Homologous Series of Chemical Compounds in Three-component Systems (Aa+ – Bb+ – Cc–) and (Zn2+ - Ge4+ - P3-) in Generalized Form

Abstract

For the first time, a method for calculating formulas of homologous series of chemical compounds of the systems (A^a+ – B^b+ – C^c–) and {Zn²⁺ – Ge⁴⁺ – P^3–} in a generalized form is presented. The calculation is confirmed by the literature experimentally obtained compounds: thirteen compounds of the system (Na⁺ – Ti⁴⁺ – O^2–), seven – systems (Li⁺ – Ti⁴⁺ – O^2–), five – systems (K⁺ – V⁵⁺ –        – O^2–), eight – systems (Ba²⁺ – Cu²⁺ – O^2–). Homological series in (A^a+ – B^b+ – C^c–) have the following generalized form: A_{{t – k·r + nr – r)bc}B_racC_{{t – k·r + nr)ab} and A_tbcB_{{r – k·t + nt – t}ac}C _{(r – k·t + nt)ab}.

In (Zn²⁺ – Ge⁴⁺ – P^3–) systems for the m-group the formulas of homologous series, that develops towards Ge₃P₄, have the following generalized form: Zn_6tGe_{(6r – 6kt + 6n – 6t)}P_{(8r – 8kt + 8n)}  and for am-homologous series – Zn₆Ge_3nP_{4(n + 1)}. A method for calculating formulas of homologous series of chemical compounds in a generalized form can be used for any system of chemical elements.

Introduction

The search for new Chemical Compounds (CC) in multicomponent systems of Chemical Elements (CE) is a difficult task. The great variety of properties of a set of Three-Component Chemical Compounds (TCC) is of great interest in solving a number of scientific and applied studies. There are known works in the literature that use the mathematical apparatus for predicting phases in multicomponent systems. Thus, semiempirical quantum-chemical methods, such as Hartree-Fock-Rutaan and Hartree-Fock-Slater methods [11Fok VA. The beginning of quantum mechanics. Science. 1976.-33Slater JC. Phys. Rev. 1051; 51:3;385. https://doi.org/10.1103/PhysRev81.385], are used to calculate the formulae of hypothetical multicomponent CC. When using these methods to describe a chemical system, many physical and chemical phenomena must be taken into account. The exact solution of the basic laws leads to overcomplicated calculations. Therefore, for practically all CE systems, the solution of the corresponding electronic equations used in quantum-chemical calculations is only possible approximately. Well-known quantum-chemical methods of calculation, allow to calculate the formula of any CC, are rather complicated and require special knowledge in the field of mathematical programming.

According to the works [44Undalov YK. Russian. Inorganic Chemistry. 1998; 43:1447.-77Undalov YК, Terukov EI, Agafonov DV. Izvestia of St. Petersburg State Institute of Technology (Technical University). 2021; 59(85):26. DOI:10.36807/1998-9849-2021-59-85-26-36], knowledge of the laws of formation of Homological Series (HS) of chemical compounds can be included in the process of searching for new CC.

Thus, the author of the work [8, p. 190] believes that “...the activated complex theory consists in the fact that in the course of any chemical reaction, the initial configuration of atoms passes to the final one as a result of continuous change of interatomic distances”, which is characterized by the formation of various critical intermediate configurations, i.e. activated complexes”. On the basis of this idea, in the works [44Undalov YK. Russian. Inorganic Chemistry. 1998; 43:1447.-77Undalov YК, Terukov EI, Agafonov DV. Izvestia of St. Petersburg State Institute of Technology (Technical University). 2021; 59(85):26. DOI:10.36807/1998-9849-2021-59-85-26-36] the method of calculation of formulae of HS of chemical compounds of three-component systems of CE ions was developed.
 It is known [99Urusov VS. Theoreticalcrystal chemistry. Moscow State University. 1987.,1010Kovba LM. Stoichiometry, defects in crystals and structural homology. Chemistry. 1996.] that multicomponent CE systems include CC, which are combined into different HS. As the HS develops, the fundamental properties of its members change in a regular way, which is determined by a regular change in the crystal structure of homologs [88Eriemin EN. Fundamentals of chemical kinetics. Visshaia Shkola. 1976.-1010Kovba LM. Stoichiometry, defects in crystals and structural homology. Chemistry. 1996.]. In turn, the regular change in the crystal structure of homologs is determined by the regular change in their composition, which contributes to the search for new CC in [44Undalov YK. Russian. Inorganic Chemistry. 1998; 43:1447.-77Undalov YК, Terukov EI, Agafonov DV. Izvestia of St. Petersburg State Institute of Technology (Technical University). 2021; 59(85):26. DOI:10.36807/1998-9849-2021-59-85-26-36]. In the works [99Urusov VS. Theoreticalcrystal chemistry. Moscow State University. 1987.-1111Butlerov A. On the chemical structure of substances. Scientific notes of Kazan University. (Dep. Physical, mathematical and medical sciences). 1862; 1:1;1.], the structural homology of inorganic CC of different systems of CE is discussed in detail.

So far, judging from the literature, the formulas of HS are known, which were obtained only empirically.

In the literature, there are known HS of systems with a single set of chemical elements:

Ba_mCu²⁺_{m + n}O [1212Klinkova LA, Barkovskii NV, Fedotov VK. Physica C: Superconductivity. 2010; 470:206. https://doi.org/1016/j.physc.2010.09.013]; or La_nNi_nO_{3n – 1} [1313Gay PL, Rao CNR. Z Naturforsch. 1975; 30a:8;1092. https://doi.org/1515/zna-1975-0831]; La_{n + 1}Ni_nO_{3n + 1}, n = 1-5 [1414Li Z, Guo W, Zhang NN. APL Matereals. 2020; 8:091112. DOI: 10.1063/5.0018934]; La_{n + 1}Ni_nO_{3n – 1}, n = 7, 9, 13 and 30 [1515Drennan J, Tavares CP, Steele BCH. Mater. Res. Bull. 1982; 17:5;621. https://doi.org/1016/0025-5408(82)90044-7]; La_{2n – 4}Ni₂O_{4n – 5}, n = 5-8 [1616Savchenko VF, Lyubkina IY. Russian. Inorganic Chemistry. 1986; 2:1483.]; or Fe²⁺nFe³⁺_2mO_{n + 3m} [1717Bykova T, Dubrovinsky L, Dubrovinskaia N. Nature Communication. 2016; 10661: https://doi.org/10.1038/neomms10661] or Sr_{n + 1}Ti_nO_{3n + 1}[1818Ruddbesden SN, Popper P.Acta Crystall. 1958; 11:1;54. https://doi.org/10.1107/S0365110X58000128]. Thus, in the (Ba²⁺ – Cu²⁺ – O^2–) system, the homologous series of CC is described by the formula Ba_mCu_{m + n}O_{3n – 1}, where 2m = 2n – 1 [1212Klinkova LA, Barkovskii NV, Fedotov VK. Physica C: Superconductivity. 2010; 470:206. https://doi.org/1016/j.physc.2010.09.013], and in [1717Bykova T, Dubrovinsky L, Dubrovinskaia N. Nature Communication. 2016; 10661: https://doi.org/10.1038/neomms10661] it is reported that a series of CC belonging to the HS were obtained, described by the formula nFeO∙mFe₂O₃ ≡ Fe²⁺_nFe³⁺_2mO^2–_{n + 3m}.

Additionally, experimentally obtained inorganic CC described by formulas that exhibit regular changes in composition are known in the literature: M_nO_{3n – 1} (M = Mo, W) [1919Beznosikov VV. Structural Chem. 1991; 32:3.], M_nO_{2n – 1} (M = Ti, V) (n = 4-10) [2020Magneli A, Blomberg B, Kihlborg H. Acta chem. scand. 1955; 9:8;1382. https://doi.org/10.3891/acta.chem.scand.09-1382-2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B], W_nO_{3n – 2} (n = 20, 38-40) [2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B].

However, in the case of “Magneli phases” in the systems (M – O), where M º Mo, W, V, Ti, [1919Beznosikov VV. Structural Chem. 1991; 32:3.-2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B] and in the system (La – Ni – O) [1313Gay PL, Rao CNR. Z Naturforsch. 1975; 30a:8;1092. https://doi.org/1515/zna-1975-0831-1616Savchenko VF, Lyubkina IY. Russian. Inorganic Chemistry. 1986; 2:1483.], all these formulas, without taking into account the existence of atom of metal with the same name with different atom valence in them, in our opinion, they cannot be considered correct. Following the electron neutrality of CC formulas, in all crystal lattices of CC belonging to the “Magnéli phases” described by formulas from [1919Beznosikov VV. Structural Chem. 1991; 32:3.-2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B], two metal ions of the same name with different atom valence should be present: for example, M⁵⁺ and M⁶⁺ in Me_nO_{3n – 1} or in W_nO_{3n – 2} (M º Mo, W), or M³⁺ and M⁴⁺ in M_nO_{2n – 1} (M º Ti, V). In the formulas described in [1313Gay PL, Rao CNR. Z Naturforsch. 1975; 30a:8;1092. https://doi.org/1515/zna-1975-0831-1616Savchenko VF, Lyubkina IY. Russian. Inorganic Chemistry. 1986; 2:1483.] two ions Ni²⁺ and Ni³⁺ with different atom valence should be present as equal chemical entities, determining the existence of the crystal lattice of CC. That is why the formulas of “Magnéli phases” from [1919Beznosikov VV. Structural Chem. 1991; 32:3.-2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B], in our opinion, should be attributed not to two-component, but to three-component CE systems. In turn, the formulas related to the (La – Ni – O) system [1616Savchenko VF, Lyubkina IY. Russian. Inorganic Chemistry. 1986; 2:1483.-1919Beznosikov VV. Structural Chem. 1991; 32:3.] belong not to a three-component system but to a four-component CE system, (La³⁺ – Ni²⁺ – Ni³⁺). For this reason, the formulas from [1919Beznosikov VV. Structural Chem. 1991; 32:3.-2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B], in our opinion, could be written as follows:

Me_nO_{3n – 1} º Me⁵⁺₂Me⁶⁺_{n – 2}O_{3n – 1}, where (Me º Mo, W) and n = 8–12, 14 [1919Beznosikov VV. Structural Chem. 1991; 32:3.]; or Me_nO_{2n – 1} º Me³⁺₂Me⁴⁺_{n – 2}O_{2n – 1}, where (Me º Ti, V) and n = 4-10 [2020Magneli A, Blomberg B, Kihlborg H. Acta chem. scand. 1955; 9:8;1382. https://doi.org/10.3891/acta.chem.scand.09-1382-2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B]; or W_nO_{3n – 2} º W⁵⁺₄W⁶⁺_{n – 4}O_{3n – 2}, where n = 20, 38-40 [2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B], and the formulas from [1313Gay PL, Rao CNR. Z Naturforsch. 1975; 30a:8;1092. https://doi.org/1515/zna-1975-0831-1616Savchenko VF, Lyubkina IY. Russian. Inorganic Chemistry. 1986; 2:1483.] are suggested to be written as follows: La_nNi_nO_{3n – 1} [1313Gay PL, Rao CNR. Z Naturforsch. 1975; 30a:8;1092. https://doi.org/1515/zna-1975-0831] º La_nNi²⁺₂Ni³⁺_{n – 2}O_{3n – 1}, La_{n + 1}Ni_nO_{3n + 1} [1414Li Z, Guo W, Zhang NN. APL Matereals. 2020; 8:091112. DOI: 10.1063/5.0018934] º La_{n + 1}Ni²⁺Ni³⁺_{n –1}O_{3n + 1}, La_{n + 1}Ni_nO_{3n – 1} [1515Drennan J, Tavares CP, Steele BCH. Mater. Res. Bull. 1982; 17:5;621. https://doi.org/1016/0025-5408(82)90044-7] º La_{n + 1}Ni²⁺Ni³⁺_{n –1}O_{3n – 1} and La_{2n – 4}Ni_nO_{4n – 5} [1616Savchenko VF, Lyubkina IY. Russian. Inorganic Chemistry. 1986; 2:1483.] º La_{2n – 4}Ni²⁺_{n – 2}Ni³⁺₂O_{4n – 5}.

In the literature, for the system (M⁺ – Ti⁴⁺ – O^2–), where M⁺ º Li⁺, Na⁺, K⁺, Rb⁺, Cs⁺, a series of formulas for experimentally obtained TCC is known, generalized for n = 1-9 in the form of M⁺₄Ti⁴⁺_nO_{2(n + 1)} [2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B].

The aim of this study is to develop a method for calculating the formulas of a homologous series of three-component systems of chemical elements in a generalized form using the geometric features of the triangle representing the system of chemical element ions.

Justification of the method for calculating homologous series of chemical compounds

The method for calculating HS of three-component systems developed in [44Undalov YK. Russian. Inorganic Chemistry. 1998; 43:1447.-77Undalov YК, Terukov EI, Agafonov DV. Izvestia of St. Petersburg State Institute of Technology (Technical University). 2021; 59(85):26. DOI:10.36807/1998-9849-2021-59-85-26-36] allows, in our opinion, for the generalized determination of HS formulas. This conclusion is drawn from the findings presented in [66Undalov YK. Russian. Inorganic Chemistry. 1999; 44:1479.,77Undalov YК, Terukov EI, Agafonov DV. Izvestia of St. Petersburg State Institute of Technology (Technical University). 2021; 59(85):26. DOI:10.36807/1998-9849-2021-59-85-26-36], which provide numerous confirmations of the accuracy of the HS calculation method for a range of three-component systems through experimental results taken from the literature. It should be noted, however, that the laws governing the formation of individual CC and the laws governing the formation of HS are different. Before the publications of [44Undalov YK. Russian. Inorganic Chemistry. 1998; 43:1447.-77Undalov YК, Terukov EI, Agafonov DV. Izvestia of St. Petersburg State Institute of Technology (Technical University). 2021; 59(85):26. DOI:10.36807/1998-9849-2021-59-85-26-36], there was no possibility in the literature to predict the formation scheme of HS. In some cases, HS formulas were only experimentally determined in works [1313Gay PL, Rao CNR. Z Naturforsch. 1975; 30a:8;1092. https://doi.org/1515/zna-1975-0831-2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B].

Based on the works [44Undalov YK. Russian. Inorganic Chemistry. 1998; 43:1447.-77Undalov YК, Terukov EI, Agafonov DV. Izvestia of St. Petersburg State Institute of Technology (Technical University). 2021; 59(85):26. DOI:10.36807/1998-9849-2021-59-85-26-36], the justification for the generalized method of calculating HS can be formulated as follows:

1)      In our view, the rule (or scheme) for forming HS of chemical compounds can be formulated by considering all possible directions of chemical interactions between the components of the system, as allowed by the combination of the valence electron count of chemical elements and the composition of complex atomic clusters in the system. In reality, only ions of elementary substances and “certain intermediate atom configurations critical for a given reaction” participate in chemical interactions [88Eriemin EN. Fundamentals of chemical kinetics. Visshaia Shkola. 1976.]. In our case, these are activated CC and charge clusters (ChCl). It is clear that determining the formula of HS is possible if, among the many presumed directions of chemical interaction between the components of the CE system, we can select those responsible for the formation of HS. Therefore, to solve this problem, the CE system must be represented as ions, as only ions can chemically interact with each other, producing intermediate, more complex, compositionally charged clusters and activated multi-component CC. To do this, the system is represented as a triangle (Figures 1-3), with CE ions placed at its vertices.

2)      It was found that the geometric features of the triangle representing the system of CE ions consist of representing the reaction of the interaction between any pair of reacting system components as a line segment, where each pair of reactants and the product of their interaction lie on a line segment unique to them. In cases where line segments connecting different pairs of chemically interacting system components intersect at a single point, the common product of the interaction, ChCl or activated CC, must be located at that point. This feature is explained by the difference in the laws governing the interaction of different pairs of reactants, which are determined by the different combinations of valence ions of CE and the composition of interacting pairs of reactants represented by different line segments: at the point of intersection of these line segments, different laws of chemical interaction of different pairs of chemical entities cannot coexist simultaneously.

Thus, the calculation of HS formulas, i.e., the search for new HS homologs, is based on the premise that homologs are located within a triangle at the intersection of line segments connecting different pairs of chemically interacting components of the system, including ions, CC and ChCl.

3)      Next, we will consider that homologs enriched with the two-component CC (TwCC), A_cC_a, belong to the p-groups of HS, while homologs enriched with the B_cC_b cluster belong to the m-groups of HS – Figure 1-3. HS are formed depending on the direction of development through a chain of sequentially occurring interactions of TCC with cation A^a+ – the development direction of p-groups of HS is A_cC_a. When TCC interacts with cation B^b+ – the m-groups of HS develop towards B_cC_b.

The formation of HS involves three-component charge clusters (TChCl_n) through interaction with anion С^с– (Figures 1-3).

Figure 1: System (Aa+ – Bb+ – Cc–): αp-HS (AcCa direction) and αm-HS (BcCb direction). αp-HS: (p. 1 = TChCln(bas) = 1 = [AbcBacCab]abc+), (p. 2 = TCCn(bas) = 1 = AbcBacC2ab) (p. 3 = TwChCln(bas) = 1 = [AbcBac]2abc+), (p. 4 = TChCln = 2 = [A2bcBacC2ab]abc+), (p. 5 = TCCn = 2 = A2bcBacC3ab, (p. 6 = TwChCln = 2 = [A2bcBac]3abc+); αm-HS: (p. 1 = TChCln(bas) = 1 = [AbcBacCab]abc+), (p. 2 = TCCn(bas) = 1 = AbcBacC2ab), (p. 3 = TwChCln(bas) = 1 = [AbcBac]2abc+), (p. 7 = TChCln = 2 = [AbcB2acC2ab]abc+), (p. 8 = TCCn = 2 = AbcB2acC3ab, (p. 9 = TwChCln = 2 = [AbcB2ac]3abc+).

Figure 2: System (Aa+ – Bb+ – Cc–): p-group HS (AcCa direction). (p. 10 = TCCn(bas) = 1 = AtbcBracC(t + r)ab), (p. 11 = TChCln(bas) = 1 = [AtbcBracCt ab]rabc+), (p. 12 = TwChCln(bas) = 1 = [AtbcBrac](t + r)abc+), (p. 13 = TChCln = 2 = [A(t + r)bcBracC(t + r)ab]rabc+), (p. 14 = TCCn = 2 = A(t + r)bcBracC(t + 2r)ab, (p. 15 = TwChCln = 2 = [A(t + r)bcBrac](t + 2r)abc+), (p. 16 = TChCln = 3 = A(t + 2r)bcBracC(t +2 r)ab]rabc+), (p. 17 = TCCn = 3 = A(t + 2r)bcBracC(t + 3r)ab, (p. 18 = TwChCln = 3 = [A(t + 2r)bcBrac](t + 3r)abc+).

Figure 3: System (Aa+ – Bb+ – Cc–): m-group HS (BcCb direction). (p. 19 = TCCn(bas) = 1 = AtbcBracC(t + r)ab), (p. 20 = TChCln(bas) = 1 = [AtbcBracCrab]tabc+), (p. 21 = TwChCln(bas) = 1 = [AtbcBrac](t + r)abc+), (p. 22 = TChCln = 2 = [AtbcB(t + r)acC(t + r)ab]tabc+),(p. 23 = TCCn = 2 = AtbcB(t + r)acC(2t + r)ab, (p. 24 = TwChCln = 2 = [AtbcB(t + r)ac](2t + r)abc+), (p. 25 = TChCln = 3 = [AtbcB(2t + r)acC(2t + r)ab]tabc+), (p. 26 = TCCn = 3 = AtbcB(2t + r)acC(3t + r)ab, (p. 27 = TwChCln = 3 = [AtbcB(2t + r)ac](3t + r)abc+).

The formation of HS is described by the following scheme:

TChCl _{n = 1} + C^c- ® TCC _{n = 1}, TCC _{n = 1} + A^a+ (or B^b+) ® TChCl _{n = 2}, TChCl _{n = 2} + C^c- ® TCC _{n = 2}, TCC _{n = 2} + A^a+ (or B^b+) ® TChCl _{n = 3}, TChCl _{n = 3} + C^c- ® TCC _{n = 3}…………………………….(1)

In the text, the determined formulas of CC and ChCl as reactants and products of their interaction are highlighted in bold. The value of n is determined experimentally, where n represents the position of the homolog in HS, (1 ≤ n).

4)      Activated ТХС_n are located on line segments (A_cC_a – B_cC_b) – Figures 1-3. In works [44Undalov YK. Russian. Inorganic Chemistry. 1998; 43:1447.-66Undalov YK. Russian. Inorganic Chemistry. 1999; 44:1479.], the nature of the connection of the same TCC_n, which simultaneously belongs to both the p-groups and m-groups of HS, is described. Following the conditions outlined in paragraph 3 regarding the membership of TCC in p-groups and m-groups of HS, it can be concluded that in the case of p-groups of HS, its first homologs, TCC_{n = 1}, are located on the line segment (p. 2 – B_cC_b), excluding B_cC_b, and the clusters TCC_{n > 1} are located on the line segment (p. 2 – A_cC_a), excluding point 2 and A_cC_a (Figures 1,2).

In the case where the same TCC_n cluster belongs to the m-group of HS, its first homologs, TCC_{n = 1}, are located on the line segment (p. 2 – A_cC_a), excluding A_cC_a. The clusters TCC_{n > 1} are located on the line segment (p. 2 – B_cC_b), excluding p. 2 and B_cC_b (Figures 1,3).

5)      Clusters TChCl_{n = 1}, belonging to p-groups of HS, are located on the line segment (p. 1 – B^b+), excluding B^b+, and clusters TChCl_{n > 1} are located on the line segment (p, 1 – A_cC_a), excluding p. 1 and A_cC_a – Figures 1,2. In the text and in Figure 1, the following notations are used: p. 1 º point 1, p. 2 º point 2, p. 3 º point 3, and so on.

In the case where the TChCl_n cluster belongs to the m-group of HS, its first homologs, TChCl_{n = 1}, are located on the line segment (p. 1 – A^a+), excluding A^a+, and clusters TChCl_{n > 1} are located on the line segment (p, 1 – B_cC_b), excluding p. 1 and B_cC_b (Figures 1,3).

6)      TCC_n and TChCl_n, occupying the same position in the same HS, are linked by the reaction:
TChCl_n + С^с– ® TCC_n……………………….......……………………(2)

7)      Any known TCC is necessarily a member of some HS.

8)     Any HS consists of a CC branch and a ChCl branch, the members of which are linked by reaction (2). Each branch of the same HS develops towards enrichment with only one Two-component CC (TwCC), either A_cC_a or B_cC_b.

9)      The geometric features of the triangle, if following scheme (1), ensure a systematic and periodic The geometric features of the triangle, if following scheme (1), ensure a systematic and periodic change in the composition of homologs in HS.

10)   The difference in compositions, ∆, of any adjacent homologs in the same HS is constant:
∆_m(p) = TCC_{n + 1} – TCC_n = TChCl_{n + 1} – TChCl_n  ........................,.(3)

11)    The charges of all ChCl in the same HS are identical.

12)    In the case of determining the formulas of HS to which some known (basic) TCC_n(bas) belongs, the calculation of HS formulas is carried out as follows: first, the formulas of TChCl_n(bas), TCC_{n(bas) + 1}, and TChCl_{n(bas) + 1} are calculated, then the formulas of ∆ and the first terms, TCC_{n = 1} and TChCl_{n = 1}, of the considered HS are determined. Formulas TCC_{n = 1} and TChCl_{n = 1} are calculated by subtracting the maximum number of times the formula ∆_m(p) from the formulas of basic clusters while retaining the minimum number of the cation contained in the formula ∆_m(p) in their composition:

TCC_n(bas) – k∙∆_m(p) = TCC_{n = 1}…………...………………………. (4)

TChCl_n(bas) – k∙∆_m(p) = TChCl_{n = 1}…...……….......................(5)

where (0 ≤ k). If k = 0, then n(bas) = 1.

13)    The formula of any homolog in the same HS, i.e., the formula of the HS, is determined as follows:

CC branch: TCC_{n = 1} + (n – 1)·∆_m(p) = TCC_n..………………(6)

ChCl branch: TChCl_{n = 1} + (n – 1)·∆_m(p) = TChCl_n….…….(7)

14)    When calculating HS, it should be taken into account that one of the CE may have different valencies, being not only a component that does not introduce impurities into the crystalline lattice of TCC but also one of the main CE in the crystalline lattice.

In practice, researchers often need to determine the formulas of HS to which a known TCC_n(bas) already belongs. In the formation of these HS, any TCC_n can participate, including TCC_{n = 1} and TCC_{n > 1}.

Calculation of homologous series in the (A^a+ – B^b+ – C^c–) system

Let’s examine the sequentially occurring chemical reactions in a three-component system.
The formation of HS occurs according to scheme (1). The system (A^a+ – B^b+ – C^c–) is initially in a state where the interaction of a positively charged ion with an anion will lead to the formation of activated TwCC, A_cC_a, and B_cC_b (Figure 1).

cA^a+ + aC^c- = A_cC_a…………………………………………………. (8)

cB^b+ + bC^c- = B_cC_b…………………………………………………. (9)

Activated TwCC, A_cC_a and B_cC_b can interact with each other to form TCC_n based on the ratios of ions A^a+, B^b+ and C^c- in TCC_n based on the ratios of ions A^a+, B^b+ and C^c- in TChCl_n, which participate in reaction (2):

x∙A_cC_a + y∙B_cC_b = (A_xcB_ycC_{(x + y)ab} = TCC_n)0………………...(10)

Furthermore, according to the initial ratios of ions in TCC_n and TChCl_n, the interaction of positively charged ions with each other leads to the formation of two-component charged clusters (TwChCl_n), which are connected to TCC_n and TChCl_n by reaction (2), where (1 ≤ n):

x∙A^a+ + y∙B^b+ = ([A_xbB_ya]^{(x + y)ab+} = TwChCl_n)……………... (11)

The combined interaction of (TChCl_{n = 1} = p. 1) and TwChCl_{n = 1} with anion C^c- according to (2), as well as activated A_cC_a and B_cC_b with each other leads to the formation of a cluster of (TCC_{n = 1} = p. 1) – Figure 1. Thus, the clusters (TChCl_{n = 1} = p. 1), (TwChCl_{n = 1} = p. 3) and (TCC_{n = 1} = p. 2) turn out to be the founders of two groups of HS, αm-group and αp-group, since (TChCl_{n = 1} = p. 1) and (TCC_{n = 1} = p. 2) can interact with both B^b+ and A^a+, respectively. The interactions mentioned are characterized by the intersection of segments corresponding to the reacting components of the CE system. Thus, the formation of different clusters TChCl_n are characterized by the intersection of segments (A_cC_a – B^b+) and (B_cC_b – A^a+), and the formation of cluster TCC_n is characterized by the intersection of segments (TChCl_n – C^c–), (A^a+ – B^b+) and (A_cC_a – B_cC_b), which is described by the following reaction equations in the formation of the αm-group of HS and αp-group of HS (Figure 1).

bA_cC_a + acB^b+ = aB_cC_b + bA^a+ = ([A_bcB_acC_ab]^abc+ = TChCl_{n = 1} = p. 1)………………………….(12)

([A_bcB_acC_ab]^abc+ = TChCl_{n = 1} = p. 1) + abC^c- = {bcA^a+ + acB^b+ = ([A_bcB_ac]^2abc+ =TwChCl_{n = 1} = p. 3) + 2abC^c- = bA_cC_a + aB_cC_b = (A_bcB_acC_2ab = TCC_{n = 1} = p. 2)…………………………….(13)

Calculation of the formula for αp-HS developing towards A_cC_a: The continuous nature of the chemical interaction of the components of the system (A^a+ – B^b+ – C^c-) and the formation of HS implies that the formation of p-groups of HS is described by the interaction of TCC_n with A^a+. The interaction of TChCl_n and TwChCl_n with C^c- is characterized by the intersection of segments (TCC_n – A^a+) and (B^b+ – A_cC_a) at the point corresponding to TChCl_{n + 1}, including the intersection of segments (TCC_{n + 1} – C^c-) and (TwChCl_n – C^c-) with the segment (A_cC_a – B_cC_b) at the point corresponding to TCC_{n + 1} – Figure 1. In turn, clusters (A_bcB_acC_2ab = TCC_{n = 1} = p. 2) interact with A^a+, and activated A_cC_a reacting with B^b+ to form (TChCl_{n = 2}=p. 4), and clusters (TwChCl_{n + 1} = p. 6) and (TChCl_{n + 1} = p. 4) interact with C^c- and form (TCC_{n + 1} = p. 5) – Figure 1. These interactions begin to form αp-HS, the members of which, as this HS develops, are enriched with the A_cC_a component, which for αp-HS is described by the reaction equation (Figure 1).

(A_bcB_acC_2ab = TCC_{n = 1} = p. 2) + bcA^a+ = acB^b+ + 2bA_cC_a = ([A_2bcB_acC_2ab]^abc+ = TChCl_{n = 2} = p. 4).…..............................(14)

([A_2bcB_acC_2ab]^abc+ = TChCl_{n = 2} = p. 4) + abC^c- = {2bcA^a+ + acB^b+ = ([A_2bcB_ac]^3abc+ = TwChCl_{n = 2} = p. 6)} = 2bA_cC_a + aB_cC_b = aB_cC_b = (A_2bcB_2acC_3ab = TCC_{n = 2} = p. 5)……………………….(15)

According to (3), (6) and (7) the formula ∆_αp is determined as follows:

∆_αp = (A_2bcB_acC_3ab = TCC_{n = 2} = p. 5) – (A_bcB_acC_2ab= TCC_{n = 1} = p. 2) = [A_2bcB_acC_2ab]^abc+ =TChCl_{n = 2} = p. 4) – ([A_bcB_acC_ab]^abc+ = TChCl_{n = 1} = p. 1) = A_bcC_ab…………………………………………(16)

Knowledge of the formulas of the first homologs (TCC_{n = 1} = p. 2), (TChCl_{n = 1} = p. 1), and ∆_αp allows us to determine the formulas of both branches of αp-HS:

CC branch in αp-HS: (A_bcB_acC_2ab= TCC_{n = 1} = p. 2) + (n – 1)A_bcC_ab = A_nbcB_acC_{(n + 1)ab} ………..............................................(17)

ChCl branch in αp-HS: ([A_bcB_acC_ab]^abc+ = TChCl_{n = 1} = p. 1) + (n – 1)A_bcC_ab = [A_nbcB_acC_nab]^abc+…………………………………...(18)

The formulas for both branches of HS in other p-group HS are determined in accordance with dependencies (4)-(7).

Calculation of the formula for αm-HS developing towards B_cC_b: In the case where the cluster (TCC_{n = 1} = p. 2) reacts with cation B^b+ to form (TChCl_{n = 2} = p. 7), and cluster (TChCl_{n = 2} = p. 7) reacts with anion C^c-, and to form (TCC_{n = 2} = p. 8) but they participate in the formation of the αm-group of HS (Figure 1).

(A_bcB_acC_2ab = TCC_{n = 1} = p. 2) + acB^b+ = bcA^a+ + 2aB_cC_b = ([A_bcB_2acC_2ab]^abc+ = TChCl_{n = 2} = p. 7)……………………………..(19)

([A_bcB_2acC_2ab]^abc+ = TChCl_{n = 2} = p. 7) + abC^c- = {2bcA^a+ + acB^b+ = ([A_bcB_2ac]^3abc+ = TwChCl_{n = 2} = p. 9) + 2abC^c- = bA_cC_a + 2aB_cC_b = aB_cC_b = (A_bcB_2acC_3ab = TCC_{n = 2} = p. 8)…………….(20)

According to (3), (6), and (7) the formula ∆_αm, TCC_n, and TChCl_n are determined by the formulas of both branches of αm-HS:

∆_αm = (A_bcB_2acC_3ab = TCC_{n = 2} = p. 5) – (A_bcB_acC_2ab= TCC_{n = 1} = p. 2) = [A_bcB_2acC_2ab]^abc+ = TChCl_{n = 2} = p. 4) – ([A_bcB_acC_ab]^abc+ = TChCl_{n = 1} = p. 1) = B_acC_ab……..........………………………(21)

CC branch in αm-HS: (A_bcB_acC_2ab= TCC_{n = 1} = p. 2) + (n – 1)B_acC_ab = A_bcB_nacC_{(n + 1)ab} ………..............................................(22)

ChCl branch in αm-HS: ([A_bcB_acC_ab]^abc+ =TChCl_{n = 1} = p. 1) + (n – 1)B_acC_ab = [A_bcB_nacC_nab]^abc+…………………………………(23)

The formulas for both branches of HS in other m-group HS are determined in accordance with dependencies (4)-(7).

Calculation of the generalized formula for homologous series to which a known (Basic) TCC_n(bas) belong: In the process of designing various devices, researchers often face the task of determining the formulas of TCC that would possess more suitable properties compared to the currently used TCC in the device. In situations where the composition of the utilized TCC is known and experimentally confirmed, we will consider this cluster as the basic one TCC_n(bas). The problem of finding new TCC_{n ≠ n(bas)} can be solved by identifying the formulas of HS to which the basic TCC_n(bas) belongs. In the desired HS, the fundamental properties of homologs, including TCC_n(bas), change periodically and systematically [88Eriemin EN. Fundamentals of chemical kinetics. Visshaia Shkola. 1976.,99Urusov VS. Theoreticalcrystal chemistry. Moscow State University. 1987.]. To determine the regularity of HS formation in a generalized form, it is necessary to represent the formula of TCC_n(bas) appropriately. Thus, the formula of any TCC, including TCC_n(bas), can be expressed in a generalized form as follows:

tA_cC_a + rB_cC_b = A_tbcB_racC_{(t + r)ab}…………………………………(24)

Where (0 < t, r).

Calculation of the generalized formula for the p-Group of ГС to which a known (Basic) (TCC_n(bas) = A_tbcB_racC_{(t + r)ab}) belongs: In accordance with paragraths 3-5, the basic cluster, represented in a generalized form as (A_tbcB_racC_{(t + r)ab} = TCC_n(bas)), begins to form the p-group of HS by interacting with A^a+. In this case, the formulas of TChCl_n(bas) and TwChCl_n(bas), associated with TCC_n(bas) by reaction (2), are determined by the intersection of segments {((A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 10) – p. 12) – A^a+}, and {(TwChCl_n(bas) = p. 12) – C^c-} and (B^b+ – A_cC_a), which is described by the reaction equation (Figure 2).

{racB^b+ + tbA_cC_a = ([A_tbcB_racC_tab]^rabc+ = TChCl_n(bas) = p. 11)} + rabC^c– = {tbcA^a+ + racB^b+ = ([A_tbcB_rac]^{(t + r)abc+} = TwChCl_n(bas)) = p. 12} + (t + r)abC^c– = (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 10).....(25)

The initiation of the formation of the p-group of HS based on (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 10) is characterized by the intersection of segments (TCC_n(bas) – A^a+), and (B^b+ – A_cC_a) at a point corresponding to (TChCl_{n(bas) + 1} = p. 13), as well as the intersection of segments {(TChCl_{n(bas) + 1} = p. 13) – С^с–} and {(TwChCl_{n(bas) + 1} = p. 12) – С^с–} with the segment (A_cC_a – B_cC_b) at a point corresponding to (TCC_{n(bas) + 1} = p. 14), which is described by the reaction equations (Figure 2).

(A_tbcB_racC_{(t + r)ab} = TCC_{n(bas) = 1} = p.10) + rbcA^a+ = racB^b+ + (t + r)
bA_cC_a = ([A_{(t + r)bc}B_racC_{(t + r)ab}]^rabc+ = TChCl_{n(bas) + 1)} = p. 13)….(26)

([A_{(t + r)bc}B_racC_{(t + r)ab}]^rabc+ = TChCl_{n(bas) + 1} = p. 13) + rabC^c– = (t + r)bA_cC_a + raB_cC_b = {(t + r)bcA^a+ + racB^b+ = (TwChCl_{n(bas) + 1} = [A_{(t + r)b}cB_rac]^{(t + 2r)abc+} = p. 15)} + (t + 2r)abC^c- = (A_{(t + r)bc}B_racC_{(t + 2r)ab} = TCC_{n(bas) + 1} = p. 14)..……………....................(27)

The formula Δ_p, TCC_{n = 1} and TChCl_{n = 1} for the p-group of HS is determined according to (3)-(5), (25)-(27):
Δ_p = (A_{(t + r)bc}B_racC_{(t + 2r)ab} = TCC_{n(bas) + 1} = p. 14) – (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 10) = ([A_{(t + r)bc}B_racC_{(t + r)ab}]^rabc+ = TChCl_{n(bas) + 1} = p. 13) – ([A_tbcB_racC_rab]^tabc+ = TChCl_n(bas) = p. 11) = A_rbcC_rab .................(28)
TCC_{n = 1} = (A_tbcB_racC_{(t + r)ab}= TCC_n(bas) = p. 10) – k·A_rbcC_rab = A_{(t – k·r)bc}B_racC_{(t + r – k·r))ab}…….…...(29)
TChCl_{n = 1} = ([A_tbcB_racC_tab]^rabc+ = TChCl_n(bas) = p. 11) – k·A_rbcC_rab =[A_{(t – k·r)bc}B_racC_{(t –k·r)ab}]^rabc+......……………………….....………...(30)

 Where (0 ≤ k) and k characterizes the positioning of TCC_n(bas) in HS, and for αp-HS corresponds n(bas) = 1 and k = 0.

As can be seen, the determination of the composition of the first homologs and the formulas of both branches of the sought p-group of HS according to the formulas (29), (30), (6), and (7) depends on the relationship between the values t and k·r. In this case, two variants are possible:

1)      Inequality (t ≤ k·r) when (t, r, k > 0) does not allow obtaining formulas for TCC_{n = 1} and TChCl_{n = 1} according to (29) and (30) by subtracting the product (k·Δ_p º k·A_rbcC_rab) from the formulas (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 10) and ([A_tbcB_racC_tab]^rabc+ = TChCl_nbas) = p.11). It follows that (k = 0) and n(bas) = 1. This fast indicates that when (t ≤ k·r), the basic cluster is the first homolog (A_tbcB_racC_{(t + r)ab} = TCC_{n(bas) = 1} = p. 10) in this p-group of HS. Therefore, any homologue or both branches of the desired p-group of HS is determined according to (6), (7), and (28):

CC branch in p-group of HS – (A_tbcB_racC_{(t + r)ab} = TCC_{n(bas) = 1} = p. 10) + (n – 1)A_rbcC_rab = A_{{t + r(n – 1)}bc}B_racC_{{t + rn}ab}…............(31)

ChCl branch in p-group of HS – ([A_tbcB_racC_tab]^rabc+ = TChCl_{n(bas) = 1} = p. 11) + (n – 1)A_rbcC_rab = [A_{{t + r(n – 1)}b}cB_racC_{{t + r(n – 1)}ab}]^rabc+…………............................................................(32)

2)      In the case when the inequality (k·r < t) holds, to determine the composition of the first homologs, it is possible to subtract the product (k·∆_p º k·A_rbcC_rab) from the formulas of the basic clusters (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 10) and ([A_tbcB_racC_tab]^rabc+ = TChCl_n(bas) = p. 11), i.e. at (0 < k) and {n(bas) > 1}. Then the first members of TCC_{n = 1} and TChCl_{n = 1} of the sought p-group of HS will be determined by expressions (29) and (30). Any homologue or both branches of the p- p-group of HS will be determined according to (6), (7), (28), (29), and (30):

CC branch in p-group of HS – (A_{(t – k·r)bc}B_racC_{(t + r – k·r))ab} = TCC_{n = 1}) + (n – 1)A_rbcC_rab = A_{{t – k·r + (n – 1)r}bc}B_racC_{{t – k·r + nr}ab}....(33)

ChCl branch in p-group of HS – ([A_{(t – k·r)bc}B_racC_{(t – k·r)ab}]^rabc+= TChCl_{n = 1}) + (n – 1)A_rbcC_rab = [A _{{t – k·r + (n – 1)r}bc}B_racC_{{t – k·r + (n – 1)r}ab}]^rabc+….........………………….……………………………….(34)

 It should be noted that for the p-group of HS in both cases (t ≤ k·r) and (kr < t) cluster TCC_{n = 1} is located in the p. 2, and cluster TCC_{n(bas) > 1} should be on the segment (p. 2 – A_cC_a) excluding p. 2 and A_cC_a. Cluster TChCl_{n = 1} is located in the on the segment (p. 1 – B^b+) excluding p. 1 and cluster TCC_{n(bas) > 1} should be on the segment (p. 1 – A_cC_a) excluding p. 1 and A_cC_a (Figure 2).

Comparing the concentration coefficients for A^a+ in the formula of the base cluster (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 10) and in the formula of the homologue in the p-group HS in the generalized form A_{t – k·r + {n(bas) – 1)r}bc}B_racC_{{t – k·r + n(bas)∙r}ab} from (33), it can be seen that there is equality {rn(bas) – r – k·r} = 0 holds. Then for (r ≠ 0) we get:

k = {n(bas) – 1}..…………………………………………………(35)

In reality, the value of k is determined during the calculation of the p-group HS in accordance with the possibility of subtracting the product (k·∆_p º k·A_rbcC_rab) from the formula {A_tbcB_racC_{(t + r)ab} = TCC_n(bas)}: k·r and t are compared. The values of t and r are determined by the formula TCC_n(bas). By setting the value of n, it is possible, in accordance with the formulas (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 10), (33) and (35), to calculate the formula of any homologue of TCC_n belonging to the p-group of HS.

Calculation of the generalized formula for the m-group of hs to which the known (basic) (TCC_n(bas) = A_tbcB_racC_{(t + r)ab})
belongs: Let the cluster TCC_n(bas), the formula of which is represented in a generalized form as (A_tbcB_racC_{(t +r)ab} = TCC_n(bas) = p. 19), interact with B^b+ to start forming the HS, which in this case will belong to the m-group of HS. In this process, the formulas for (TChCl_n(bas) = p. 20) and (TwChCl_n(bas) = p. 21) related to TCC_n(bas) by reaction (2) are determined by the composition of TCC_n(bas), which is described by the reaction equation:

{tbcA^a+ + racB^b+ = ([A_tbcB_rac]^{(t + r)abc+} = TwChCl_n(bas) = p. 21)} + (t + r)abC^c- = {tbcA^a+ + raB_cC_b = ([A_tbcB_racC_rab]^tabc+ = TChCl_n(bas) = p. 20) + tabC^c– = (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 19)……(36)

The formation of the m-group of HS based on (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 19) is characterized by the intersection of the segments (TCC_n(bas) – B^b+) and (A^a+ – B_cC_b) at the point corresponding to (TChCl_{n(bas) + 1} = p. 22), as well as the intersection of the segments {(TChCl_{n(bas) + 1} = p. 22) – С^с-} and {(TwChCl_{n(bas) + 1} = p. 24) – С^с-} with the segment (A_cC_a – B_cC_b) at the point corresponding to (TCC_{n(bas) + 1} = p. 23) – Figure 3, which is described by the following reaction equations:

(A_tbcB_racC_{(t + r)ab} = TCC_{n(bas) = 1} = p.19) + tacB^b+ = tbcA^a+ + (t + r)aB_cC_b = ([A_tbcB_{(t + r)ac}C_{(t + r)ab}]^tabc+ = TChCl_{n(bas) + 1 = 2} = p.22)………………..…………………………(37)

([A_tbcB_{(t + r)ac}C_{(t + r)ab}]^tabc+ = TChCl_{n(bas) + 1 = 2} = p.22) + tabC^c– = tbA_cC_a + (t + r)aB_cC_b = {tbcA^a+ + (t + r)acB^b+ = (TwChCl_{n(bas) + 1 = 2} = [A_tbcB_{(t + r)ac}]^{(2t + r)abc+} = p. 24)} + (t + r)abC^c- = (A_tbcB_{(t + r)ac}C_{(2t + r)ab} = TCC_{n(bas) + 1 = 2} = p. 23)….……………………...........(38)

The formula Δ_m, TCC_{n = 1,} and TChCl_{n = 1} are determined according to (3), (4), (5), (36)-(38):

Δ_m = (A_tbcB_{(t + r)ac}C_{(2t + r)ab} = TCC_{n(bas) + 1}) – (A_tbcB_racC_{(t + r)ab} = TCC_n(bas)) = ([A_tbcB_{(t + r)ac}C_{(t + r)ab}]^tabc+ = TChCl_{n(bas) + 1} = TChCl_{n = 2}) – ([A_tbcB_racC_rab]^tabc+ = TChCl_n(bas)) = B_tacC_t…………………...(39)

TCC_{n = 1} = (A_tbcB_racC_{(t +r)ab}= TCC_n(bas) = p. 19) – k·B_tacC_tab = A_tbcB_{(r – k·t)ac}C_{(t + r – k·t))ab}.………….(40)

TChCl_{n = 1}= ([A_tbcB_racC_rab]^tabc+ = TChCl_nbas) = p. 20) – k·B_tacC_tab = [A_tbcB_{(r – k·t)ac}C_{(r – k·t)ab}]^tabc+…(41)

As can be seen from equations (40) and (41), determining the composition of the first homologs and formulas of both branches of the sought m-group of HS depends on the relationship between the values of r and k·t. In this regard, two variants are possible:

1)      In the case of the inequality (r ≤ k·t) with {0 < t, r, k}, it is not possible to obtain the compositions of TCC_{n = 1} and TChCl_{n = 1} according to (4) and (5) by subtracting the formula (Δ_m = B_tacC_tab) from the formulas (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 19) and ([A_tbcB_racC_rab]^tabc+ = TChCl_n(bas) = p. 20). In this scenario, it implies that (k = 0), and {n(bas) = 1}. Consequently, both branches of the m-group of HS can be determined according to (6), (7), and (39)-(41):

CC branch of m-group of HS – (A_tbcB_racC_{(t + r)ab} = TCC_{n(bas) = 1} = p. 19) + (n – 1)B_tacC_tab = A_tbcB_{{r + t(n – 1)}ac}C_{{r + tn}ab}……....(42)

ChCl branch of m-group of HS – ([A_tbcB_racC_rab]^tabc+ = TChCl_{n(bas) = 1} = p. 20) + (n – 1)B_tacC_tab = [A_tbcB_{{r + t(n – 1)}ac}C_{{r + t(n – 1)}ab}]^tabc+………….............................................................(43)

2)      In the case of the inequality (k·t < r), to determine the formulas of the first homologs, it is possible to subtract the product (k·∆_m º k·B_tacC_tab) from the composition of the basic clusters (A_tbcB_racC(_{t +r)a}b = TCC_n(bas) = p. 19) and ([A_tbcB_racC_rab]^tabc+ = TChCl_{n(bas) = 1} = p. 20), i.e., (0 < k). In this scenario, the first members of the sought m-group of HS will be determined by equations (37) and (38). Both branches of the m-group of HS will be determined according to (6), (7), (36), (37), and (38):

CC branch of m-group of HS – A_tbcB_{{r – k·t}ac}C_{{t + r – k·t}ab} = TCC_{n = 1}) + (n –1)B_tacC_tab = A_tbcB_{{r – kt + (n – 1)t}ac}C_{{r – k·t + nt}ab} ….(44)

ChCl branch of m-group of HS – ([A_tbcB_{(r – k·t)ac}C_(r–k·t)ab]^tabc+= TChCl_{n = 1}) + (n – 1)B_tacC_tab = [A_tbcB_{{r – k·t + (n – 1)t}ac}C_{{r – k·t + (n – 1)t}ab}]^tabc+…………………………………………………….........….(45)

It should be noted that for the m-group of HS in both cases (r ≤ k·t) and (k·t < r) cluster TCC_{n = 1} is located in the p. 2, and cluster TCC_{n(bas) > 1} should be on the segment (p. 2 – B_cC_b) excluding p. 2 and B_cC_b. Cluster TChCl_{n = 1} is located in the on the segment (p. 1 – A^a+) excluding p. 1 and cluster TCC_{n(bas) > 1} should be on the segment (p. 1 – B_cC_b) excluding p. 1 and B_cC_b (Figure 3).

Comparing the concentration coefficients for B^b+ in the formula of the base cluster (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 19) and in the formula of the homologue in the m-group HS in the generalized form A_tbcB_{{r – kt + (n – 1)t}ac}C_{{r – k·t + nt}ab} from (44), it can be seen that there is equality {t∙n(bas) – t – k·t = 0} holds. Then for (t ≠ 0) we get:

{k = {n(bas) – 1}……………………………………....…………(46)

In reality, the value of k is determined during the calculation of the m-group HS in accordance with the possibility of subtracting the product (k·∆_m º k·A_rbcC_rab) from the formula {A_tbcB_racC_{(t + r)ab} = TCC_n(bas)}: k·t and r are compared. The values of t and r are determined by the formula TCC_n(bas). By setting the value of n, it is possible, in accordance with the formulas (A_tbcB_racC_{(t + r)ab} = TCC_n(bas) = p. 19), (44), and (45), to calculate the formula of any homologue of TCC_n belonging to the m-group of HS.

Calculation of the generalized formula for the m-group of HS developing towards Ge₃P₄ in the system (Zn²⁺ - Ge⁴⁺ - P^3-) to which the known (basic) (TCC_n(bas) = ZnGeP₂) belongs:

As an example of the practical application of the above HS formulas in a generalized form, we calculate the formula of the m-group of HS, which belongs to the well-known (basic) (TCC_n(bas) = ZnGeP₂ º A_tbcB_racC_{(t + r)ab}). Attention to the system (Zn²⁺ – Ge⁴⁺ – P^3-) and, in particular, to ZnGeP₂ is caused by the fact that the ZnGeP₂ compound, having a chalcopyrite crystal structure, has highly effective nonlinear optical characteristics that are used for parametric conversion of medium and far infrared radiation range, as well as for terahertz radiation [2525Zhang S, Xie LH, Ouyang SD. Phys. Scr. 2016; 91:015801;6. DOI: 10.1088/0031-8949/91/1/015801]. The calculated HS formula based on (TCC_n(bas) = ZnGeP₂) will allow us to determine the formulas of other unknown TCC-homologues that may be potential substitutes for the compound ZnGeP₂.

When solving this problem, we agree that homologues, as they develop m-HS, enriched by a cluster (Ge₃P₄ º B_cC_b).

So, for (TCC_n(bas) = ZnGeP₂ º A_tbcB_racC_{(t + r)ab}) we get: A^a+ º Zn²⁺, B^b+ º Ge⁴⁺, C^c– º P^3–, bc = 12, ac = 6, ab = 8, abc = 24, tbc = 1, t = 1/12, rac = 1, r = 1/6, (t + r)ab = 2, (t + r) = 2/8. Then the following is true:

(ТCС_n(bas) = A_tbcB_racC_{(t +r)ab} º ZnGeP₂ º Zn_12/12Ge_6/6P_16/8)...(47)

In order to determine the position of the base cluster in the HS m-group, it is necessary to determine the formula Δ_m (direction Ge₃P₄) in accordance with (39):

Δ_m = B_tacC_tab = Ge_6tP_8t º Ge_6/12P_8/12………………………… (48)

To determine the value of k, it is necessary to compare the formula (Δ_m = Ge_6/12P_8/12) with the cluster formula (TCC_n(bas) = Zn_12/12Ge_6/6P_16/8) from (47): when comparing, it can be seen that (r = = 1/6) > (t = 1/12). Therefore, the formula {Δ_m = Ge_6/12P_8/12} from the formula (TCC_n(bas) = Zn_12/12Ge_6/6P_16/8) can be subtracted only once. In this case (k = 1) and {n(bass) = 2}. Then we get the formula TCC_{n = 1} according to (40):

(ТCС_{n = 1} = A_tbcB_{(r – k·t)ac}C_{(t + r – k·t))ab} = Zn_12/12Ge_6/12P_16/12 º Zn₆Ge₃P₈)
........(49)

The CC branch of the m-group of HS, to which ZnGeP₂ belongs, can be represented as follows:

The branch of the CC m-group is A_tbcB_{{r – k∙t + (n – 1)t}ac}C_{{r – k∙t + nt}ab} º Zn_12tGe_{3(n∙t – t + r – t∙k)}P_{8{(n∙t + r – t∙k}} º Zn_12/12Ge_{(3n + 3 – 3k)/12}P_{(8n + 16 – 8k)/12} º Zn₆Ge_3nP_{(n + 1)}   .........................................................(50)

Since the compound experimentally obtained in [2525Zhang S, Xie LH, Ouyang SD. Phys. Scr. 2016; 91:015801;6. DOI: 10.1088/0031-8949/91/1/015801] (TCC_{n = 2} = ZnGeP₂) in the desired HS m-group turned out to be the second member, given the continuity of the HS cluster (Zn₆Ge₃P₈ = TCC_{n = 1}) must exist. As can be seen, the relative germanium content in the cluster (TCC_{n = 1} = Zn₆Ge₃P₈) is less than in the cluster (TCC_{n = 2} = ZnGeP₂), which in the synthesis of Zn₆Ge₃P₈ can give some advantages over ZnGeP₂ while preserving the basic nonlinear optical characteristics of the material.

According to (19), the formula of the first homologue (TCC_{n = 1} = p. 2 = A_bcB_acC_2ab = Zn₁₂Ge₆P₁₆) in αm-HS and the formula (TCC_{n = 1}= A_tbcB_{(r – k∙t)ac}C_{(t + r – k∙t))ab} = Zn_12/12Ge_6/12P_16/12 º Zn₁₂Ge₆P₁₆ º Zn₆Ge₃P₈) coincide, which indicates that formula (50) belongs to the CC branch of the αm-HS, and the base cluster (ZnGeP₂ = TCC_{n(bas) = 2}) belongs to the αm-HS.

Comparison of experiments (known from the literature) with the results of the calculation of HS

The operability of the method of calculating the HS of chemical compounds presented here and in [44Undalov YK. Russian. Inorganic Chemistry. 1998; 43:1447.]-[77Undalov YК, Terukov EI, Agafonov DV. Izvestia of St. Petersburg State Institute of Technology (Technical University). 2021; 59(85):26. DOI:10.36807/1998-9849-2021-59-85-26-36] is confirmed by numerous experiments taken from the literature [1010Kovba LM. Stoichiometry, defects in crystals and structural homology. Chemistry. 1996.]: so, if in formula (25), A_bcB_nacC_{(n + 1)ab}, replace A^a+ º Li⁺, Na⁺, K⁺, Rb⁺, etc., B^b+ º Ti⁴⁺, C^c– º O^2–, then for the systems (Na⁺ – Ti⁴⁺– O^2–) and (Li⁺ – Ti⁴⁺– O^2–), the following can be obtained:

1)      Thirteen experimentally obtained compounds of the system (Na⁺ –Ti⁴⁺– O^2–): n = 1 [21-2321-23Andersson S, Collen B, Kuylenstierna U. Acta chem. scand. 1957; 11:10; 1641. https://doi.org/10.3891/acta.chem.scand.11-1641,2626El-Naggar IM, Mowafy E. Adsorption. 2002; 87:3;225.], 2 [21-2321-23Andersson S, Collen B, Kuylenstierna U. Acta chem. scand. 1957; 11:10; 1641. https://doi.org/10.3891/acta.chem.scand.11-1641,2626El-Naggar IM, Mowafy E. Adsorption. 2002; 87:3;225.], 3 [2121Andersson S, Collen B, Kuylenstierna U. Acta chem. scand. 1957; 11:10; 1641. https://doi.org/10.3891/acta.chem.scand.11-1641,2222Andersson S. Acta chem. scand. 1954; 8:9;1599. https://doi.org/10.3891/acta.chem.scand.08-1599], 4 [2121Andersson S, Collen B, Kuylenstierna U. Acta chem. scand. 1957; 11:10; 1641. https://doi.org/10.3891/acta.chem.scand.11-1641,2222Andersson S. Acta chem. scand. 1954; 8:9;1599. https://doi.org/10.3891/acta.chem.scand.08-1599,2727Wefer K. Naturwissenschaften. 1967; 54:1:18.], 5 [2121Andersson S, Collen B, Kuylenstierna U. Acta chem. scand. 1957; 11:10; 1641. https://doi.org/10.3891/acta.chem.scand.11-1641,2222Andersson S. Acta chem. scand. 1954; 8:9;1599. https://doi.org/10.3891/acta.chem.scand.08-1599], 6 [2222Andersson S. Acta chem. scand. 1954; 8:9;1599. https://doi.org/10.3891/acta.chem.scand.08-1599,27-2927-29Wefer K. Naturwissenschaften. 1967; 54:1:18.], 7 [3030Clearfield A, Lehto. Solid State Chem. 1988; 73:98.], 8 [3131Dion M, Piffard Y, Tournoux M. Inorg. Nucl.Chem. 1978; 140:917.], 9 [3030Clearfield A, Lehto. Solid State Chem. 1988; 73:98.], 12 [27-29,32,], 14 [3030Clearfield A, Lehto. Solid State Chem. 1988; 73:98.], 16 [3030Clearfield A, Lehto. Solid State Chem. 1988; 73:98.], 18 [3333Bando Y. Acta. Cryst. 1982; A38:211. DOI: 10.1107/S0567739482000473,3434Watanabe M, Bando Y, Tsatsum M. Solid State Chem. 1979; 28:3;397. DOI: 10.1016/0022-4596(79)90091-4].

2)      Seven compounds of the system (Li⁺ – Ti⁴⁺ – O^2–) obey the calculated formula (22) (direction of development of HS – TiO₂) – Li₄Ti⁴⁺_nO_{2(n + 1)}: n = 1 [2424Du MH. J Materials Chemistry C. 2019; 7:19;5710. https://doi.org/10.1039/C9TC00197B,3535Kataoka K, Takahashi Y, Kijima N, Ohshima K.Mater. Res. Bull. 2009; 44: N1; 168. DOI: 10.1016/j.materresbull.2008.03.015,3636Luchinskii GP. Chemistry of titanium. Chemistry. 1971; 472 с.], 2 [3535Kataoka K, Takahashi Y, Kijima N, Ohshima K.Mater. Res. Bull. 2009; 44: N1; 168. DOI: 10.1016/j.materresbull.2008.03.015], 3 [3636Luchinskii GP. Chemistry of titanium. Chemistry. 1971; 472 с.], 4 [3737Yi T, Yang Z, XieY. Mater. Chem. 2015; 3:5750. DOI: 10.1039/c4ta06882c.], 5 [3636Luchinskii GP. Chemistry of titanium. Chemistry. 1971; 472 с.-3838Hawthorne FC, Calvo C. Solid State Chem. 1977;22: N1;157.], 6 [3636Luchinskii GP. Chemistry of titanium. Chemistry. 1971; 472 с.,3939Ogihara T, Kodera T. Matereals. 2013; 6:2285. DOI: 10.1039/ma6062285], 7 [3737Yi T, Yang Z, XieY. Mater. Chem. 2015; 3:5750. DOI: 10.1039/c4ta06882c.], 8 [3030Clearfield A, Lehto. Solid State Chem. 1988; 73:98.,3636Luchinskii GP. Chemistry of titanium. Chemistry. 1971; 472 с.,3737Yi T, Yang Z, XieY. Mater. Chem. 2015; 3:5750. DOI: 10.1039/c4ta06882c.], 12 [3636Luchinskii GP. Chemistry of titanium. Chemistry. 1971; 472 с.]. Based on the calculations of the systems (Li⁺ – Ti⁴⁺ – O^2–) and (Na⁺ – Ti⁴⁺ – O^2–), formula (22) can be written for the system (M⁺ – Ti⁴⁺ – O^2–), where M⁺ º Li⁺, Na⁺, K⁺, Rb⁺, so:

M⁺₄Ti⁴⁺_nO_{2(n + 1)}……………....…………………………………… (51)

The formula (51) obtained here completely coincides with the formula obtained in [2222Andersson S. Acta chem. scand. 1954; 8:9;1599. https://doi.org/10.3891/acta.chem.scand.08-1599] only on the basis of experiment.

3)      When replacing A^a+ º K⁺, B^b+ º V⁵⁺ and C^c– º O^2– five compounds of the system (K⁺ – V⁵⁺ – O^2–) obey the calculated formula (22) (direction of development of HS – V₂O₅): K₁₀V⁵⁺_2nO_{5(n + 1)}: n = 3, 5, 15, 20, 25 [3838Hawthorne FC, Calvo C. Solid State Chem. 1977;22: N1;157.,4040Fotiev АА, Slobodin BV, Hodos M.Ya. Vanadatiy. Nauka. 1988; 267s.,4141An JN, Xu Y, Zhen L, Yuang YD. Ceram. International. 2010; 36:6; 1825. DOI: 10.1016/J.CERAMINT.2010.03.031];

4)      When replacing A^a+ º Ba²⁺, B^b+ º Cu²⁺ and C^c– º O^2– five compounds of the system (Ba²⁺ – Cu²⁺ – O^2–) obey the calculated formula (22) (direction of development of HS – CuO) – BaCu²⁺_nO_{n + 1}: n = 1 [1212Klinkova LA, Barkovskii NV, Fedotov VK. Physica C: Superconductivity. 2010; 470:206. https://doi.org/1016/j.physc.2010.09.013, 42-44 42-44Poulus EF, Miehe G, Fuess H, Yehia I. Solid State Chem. 1991; 90:1;17. DOI: 10.1016/0022-4596(91)90166-F], n = 2 [4444Maeda M, Kadoi M, Ikeda T. Jpn. J. Appl. Phys. 1989; 28:8;1417. DOI:10.1143/JJAP.28.1417]; n = 3, 7 [4545Zandbergen HW, Jansen J, Svetchnikov VL. Physica C: Superconductivity and its applications. 1999; 328:3-4;211. https://doi.org/10.1016/S021-4534(99)00537-7]; n = 4 [1212Klinkova LA, Barkovskii NV, Fedotov VK. Physica C: Superconductivity. 2010; 470:206. https://doi.org/1016/j.physc.2010.09.013];

5)            When replacing A^a+ º Ba²⁺, B^b+ º Cu²⁺ and C^c– º O^2– three compounds of the system (Ba²⁺ – Cu²⁺ – O^2–) obey the calculated formula (17) (direction of development of HS – BaO) – Ba_nCu²⁺_{On + 1}: n = 1 [1212Klinkova LA, Barkovskii NV, Fedotov VK. Physica C: Superconductivity. 2010; 470:206. https://doi.org/1016/j.physc.2010.09.013, 42 42Poulus EF, Miehe G, Fuess H, Yehia I. Solid State Chem. 1991; 90:1;17. DOI: 10.1016/0022-4596(91)90166-F,4343Jiang XP,Huang JG, Yu Y. Supercond. Sci. Technol. 1988; 1:2;102. DOI: 10.1088/0953-2048/1/2/012], n = 2 [1414Li Z, Guo W, Zhang NN. APL Matereals. 2020; 8:091112. DOI: 10.1063/5.0018934, 45 45Zandbergen HW, Jansen J, Svetchnikov VL. Physica C: Superconductivity and its applications. 1999; 328:3-4;211. https://doi.org/10.1016/S021-4534(99)00537-7], n = 3 [4444Maeda M, Kadoi M, Ikeda T. Jpn. J. Appl. Phys. 1989; 28:8;1417. DOI:10.1143/JJAP.28.1417].

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Undalov Yu K. Homologous Series of Chemical Compounds in Three-component Systems (Aa+ – Bb+ – Cc–) and (Zn2+ - Ge4+ - P3-) in Generalized Form. IgMin Res. Nov 21, 2023; 1(1): 060-069. IgMin ID: igmin117; DOI: 10.61927/igmin117; Available at: www.igminresearch.com/articles/pdf/igmin117.pdf

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