 17, and 19. We should mention that crystals 9, 10, and 11 have negative peaks at energy E ≈ 19.0 eV, and these peaks can identify the reflection of incident photons.

The imaginary part of the dielectric function (ε₂(ω)) shows the energy needed to excite electrons from the valence band to the conduction band [93]. It is implied with the absorption of incident photon. The peaks in the ε₂(ω) spectra are caused by transitions from the occupied valence band (VB) states to the unoccupied conduction band (CB) states, which originally occurred due to inter-band transitions. The ε₂(ω) curve has different threshold values depending on the energy gap of each crystal. Threshold values of crystals 1, 6, 8, 12, 13, 17, and 19 are 4.5 eV, 3.0 eV, 5.4 eV, 4.9 eV, 5.0 eV, 3.1 eV, and 3.2 eV, respectively. In most of the 21 crystals, the absorption spectrum is observed to diminish after 30 eV. From Supplemental Figure S6 (1–21), it can be noticed clearly that for most of the 21 crystals, the highest absorption of incident photon occurs in the energy range from 9.0 eV to 10.0 eV, which can be considered to be in the ultraviolet (UV) region. These crystals can therefore be used for protection from ultraviolet radiation. In crystals 3 and 17–21, the highest absorption of incident photon occurs in the energy range from 3.5 eV to 4.0 eV, which can be considered to be in the visible light region. The refractive index is given by:

$$n=\sqrt{{\epsilon }_r{\mu }_r}$$

(4)

In Equation (4), ε_r is the relative permittivity or the dielectric function and µ_r is the relative permeability. The relative permeability (µ_r) is very close to unity (with the absence of magnetic effects), so that the refractive index can be approximated as $n=\sqrt{{\epsilon }_r}$. The real part of the dielectric function ε₁(ω) can be obtained from ε₂(ω) using Kramers–Kronig transformation. The static refractive index n can be obtained from the square root of ε₁(ω) in the zero frequency limit, i.e., $n=\sqrt{{\epsilon }_1\left(0\right)}$. The calculated refractive indices are listed in Table 2. Crystals 11, 17, 20, and 21 have the highest values of n: 1.718, 1.703, 1.844, and 2.098, respectively. Another important optical parameter that can be calculated from the complex dielectric function is the energy loss function (ELF). ELFs represent the collective excitation of electrons at high frequency. The ELF is related to the loss of fast electrons [93]. Generally speaking, the peaks of the ELF are plasmon oscillations that are created due to valence band maxima and conduction band minima at different values of energy depending on the type of each crystal. The position of ELF’s main peak is interpreted as the plasmon frequency ω_p, which usually occurs at the frequency when the real part of the dielectric function vanishes [ε₁(ω_p) = 0)]. Below ω_p, the incident waves will be mostly reflected. Above ω_p, the material behaves like plasma, or all the excited electrons are nearly free electron. The calculated energy loss functions are shown in Supplemental Figure S7 in the SI, and the calculated values of ω_p are listed in Table 2. Supplemental Figure S7 indicates isotropic character in the energy range from 0 to 50 eV for most crystals. The plasmon oscillations of 21 pyrophosphate crystals occur in the energy range from 20 eV to 23 eV.

3.5. Spin-Polarized Calculations

The interaction of magnetism and electronic structure characteristics in the same material may give rise to half-metallic ferromagnetism (HMF). The half-metallic materials can have metallic behavior for electrons of one spin direction while there could be semi-conducting behavior in the opposite spin direction. Half-metallic ferromagnetic materials have wide application in spintronic and spin caloritronic devices [94,95]. We have calculated the TDOS and optical dielectric functions for the crystals 3 (K₂Cu-3), 4 (K₂Cu-4), 5 (K₂Ni-5), 6 (K₂Ni-6), 7 (K₂Co-7), 11 (Rb₂Co-11), 14 ((NH₄)₂Ni-14), 15 ((NH₄)₂Co-15), 16 ((NH₄)₂Mn-16), 18 (Tl₂Ni-18), 20 (Tl₂Mn-20), and 21 (Tl₂Co-21) by considering the spin-polarized effect within the local density approximation (LDA) using OLCAO code. Energy gaps and refractive indices are presented in

Table 4. Calculated spin-up and spin-down energy gap and refractive index.

#	Crystal	Eg (eV)	n
3	K2Cu-3	1.39 (spin-up)	1.342 (spin-up), 1.483 (spin-down)
4	K2Cu-4	3.8 (spin-up), 0.2 (spin-down)	1.265 (spin-up), 1.342 (spin-down)
7	K2Co-7	0.7 (spin-up)	1.273 (spin-up)
11	Rb2Co-11	0.6 (spin-down)	1.483 (spin-up), 1.295 (spin-down)
15	(NH4)2Co-15	0.4 (spin-up)	1.265 (spin-up, spin-down)
16	(NH4)2Mn-16	0.1 (spin-up)	1.265 (spin-up, spin-down)
20	Tl2Mn-20	0.1 (spin-up)	1.449 (spin-up), 1.643 (spin-down)
21	Tl2Co-21	0.6 (spin-down)	1.789 (spin-up), 1.449 (spin-down)

. The calculated TDOS and optical dielectric functions are displayed in Supplemental Figures S8 and S9 in the SI. Positive (negative) TDOS are plotted for majority (minority) electrons. From Supplemental Figure S8, K₂Cu-3 has semi-conducting behavior in the majority spin channel (spin-up), with E_g being about 1.3 eV, while it has semi-metallic behavior in the minority spin channel (spin-down) due to the slight overlap of the valence bands above the Fermi level. However, on closer inspection, K₂Cu-3 has semi-metallic behavior in the majority spin channel due to a very slight overlap of the valence bands. K₂Cu-4 has semi-conducting behavior in the majority spin channel with E_g being about 4.0 eV and semi-conducting behavior in the minority spin channel with E_g equal to 0.2 eV. However, on closer inspection, K₂Cu-4 has semi-metallic behavior in the majority spin channel due to the slight overlap of the valence bands above Fermi level. Therefore, based on the more accurate analysis of the electronic structure of crystals 3 and 4, we say that they do not have half-metallic (HM) behavior. K₂Ni-5 has semi-metallic behavior in both the majority and minority spin channels because the energy states near the valence band top are metallic bands with both occupied and unoccupied bands below and above the Fermi level.

TDOS for K₂Ni-6 is clearly identical for both the majority and minority spin channels. K₂Co-7 has semi-conducting behavior in the majority spin channel with E_g being about 0.6 eV, while it has semi-metallic behavior in the minority spin channel due to the overlap of the valence bands above the Fermi level. Rb₂Co-11 has semi-metallic behavior in the majority spin channel due to the slight overlap of the valence bands above the Fermi level, while it has semi-conducting behavior in the minority spin channel with E_g being about 0.2 eV. TDOS for (NH₄)₂Ni-14 is identical for both the majority and minority spin channels. (NH₄)₂Co-15 has semi-conducting behavior in the majority and minority spin channels, however, on closer inspection, it appears that (NH₄)₂Co-15 has semi-metallic behavior in the minority spin channel due to the slight overlap of the valence bands above Fermi level. (NH₄)₂Mn-16 has semi-conducting behavior in the majority spin channel with E_g being about 0.2 eV, while it has semi-metallic behavior in the minority spin channel because the energy states near the valence band top are metallic bands with both occupied and unoccupied band below and above the Fermi level. Tl₂Co-21 has semi-metallic behavior in the majority spin channel and semi-conducting behavior in the majority spin channel with clear E_g. In conclusion, our calculations showed that the crystals 3, 4, 5, 7, 11, 15, 16, 20, and 21 have half-metallic (HM) behavior, while the crystals 6, 14, and 18 have identical electronic behavior.

3.6. Mechanical Properties

Although pyrophosphate crystals have many structural applications, their mechanical properties are much less studied compared to other properties. Here, we have calculated the mechanical properties of the 21 pyrophosphate crystals, using the method described in Section 2 in the Supplementary Material. The calculated elastic constants are listed in Table S2 in the SI, which provides a profusion of information about the stability, stiffness, brittleness, ductility, and anisotropy of these crystals. C₁₁, C₂₂, and C₃₃ coefficients reflect the isotropic elasticity of crystals, and they have very close values in the cubic crystals. C₁₁, C₂₂, and C₃₃ are associated with unidirectional compression along the principal x, y, and z directions [96]. Synonymously, C₁₁, C₂₂, and C₃₃ represent the resistance of a crystal to the deformation along x, y, and z directions. Large C₁₁, C₂₂, and C₃₃ elastic constants of a material indicate that it is very incompressible under uniaxial stress along x, y, or z axes, respectively. The value of C₁₁ for triclinic crystals: 2, 4, and 6 is much larger than C₁₁ of the orthorhombic crystals: 1, 3, and 5, which indicates that triclinic crystals: 2, 4, and 6 are much more incompressible under uniaxial stress along x than the orthorhombic ones. The values of the elastic constants depend on the crystal structure (lattice parameters) and the strength of the bonds between the elements. Table S2 shows that C₃₃ is higher than C₁₁ and C₂₂ in crystal 1, which implies that this crystal is more compressible along x- and y-axes than along the z-axis. C₂₂ is higher than C₁₁ and C₃₃ in crystal 3, which implies that this crystal is more compressible along x- and z-axes than along the y-axis. C₁₁ is higher than C₂₂ and C₃₃ in crystal 2 and crystals 4–21, so these crystals are more compressible along y- and z-axes than along the x-axis. The 21 crystals have lower symmetry and a larger number of independent elastic constants (C_ij). For the orthorhombic crystal structure, the mechanical stability criteria are given by [97,98]:

C₁₁ > 0; C₂₂ > 0; C₃₃ > 0; C₄₄ > 0; C₅₅ > 0; C₆₆ > 0; C₁₁C₂₂ > C²₁₂; C₁₁ + C₂₂ − 2C₁₂ > 0

The calculated elastic constants for the orthorhombic phase crystals: 1, 3, and 5, fulfill the stability criteria above. For the other triclinic crystals which have even lower symmetry and a larger number of independent elastic constants than the orthorhombic crystals, the mechanical stability criterion requires all values of C_ij to be positive [97]. All elastic constant values are presented in Tables S2–S4 in the SI. From these values, we conclude that the crystals 4, 6, 7, 9, 11, 12, 14, 15, 18, and 21 do not match this criterion and they are mechanically unstable crystals.

From the elastic coefficient C_ij and compliance tensor S_ij (S_ij = 1/C_ij), the other mechanical parameters: bulk modulus (K), Young’s modulus (E), shear modulus (G), and Poisson’s ratio (η) can be obtained under the Voigt–Reuss–Hill (VRH) approximation for polycrystals as explained in Section 2 in the Supplementary Material [99,100]. These parameters are listed in Table S1. Generally speaking, Young’s modulus can measure the stiffness of the materials, or Young’s modulus measures the change in length. Bulk modulus is a measure of resistance to compressibility, or bulk modulus measures the change in volume when a pressure is applied. Shear modulus represents the resistance against shear distortion. Supplemental Figure S10a–d show the distribution of the bulk modulus, Young’s modulus, shear modulus, and Poisson’s ration (η) of the 21 crystals. The 21 crystals were divided into four groups regarding the A element, similar to Figure 5 for the TBOD. We can notice from Supplemental Figure S10a,b that the crystals 12–16 have the highest Young’s modulus and bulk modulus. As can be seen in Table S1, most pyrophosphate crystals have a bulk modulus less than 30 GPa, except for crystals 4 and 12–16. In general, if the obtained bulk moduli are quite small, i.e., <30 GPa, the materials can be classified as relatively soft materials with high compressibility. Another useful parameter that can be used to classify the brittle and ductile behaviors of a material is the ratio of the shear modulus to the bulk modulus, G/K. It is an empirical relationship related to the plastic and elastic properties of the material, and it is called Pugh’s modulus ratio [101,102]. G/K values for the 21 crystals are represented in Table S1. According to Pugh’s criterion, crystals with G/K larger than 0.571 tend to be brittle and those with G/K less than 0.571 tend to be more ductile [102,103,104,105]. The crystals 2, 7, 8, 10, and 12 have G/K larger than 0.571, which indicates that these five crystals tend to be more brittle. All remaining crystals have G/K less than 0.571, which indicates that these crystals tend to be more ductile. Further, the Vickers hardness (H_V) formula was used to calculate the hardness values of pyrophosphate crystals under study. Vickers hardness formula, which was formulated by Chen et al. [106] and Tian et al. [107], can be derived from the elastic constants, as shown earlier [108,109,110]. In this work, we used the formula of Tian et al. [107]:

$$H_V=0.92{\left(\frac{G}{K}\right)}^{1.137}{G}^{0.708}$$

(5)

It is assumed that materials with Vickers hardness larger than 40 GPa can be classified as super-hard materials [111]. The calculated values of H_V of all 21 crystals are presented in Table S1 in the SI. As can be seen in this table, H_V has value less than 10 GPa for all crystals, so these 21 crystals cannot be considered as hard materials, or they are soft materials.

The elastic anisotropy parameter can be characterized by the universal anisotropic index A^U [112]:

$$A^U=5\frac{G_V}{G_R}+\frac{K_V}{K_R}-6$$

(6)

where G_V, G_R, K_V, and V_R are the shear and bulk modulus’ Voigt and Reuss estimates respectively. The calculated values of A^U are given in Table S5. If A^U has a value of unity, the crystal exhibits an isotropic characteristic, while values other than unity represent varying degrees of anisotropy [113]. After calculation, A^U for all 21 crystals varies from unity (in different degrees), indicating anisotropic behavior for all crystals. However, crystals 5 and 12 show weak anisotropy. The percentage of anisotropy in the compression and shear is another way of measuring the elastic anisotropy. They are given by [114]:

$$A_{comp}=\frac{K_V-K_R}{K_V+K_R}$$

(7)

$$A_{shear}=\frac{G_V-G_R}{G_V+G_R}$$

(8)

The calculated values of A_comp and A_shear are given in Table S5. The value of zero for A_comp and A_shear indicates isotropic elastic behavior, while the variation from zero refers to varying degrees of anisotropy, and a value of 1 (100%) is the largest possible anisotropy. To characterize the materials as ductile or brittle, Frantsevich’s rule of Poisson’s ratio [115] can be applied here. According to this rule, if the Poisson’s ratio (η) is less than 0.26, the material tends to be brittle in nature; otherwise, the material will be ductile. As can be seen in Table S1, most crystals have η higher than 0.26, so they can be considered as ductile materials, except for crystals 2, 7, 8, 10, and 12, which have η less than 0.26, so they can be considered as brittle materials, and this result was exactly the same when we applied the Pugh’s ratio rule. To reveal the correlation between the electronic structure/bonding characteristics and the mechanical properties of the pyrophosphate crystals, we plotted bulk modulus (Figure 6a), Young’s modulus (Figure 6b), and shear modulus (Figure 6c) versus TBOD for the 21 crystals. As can be seen in these figures, E, K, and G steadily increase with the TBOD, which allows us to conclude that the mechanical properties of these 21 crystals are strongly correlated to the strength of the interatomic bonds which are collectively represented by the single metric TBOD.

4. Conclusions

We have studied 21 pyrophosphate crystals in the framework of density functional theory. By using first-principles calculations, the electronic structure, bonding, charge transfer, optical properties, and mechanical properties were calculated. Electronic structure calculations show the semi-metallic nature of some crystals and semi-conducting behavior for others. Bonding characteristics show that the two most common bonds in all crystal are the P-O and H-O bonds. The polar covalent bonds P-O are found to be the strongest bonds in most crystals, expect for crystals numbered from 12 to 16, which contain N-H bonds that are a little stronger than P-O bonds. P-O and N-H bonds come from H₂P₂O₇ and NH₄ units. The metal-oxygen bonds: Zn-O, Cu-O, Ni-O, Mg-O, Co-O, and Mn-O in all crystals are comparable. NH₄ is a unique group of atoms that replace the metallic ion (M) in the structure formula: A₂M (H₂P₂O₇)₂•2H₂O, and this replacement caused the crystals 12 to 16 to have the largest values of TBOD. The electronic structure and bonding in these crystals appear to be quite similar, but there are subtle differences here and there due to the differences in their compositions in different metallic elements and crystal geometry. The optical dispersion analysis of complex dielectric function ε, energy loss function ELF, and refractive index n was also conducted in the energy range of 0–40 eV for ε and 0–50 eV for ELF. The calculated optical results of all investigated materials show that the highest absorption of incident photon occurs in the ultraviolet (UV) region, and these crystals have medium values of n ranging from 1.446 to 2.098. Spin-polarized DFT calculations of the electronic structure were conducted for the crystals with (M = Cu, Ni, Co, Mn) to check if there are half-metallic ferromagnetic characteristics. Our calculations show that crystals 3, 4, 5, 7, 11, 15, 16, 20, and 21 have half-metallic (HM) behavior, while the crystals 6, 14, and 18 have identical electronic behavior in both spin-up and spin-down directions of electrons. Mechanical properties calculations show that most of these 21 crystals are ductile, soft, and anisotropic materials.