Analytic dependence of the Madelung constant on lattice parameters for 2D and 3D metal diiodides (MI2) with CdI2 (2H polytype) layered structure

In this theoretical study, the Madelung constant (AM) both for a 2D layer and parent 3D bulk crystal of metal diiodides MI2 (M = Mg, Ca, Mn, Fe, Cd, Pb) with CdI2(2 H polytype) structure is calculated on the basis of the lattice summation method proposed in author’s earlier work. This method enabled, both for a 2D layer and 3D bulk crystal of these compounds, to obtain an analytic dependence of the Madelung constant, AM (a, c, u), on the main crystallographic parameters a, c, and u. The dependence AM (a, c, u) reproduces with a high accuracy the value of the constant AM not only for metal diiodides MI2 with CdI2(2 H polytype) structure, but also for metal dihalides (MX2) and metal dihydroxides [M(OH)2] with the same structure. With the use of the high-pressure experimental results available in literature particularly for FeI2, it is demonstrated that the above analytic dependence AM (a, c, u) is also valid for direct and precise analysis of the pressure-dependent variation of the Madelung constant.

Especially in applications of the exfoliation methods (through-solution, mechanical, intercalation) for preparation of 2D nanolayers of MI 2 layered compounds [2,16,[19][20][21], it is of technological interest the theoretical quantification of the cohesive energy (intralayer and interlayer) both in 2D layers/nanolayers and parent 3D bulk crystals. In metal dihalides (MX 2 ) and particularly in MI 2 compounds, the chemical bonds are partially ionic and partially covalent, with dominating iconicity [22][23][24][25][26][27][28], and this results in dominating contribution of the Coulomb electrostatic energy to the total cohesive energy (cohesive energy includes also the repulsive, polarization, van der Waals, and covalent energies) [22][23][24][25]29]. In its turn, the evaluation of the Coulomb electrostatic energy assumes, besides the partial (non-formal) ionic charges, also the precise knowledge of the Madelung constant, which characterizes the type of crystallographic arrangement of constituent ions in the crystal lattice [22,24,30].
There are two main approaches for calculation of the Madelung constant. The first approach assumes the direct summation in real space, and for this approach a number of effective methods have been proposed for achieving the fast convergence of the Madelung sum [31][32][33][34]. The second approach is more complex and is based on the Ewald's method that assumes summation both in real and reciprocal spaces [35][36][37].
The Madelung constant (A M ) of metal dihalides (MX 2 ) and particularly of metal diiodides (MI 2 ), with the crystallographic structure of CdI 2 polytypes (2 H, 4 H, P6 3 mc, etc), is dependent on the three crystallographic lattice parameters: the lattice constants along the a and c crystallographic axes, a and c, and the fractional out-ofplane coordinate u of halogen ion (X) in a unit cell [24,25,[29][30][31][32][33][34][35][36][37][38]. It is not a simple task to obtain an analytic dependence (i.e., analytic functional form) A a c u , , M ( )from corresponding lattice summation. Therefore, in comparative analysis of the Madelung constants of different compounds, the dependence A a c u , , M ( )is analyzed by calculating the numerical limit of the lattice sum (i.e., A M ) for each compound of interest with the use of corresponding values of a c u , , { } parameters [24,30]. For CdI 2 type structure with a fixed parameter = u 0.25, a rather complex method for determining the functional dependence of the Madelung constant on c a / ratio, was proposed by Herzig and Neckel [39]. However, in the case of compounds with parameters u considerably deviating form a value = u 0.25, the above dependence cannot result in precise value for constant A , M since A M is very sensitive to variations of the parameter u [24,29]. De Haan conducted a numerical analysis [40,41] for the dependence of the Madelung constant of some metal dihalides MX 2 , with CdI 2 (2 H polytype) structure, on the c a / ratio and the geometrical parameter characterizing the degree of inclination of M-X bonds with respect to the crystallographic c axis. Meanwhile, it would be useful, for 2D layers/nanolayers and parent 3D bulk crystals of metal diiodides (MI 2 ), to obtain the analytic dependence A a c u , , M ( )incorporating also the dependence on parameter u. The knowledge of above analytic dependence A a c u , , M ( )may considerably simplify the mathematical and physical analysis of the interdependence between the constant A M and lattice parameters. The analytic dependence A a c u , , M ( )may be of a special interest also in high temperature [42,43] and high pressure [44,45] experiments, when the constant A M depending on temperature and/or pressure changes considerably.
Within the overall program of the study of structural, energy, and elastic characteristics of layered compounds, our previous publications [10,[46][47][48] were devoted to investigations of alkaline earth metal dihydroxides Mg(OH) 2 and Ca(OH) 2 [10,46,47] and iron diiodide FeI 2 [48] having CdI 2 (2 H polytype) structure. In study [48], it is analyzed how under high-pressure conditions the variation of the pressuredependent Madelung constant of the FeI 2 is associated with the structural phase transition of this compound. The most part of results in above studies [10,47,48] were obtained on the basis of the method for calculation of the Madelung lattice sums proposed in study [10]. The main idea of this approach is that, in the Madelung lattice sum, the summation is performed not over contributions corresponding to 'ion-ion' pair interactions, but over those corresponding to 'molecule-molecule' pair interactions (here, 'molecule' is defined through a formula unit MX 2 , with M and X being a cation and an anion, respectively). Unlike the consideration of the 'ion-ion' pair interactions that result in conditional convergence of the lattice sum, the account of the 'molecule-molecule' pair interactions results in absolute and fast convergence of this sum. Therefore, it is of interest to apply this approach to 2D and 3D metal diiodides MI 2 with the CdI 2 (2 H polytype) structure particularly for determination of the above specified analytic dependence A a c u , , M ( )for the Madelung constant. In this theoretical study, the Madelung constant A M ( ) of metal diiodides MI 2 (M=Mg, Ca, Mn, Fe, Cd, Pb) with CdI 2 (2 H polytype) structure is calculated with the use of the approach proposed in author's earlier work [10]. On the basis of the mathematical and graphical analysis of the dependence of the lattice sum (for constant A M ) on the crystallographic parameters a, c, and u, an analytic dependence A a c u , , M ( )is obtained both for a 2D layer and parent 3D bulk crystal. The obtained analytic dependence A a c u , , M ( )is also analyzed in terms of its (i) applicability to some other metal dihalides (MX 2 ) and metal dihydroxides [M(OH) 2 ] with the same polytypic structure, (ii) extreme values and character of variation depending on variation of individual crystallographic parameters a, c, and u, and (iii) validness for direct and precise analysis of the pressuredependent variation of the Madelung constant.

MI 2 compounds with CdI 2 (2 H polytype) structure
The crystallographic structure of the 2 H polytype of metal diiodides MI 2 (M=Mg, Ca, Mn, Fe, Cd, Pb) is trigonal with space group P m 3 1.In a MI 2 crystal, each positively charged metal (M) ion is octahedrally coordinated by the six negatively charged nearest-neighbor I ions in equal distances ) as shown in figures 1(a) and (b). These octahedral MI 6 complexes form ionic layers with (0001) orientation and stacking along the [0001] crystallographic direction ( figure 1(c)). Each of these ionic layers is composed of the three ionic sublayers: a sublayer of M ions sandwiched between two sublayers of I ions (figure 1(c)). In above all three sublayers, the ions are arranged with a hexagonal close-packed configuration as depicted in figure 1(d). The physical rectangular coordinate system, x y z, is chosen in such a way that the x, y, and z axes are parallel to    ( ) is the permittivity of vacuum, R is the nearest-neighbor interionic distance M −X, and A M is the Madelung constant. As mentioned above, in metal dihalides (MX 2 ) and particularly in MI 2 compounds, the chemical bonds are partially ionic and partially covalent [22][23][24][25][26][27][28]. Therefore, in equation (1), the charge numbers Z c and Z a are fractional and do not coincide with (are smaller than) their corresponding formal values = Z 2 The data for the Madelung constants of metal diiodides MI 2 (M=Mg, Ca, Mn, Fe, Cd, Pb), with CdI 2 (2 H polytype) structure, are available in a number of publications [24,29,30,[39][40][41]. However, the parameter A M for different MI 2 compounds was calculated by applying alternative methods, with different precisions. For correctness of the comparative analysis it is necessary that the parameters A M for these compounds were evaluated by the same method. Therefore, for this purpose, in the present study it is used the approach proposed in author's earlier work [10]. This approach has been proposed for calculation of the Madelung constant associated with the Coulomb energy of interaction of an adsorbed Ca(OH) 2 molecule with constituent ions of a Ca(OH) 2 crystal. The Ca(OH) 2 is isostructural with the 2 H polytype of CdI 2 , and this enables to apply the above approach in the calculations of the Madelung constants of MI 2 compounds. As it was mentioned in section 1, the main idea of this approach is that, in the Madelung lattice sum, the summation is performed not over contributions corresponding to 'ion-ion' pair interactions (in this case, the convergence of the lattice sum is conditional), but over those corresponding to 'molecule-molecule' pair interactions schematically shown in figure   m 0, except with those belonging to above specified 1D molecular chain (figure 1(d)), and A 3 is the contribution resultant from Coulomb pair interactions of the reference molecule with molecules in the 3D crystal, except with those belonging to above specified 2D molecular layer with = m 0 ( figure 1(c)). From practical standpoint, it is also important to evaluate the intralayer Coulomb energy, i.e. the contribution of 2D layer into the total energy U C expressed via equations (1) and (2). The above contribution is obtained from equations (1) and (2) by substitution = A 0 3 into equation (2) (i.e., the interlayer interaction is excluded):  (2) and (4)) presented in table A1 of the appendix, according to which This means that, in layered MI 2 compounds, the Coulomb intralayer interaction drastically exceeds the interlayer interaction, and this result is typical of ionic layered compounds with different crystallographic structures [40,41] and predicts a high structural stability for their free standing/ exfoliated 2D layers.

Analytic dependence of the Madelung constant on the crystallographic parameters for 3D bulk crystal and a 2D layer
According to analysis presented in section 2 of Supplementary material, it is expected that the analytic dependence of the Madelung constants A M and A ML on the crystallographic parameters a, c, and u may be expressed, with a high approximation, solely by the ratio of parameter a and product u c, a u c .   corresponding precise values A M obtained from the lattice summation method according to equation (2). To check how precise is equation (6) for calculation of the Madelung constant of MI 2 compounds, table 1 compares the values A M obtained from lattice summation (equation (2)) with the corresponding values A M an , calculated by equation (6) with the use of a c u , , { } parameters from table 1. The difference in these values is within the range of 0-0.1%, and such a high precision confirms the validness of equation (6). To check the applicability of the dependence given by equation (6) to other metal dihalides (MX 2 ) with CdI 2 (2 H polytype) structure, equation (6) figure 2(a)). Note that the above satisfactory agreement was achieved in spite of the f A , M { } data points of VCl 2 and FeBr 2 were not used in obtaining of equation (6). Nevertheless, this agreement could be expected since, both for VCl 2 and FeBr 2 , the value of parameter f is within the range of variation of this parameter for MI 2 compounds ( figure 2(a)). In this context, it was of interest to check the applicability of equation (6)   within the prescribed accuracy. In the whole range of variation of the parameter f for MI 2 compounds, < 2.394 < f 2.575, the dependence given by equation (8) is very close to that given by equation (6) (2) and (4) and the data for contribution A 3 presented in table A1of the appendix). For a free standing or an exfoliated layer of a layered compound (particularly, of MI 2 compounds), the lattice parameter a preserves its physical and crystallographic meaning, whereas the lattice parameter c is meaningless ( figure 1(c)). Therefore, in this case, parameter f associated with equation (8) may be expressed as a h / with the use of the definition = u h c / introduced in section 2 (parameter h is the half thickness of a MI 2 layer as shown in figure 1(c)). The results of atomistic simulation [55] show that, for MI 2 compounds, parameters a and h do not practically differ from their corresponding bulk values. In the case of experimental confirmation of the above theoretical finding [55], equation (8) with parameter f defined as = f a h / may be also applicable with a good approximation to free standing/exfoliated layers of MI 2 compounds.
( )respectively, where the partial sum A i k , relates to A i contribution (i=1, 2, 3). The plots in figure 2(d)  at k=15 and 7, respectively. This means that, for MI 2 compounds, the Coulomb energy of 3D nanocrystals in comparison to that of 2D nanolayers (here, it is implied that the 2D layer is nanosized also in in-plane directions) is influenced by the size effect in a higher degree.
Equation ( In the dependence given by equation (6), the maximal value of the Madelung constant with respect to parameter f , is achieved at = f 3.128, and these data show that the Madelung constants of the compounds Mg(OH) 2 and Co(OH) 2 are on maximal level (see table 1 and figure 2(b)). Correspondingly, in the dependence given by equation (8)

Applicability of the analytic dependence A f M an
, ( )to high-pressure structural states The aim of this section is to demonstrate that that analytic dependence A f M an , ( ) given by equation (6) is also valid for calculation (tracing of variation) of the Madelung constant of MI 2 compounds under high-pressure conditions.
With the use of the experimental results reported in literature [44], in author's recent study [48] it is demonstrated that equation (6) with coefficients k 1 =−0.251642, k 2 =1.574089, and k 3 =−0.163095 is also valid to a high approximation for calculation of the Madelung constant of FeI 2 at high hydrostatic pressures. Note that the above coefficients practically coincide with those achieved in this study: k 1 =−0.2516, k 2 =1.5741, and k 3 =−0.1631 (see section 3.2). Table A2 of the appendix presents : (i) the data of the lattice parameters {a, c, u} of FeI 2 measured in above mentioned experimental study [44] by x-ray diffraction technique at ambient pressure (≈0 GPa) and at different values of the applied hydrostatic pressure P=4, 7.8, 12.5, 16.9 GPa and (ii) the data for the flatness parameter f and the Madelung constant A M calculated correspondingly according to equations (7) and (2) with the use of above experimental data {a, c, u} [44]. With the use of the data from table A2, figure 4 presents for FeI 2 the plot of the Madelung constant A M versus the } data points plotted according to equation (8). The above plots are applicable to a 2D layer comprised in a 3D crystal and presumably to a free standing/exfoliated layer of MI 2 compounds.
flatness parameter f at ambient pressure, ≈0 GPa, and applied pressures P=4, 7.8, 12.5, and 16.9 GPa (black triangles). , ( )are also presented in figure 2(a)). The plots in figure 4 show that under high-pressure conditions the analytic dependence A f M an , ( ) given by equation (6) very well reproduces the pressure-dependent variation of the Madelung constant A M calculated from the lattice sums according to equation (2) (see black triangles in this figure). To check how precise is equation (6), table A2 of the appendix compares for all applied pressures the values A M obtained from lattice summation [equation (2)] with the corresponding values A M an , calculated by the analytic dependence given by equation (6). The difference in these values is within the range of 0-0.1%, and such a high precision confirms the validness of equation (6) in high-pressure experimental investigations. Besides, the data in table A2 show that the Madelung constant varies with pressure within a rather broad range of 2.081-2.191. According to results obtained in above mentioned publications [44,48], equation (6) also predicts the minimization of the Madelung constant A M at the pressure 16.9 GPa of the structural phase transition of FeI 2 (see figure 4).
Thus, equation (6) enables directly and precisely to quantify the variation of the Madelung constant of FeI 2 depending on pressure with no need in calculations of lattice sums. It is expected that equation (6) is also valid for precise analysis of the pressure-induced variations of the Madelung constant of metal diiodides MI 2 (M=Pb, Mn, Cd, Mg, Ca). Our preliminary theoretical analysis of the high pressure experimental results, reported in literature for metal dihydroxides with the CdI 2 (2H-polytype) such as Mg(OH) 2 [56], Ca(OH) 2 [57], and Co(OD) 2 [58], show that equation (6), with slightly corrected coefficients k , 1 k , 2 and k , 3 is also applicable to analysis of high-pressure dependent variations of the Madelung constant of these compounds. The results associated with above analysis will be published in our next work.

Conclusion
For metal diiodites (MI 2 ) with CdI 2 (2 H polytype) structure, an analytic dependence of the Madelung constant on the crystallographic parameters is obtained both for a 2D layer and 3D bulk crystal of these compounds. It is shown that this analytic dependence (i) reproduces with accuracy of 0%-0.1% the Madelung constants of MI 2 compounds determined from lattice summation, (ii) is also applicable with a high accuracy to other metal dihalides and metal hydroxides, with CdI 2 (2 H polytype) structure, in the variation range of the flatness parameter f up to values »3, and (iii) is very sensitive with respect to variation of the unit cell parameter u. Both the Madelung constants of a 2D layer and 3D bulk crystal of MI 2 compounds, A M and A , ML increase in the order FeI 2 <PbI 2 <MnI 2 <CdI 2 <MgI 2 <CaI 2 .
The results reported in this study may be useful in the analysis of structural changes and phase transitions in high-pressure experimental investigations of metal diiodides.
The analytic dependence A a c u , , M ( )may be incorporated into atomistic simulation programs for modeling of the cohesive energy and determination of the equilibrium crystallographic parameters a, c, and u when account of the contribution of the Coulomb energy is necessitated.  Table A1 lists the contributions A 1 , A 2 , and A 3 to the Madelung constant A in equation (2).The contributions A 1 , A 2 , and A 3 are determined correspondingly from the lattice sums given by equations (S5), (S7), and (S9) of Supplementary material.

ORCID iDs
V S Harutyunyan https://orcid.org/0000-0002-8015-5891  Table A2. The data {a, c, u} obtained for FeI 2 in study [44] from XRD measurements at different hydrostatic pressures P, the data of the flatness parameter f calculated from equation (7) with the use of the data {a, c, u}, and the Madelung constants A M and A M an , calculated according to equations (2) and (6), respectively, with the use of the data {a, c, u}.