research communications
Organic–inorganic hybrid hexachloridostannate(IV) with 2-methylimidazo[1,5-a]pyridin-2-ium cation
aDepartment of Chemistry, Taras Shevchenko National University of Kyiv, 64/13 Volodymyrska Street, Kyiv 01601, Ukraine, and bSchool of Molecular Sciences, M310, the University of Western Australia, 35 Stirling Highway, Perth, 6009, W.A., Australia
*Correspondence e-mail: vassilyeva@univ.kiev.ua
The hybrid salt bis(2-methylimidazo[1,5-a]pyridin-2-ium) hexachloridostannate(IV), (C8H9N2)2[SnCl6], crystallizes in the monoclinic P21/n with the containing an Sn0.5Cl3 fragment (Sn ) and one organic cation. The five- and six-membered rings in the cation are nearly coplanar; bond lengths in the pyridinium ring of the fused core are as expected; the C—N/C bond distances in the imidazolium entity fall in the range 1.337 (5)–1.401 (5) Å. The octahedral SnCl62– dianion is almost undistorted with the Sn–Cl distances varying from 2.4255 (9) to 2.4881 (8) Å and the cis Cl—Sn—Cl angles approaching 90°. In the crystal, π-stacked chains of cations and loosely packed SnCl62– dianions form separate sheets alternating parallel to (101). Most of the numerous C—H⋯Cl—Sn contacts between the organic and inorganic counterparts with the H⋯Cl distances above the van der Waals contact limit of 2.85 Å are considered a result of crystal packing.
Keywords: crystal structure; hybrid salt; tin(IV); 2-pyridinecarbaldehyde.
CCDC reference: 2235690
1. Chemical context
Organic–inorganic hybrid perovskites that combine discrete organic cations and rigid metal halide architectures have been considered promising materials for diverse optoelectronic applications: solar cells, light-emitting diodes, photodetectors, spintronics (Gan et al., 2021; Li et al., 2021). Most of the materials reported to date are based on PbII, SbIII, BiIII and CdII halides (Saparov & Mitzi, 2016), whose widespread application is restrained by the potential toxicity. Being in the same main group of metal atoms that Pb belongs to, Sn forms hybrid halide perovskites with similar electronic properties, which are more friendly to the environment. At the same time, the aforementioned hybrid systems suffer from high water permeability and low thermal stability, the issues being largely related to the volatility of small organic cations (Leijtens et al., 2015). The stability of hybrid perovskites can be improved by introducing larger organic cations and lowering the dimensionality of the octahedral halometallate frameworks (Zhang et al., 2016; Leblanc et al., 2019). Moreover, functional organic cations are a valuable tool for introducing useful properties into the hybrid structure. For example, the use of the photoactive zwitterion viologen N,N′-4,4′-bipyridiniodipropionate (CV) afforded the formation of the covalently bonded pillared layered bromoplumbate, [Pb3Br6(CV)]n, showing high thermal stability in air and a remarkable increase of capacitance after (Sun et al., 2019). Mono-periodic hybrid lead halides incorporating optically active protonated 1,3-bis(4-pyridyl)-propane cations exhibit dual-light emissions combined of higher energy blue and lower energy yellow light spectra, which were attributed to the individual contributions of the organic and inorganic components (Sun et al., 2021).
Multiple advantages of the organic–inorganic hybrid materials inspire the huge appeal in exploring other kinds of low-dimensional metal halide compounds templated by functional aromatic cations. Fine-tuning of the electronic structure and optoelectronic properties of the metal halide hybrids, which depend, among other things, on the anionic speciation and halogen ratio, can be achieved by mixing halide ligands in self-assembled organic–inorganic systems (Rogers et al., 2019; Askar et al., 2018).
Pursuing our research on hybrid halometalates incorporating substituted imidazo[1,5-a]pyridinium cations (Buvaylo et al., 2015; Vassilyeva et al., 2019; 2020; 2021), we attempted the synthesis of a hybrid tin mixed halide with 2-methylimidazo[1,5-a]pyridinium, L+, a product of the oxidative cyclocondensation between 2-pyridinecarbaldehyde (2-PCA), formaldehyde and CH3NH2. One necessary component of the reaction is acid, which is introduced as a hydrohalide adduct of the amine (Vassilyeva et al., 2020). Following the method of preparation used to obtain mixed-halide ZnII and CdII tetrahalometalates (Cl/I, Br/Cl) with L+ (Vassilyeva et al., 2022), SnCl2·2H2O was reacted with the solution of L+ formed in situ using 2-PCA, formaldehyde and CH3NH2·HBr. The isolated product was crystallographically characterized as [L]2[SnCl6], (I); the detrimental oxidation of SnII to SnIV appeared unavoidable leading to the formation of ubiquitous hexachloridostannate(IV) dianion. Herein, the synthesis, structural analysis and spectroscopic characterization of (I) are reported.
2. Structural commentary
The title hybrid salt, with formula (C8H9N2)2[SnCl6], crystallizes in the monoclinic P21/n. The consists of an Sn0.5Cl3 fragment (Sn ) and 2-methylimidazo[1,5-a]pyridinium cation, as shown in Fig. 1. The structural configuration of the cation is similar to those of other 2-methylimidazo[1,5-a]pyridinium hybrid salts (C8H9N2)2[ZnCl4] (GOTHAB; Vassilyeva et al., 2020) and (C8H9N2)2[CdCl4] (GOTJAD; Vassilyeva et al., 2021). Bond lengths in the pyridinium ring of the fused core are as expected; the C—N/C bond distances in the imidazolium entity fall in the range 1.337 (5)–1.401 (5) Å; N2 and N3A atoms are planar with the sums of three angles being equal to 360°. The almost coplanar five- and six-membered rings in the cation show the dihedral angle between them of 1.6 (2)°. The octahedral SnCl62– dianion in (I) is almost undistorted with the Sn—Cl distances varying from 2.4255 (9) to 2.4881 (8) Å and the cis Cl—Sn—Cl angles approaching 90° (Table 1). The geometric parameters of the dianion are normal and comparable to those of similar structure types.
3. Supramolecular features
In the crystal, cationic and anionic sheets alternate lying parallel to (101) (Fig. 2). In the sheets, pairs of centrosymmetically related trans-oriented L+ cations demonstrate offset 10πe–10πe stacking with a centroid–centroid distance of 3.530 (2) Å (Fig. 3). The pairs further form π-bonded chains with a distance of 3.713 (2) Å between neighbouring pyridinium ring centroids. In the anion sheet, loose packing of SnCl62– dianions that are identically stacked one above the other with the shortest Sn–Cl⋯Cl–Sn distance being 4.4433 (12) Å, results in a closest separation of 7.7926 (1) Å between the metal atoms. The hybrid salt lacks classical hydrogen-bonding interactions but shows a variety of C—H⋯Cl—Sn contacts between the organic and inorganic counterparts (Table 2), a feature common to hybrid chlorometalates with nitrogen-containing aromatic cations (Coleman et al., 2013). Most of these contacts are longer than the van der Waals contact limit of 2.85 Å (Cl) (Mantina et al., 2009) and can be considered a result of crystal packing.
4. Database survey
A search of the Cambridge Structural Database (CSD, Version 5.42; Groom et al., 2016) for structures including substituted imidazo[1,5-a]pyridinium cations gave 53 salts with about a half (23) reported by our research group. The latter comprise organic–inorganic hybrids with the L+ cation or its derivatives [2-methyl-3-(pyridin-2-yl)imidazo[1,5-a]pyridin-2-ium and 2,2′-(ethane-1,2-diyl)bis(imidazo[1,5-a]pyridin-2-ium)] counterbalanced by transition and main-group (Mn, Cu, Zn, Cd, Pb) halometalates. The other compounds in the CSD with cations similar to L+ are mostly organic salts with the imidazo[1,5-a]pyridinium core having various substituents in the rings. Perchlorate NAKNET (Mishra, et al., 2005) and hexafluorophosphate DIWYEP (Kriechbaum, et al., 2014), which bear methylphenyl and dimethylphenyl substituents, respectively, in place of the methyl group in L+ are the most closely related. Structures of main group halometalates with substituted imidazo[1,5-a]pyridinium cations are limited to a few examples such as bis[2-(6-methylpyridin-2-yl)imidazo[1,5-a]pyridin-2-ium] dichlorogold tetrachlorogold (SUWVIR; Nandy et al., 2016) and 2-(2-ammoniocyclohexyl)-3-(pyridin-2-yl)imidazo[1,5-a]pyridin-2-ium hexabromotellurium acetonitrile solvate (TEVVIB; Vasudevan et al., 2012).
Within the variety of 279 crystal structures in the CSD comprising [SnCl6]2– dianions, the latter are mostly highly symmetrical being associated with special positions. The structures including organic counterparts can be seen as an arrangement of alternating organic and inorganic layers supported by hydrogen bonds of the N–H⋯Cl type in the case of protonated N-containing cations. An organic–inorganic hybrid compound with the structure most similar to that of the title compound is, for example, monoclinic bis[1-(prop-2-en-1-yl)-1H-imidazol-3-ium] hexachloridostannate(IV), in P21/n, with layers formed by isolated [SnCl6]2– octahedra and (C6H9N2)+ organic cations, which propagate along the a-axis direction at y = 0 and y = 1/2 (Ferjani, 2020).
5. Synthesis and crystallization
Synthesis of [L]2[SnCl6] (I). Solid CH3NH2·HBr (0.45 g, 4 mmol) was added to the warm formaldehyde solution prepared by dissolving paraform (0.13 g, 4.5 mmol) in boiling deionized water (10 ml) in a 50 ml conical flask. The solution was stirred vigorously for 1 h at room temperature and filtered. On the following day, 2-PCA (0.38 ml, 4 mmol) was introduced into the flask under stirring, followed by the addition of SnCl2·2H2O (0.22 g, 1 mmol) dissolved in ethanol (10 ml) in 30 min. The solution was kept magnetically stirred at room temperature for another hour, then filtered to remove Sn(OH)2 and allowed to evaporate. It was diluted with methanol (5 ml) since it was thickening. Pale-yellow needles of (I) suitable for X-ray crystallography formed over three months after successive addition of iPrOH (5 ml). The crystals were filtered off, washed with diethyl ether and dried in air. Yield: 12% (based on Sn). FT–IR (ν, cm−1): 3422br, 3122vs, 3092vs, 3050vs, 3014, 2956, 2914, 1654, 1566, 1544, 1454, 1374, 1352, 1328, 1258, 1222, 1148vs, 1130, 1038, 986, 920, 789vs, 764, 742, 666, 624vs, 498, 468, 434. 1H NMR (400 MHz, DMSO-d6): δ (ppm) 9.83 (s, 1H, HC3), 8.69 (d, 1H, J = 6.8 Hz, HC4), 8.24 (s, 1H, HC1), 7.82 (d, 1H, J = 9.2 Hz, HC7), 7.22 (t, 1H, J = 8.2 Hz, HC5), 7.13 (t, 1H, J = 6.7 Hz, HC6), 4.27 (s, 3H, CH3). Analysis calculated for C16H18N4SnCl6 (597.73): C 32.15; H 3.04; N 9.37%. Found: C 32.40; H 2.88; N 9.19%.
6. Refinement
Crystal data, data collection and structure . All hydrogen atoms were included in calculated positions and refined using a riding model with isotropic displacement parameters based on those of the parent atom (C—H = 0.95 Å, Uiso(H) = 1.2UeqC for CH, C—H = 0.98 Å, Uiso(H) = 1.5UeqC for CH3). Anisotropic displacement parameters were employed for the non-hydrogen atoms.
details are summarized in Table 3
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Supporting information
CCDC reference: 2235690
https://doi.org/10.1107/S2056989023000324/pk2675sup1.cif
contains datablocks I, global. DOI:Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S2056989023000324/pk2675Isup2.hkl
Data collection: CrysAlis PRO (Rigaku OD, 2016); cell
CrysAlis PRO (Rigaku OD, 2016); data reduction: CrysAlis PRO (Rigaku OD, 2016); program(s) used to solve structure: SHELXT2015/1 (Sheldrick, 2015a); program(s) used to refine structure: SHELXL2019/2 (Sheldrick, 2015b); molecular graphics: Mercury (Macrae et al., 2020); software used to prepare material for publication: WinGX (Farrugia, 2012).(C8H9N2)2[SnCl6] | F(000) = 588 |
Mr = 597.73 | Dx = 1.825 Mg m−3 |
Monoclinic, P21/n | Mo Kα radiation, λ = 0.71073 Å |
a = 7.7926 (1) Å | Cell parameters from 17399 reflections |
b = 12.1425 (1) Å | θ = 2.4–37.5° |
c = 11.7114 (1) Å | µ = 1.92 mm−1 |
β = 101.082 (1)° | T = 100 K |
V = 1087.49 (2) Å3 | Needle, pale yellow |
Z = 2 | 0.38 × 0.20 × 0.13 mm |
Oxford Diffraction Gemini-R Ultra diffractometer | 2215 independent reflections |
Radiation source: fine-focus sealed X-ray tube, Enhance (Mo) X-ray Source | 2108 reflections with I > 2σ(I) |
Graphite monochromator | Rint = 0.035 |
ω scans | θmax = 26.4°, θmin = 2.4° |
Absorption correction: analytical [CrysAlisPro (Rigaku OD, 2016); analytical numeric absorption correction using a multifaceted crystal model based on expressions derived by Clark & Reid (1995)] | h = −9→9 |
Tmin = 0.618, Tmax = 0.802 | k = −15→15 |
18730 measured reflections | l = −14→14 |
Refinement on F2 | 0 restraints |
Least-squares matrix: full | Hydrogen site location: inferred from neighbouring sites |
R[F2 > 2σ(F2)] = 0.034 | H-atom parameters constrained |
wR(F2) = 0.098 | w = 1/[σ2(Fo2) + (0.050P)2 + 6.970P] where P = (Fo2 + 2Fc2)/3 |
S = 1.00 | (Δ/σ)max < 0.001 |
2215 reflections | Δρmax = 1.54 e Å−3 |
125 parameters | Δρmin = −1.15 e Å−3 |
Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds involving l.s. planes. |
x | y | z | Uiso*/Ueq | ||
Sn1 | 0.500000 | 0.500000 | 0.500000 | 0.01497 (14) | |
Cl1 | 0.35555 (12) | 0.63407 (8) | 0.60266 (8) | 0.0195 (2) | |
Cl2 | 0.78853 (11) | 0.57935 (7) | 0.59047 (7) | 0.01241 (19) | |
Cl3 | 0.52226 (11) | 0.37088 (7) | 0.66602 (7) | 0.0141 (2) | |
C1 | 0.0945 (5) | 0.8366 (3) | 0.5329 (4) | 0.0206 (8) | |
H1 | 0.048748 | 0.798625 | 0.591356 | 0.025* | |
N2 | 0.0816 (4) | 0.8046 (3) | 0.4195 (3) | 0.0199 (7) | |
C2 | −0.0015 (6) | 0.7034 (4) | 0.3679 (4) | 0.0250 (9) | |
H2A | −0.023505 | 0.709472 | 0.282891 | 0.038* | |
H2B | −0.112456 | 0.692324 | 0.393919 | 0.038* | |
H2C | 0.075990 | 0.640636 | 0.392353 | 0.038* | |
N3A | 0.2249 (4) | 0.9583 (3) | 0.4373 (3) | 0.0190 (7) | |
C3 | 0.1607 (5) | 0.8781 (3) | 0.3620 (4) | 0.0206 (8) | |
H3 | 0.169920 | 0.874245 | 0.282421 | 0.025* | |
C4 | 0.3127 (5) | 1.0545 (3) | 0.4176 (4) | 0.0222 (8) | |
H4 | 0.334519 | 1.070978 | 0.342328 | 0.027* | |
C5 | 0.3661 (5) | 1.1237 (4) | 0.5076 (4) | 0.0232 (9) | |
H5 | 0.425896 | 1.189613 | 0.495717 | 0.028* | |
C6 | 0.3333 (6) | 1.0986 (4) | 0.6211 (4) | 0.0251 (9) | |
H6 | 0.373638 | 1.147621 | 0.683721 | 0.030* | |
C7A | 0.1860 (5) | 0.9340 (3) | 0.5465 (3) | 0.0185 (8) | |
C7 | 0.2459 (6) | 1.0064 (3) | 0.6405 (4) | 0.0214 (9) | |
H7 | 0.225017 | 0.990431 | 0.716016 | 0.026* |
U11 | U22 | U33 | U12 | U13 | U23 | |
Sn1 | 0.0167 (2) | 0.0148 (2) | 0.0133 (2) | 0.00138 (12) | 0.00244 (14) | 0.00014 (12) |
Cl1 | 0.0212 (5) | 0.0206 (5) | 0.0163 (4) | 0.0061 (3) | 0.0027 (3) | −0.0021 (3) |
Cl2 | 0.0146 (4) | 0.0125 (4) | 0.0100 (4) | −0.0003 (3) | 0.0023 (3) | −0.0006 (3) |
Cl3 | 0.0182 (4) | 0.0135 (4) | 0.0104 (4) | −0.0010 (3) | 0.0023 (3) | 0.0003 (3) |
C1 | 0.0206 (19) | 0.021 (2) | 0.0205 (19) | 0.0058 (15) | 0.0052 (15) | 0.0039 (15) |
N2 | 0.0186 (16) | 0.0202 (17) | 0.0210 (17) | 0.0036 (13) | 0.0041 (13) | 0.0026 (13) |
C2 | 0.024 (2) | 0.023 (2) | 0.027 (2) | −0.0007 (16) | 0.0010 (17) | −0.0001 (17) |
N3A | 0.0171 (16) | 0.0225 (18) | 0.0173 (16) | 0.0037 (13) | 0.0027 (13) | 0.0041 (13) |
C3 | 0.0182 (18) | 0.023 (2) | 0.0201 (19) | 0.0039 (15) | 0.0032 (15) | 0.0035 (16) |
C4 | 0.0195 (19) | 0.022 (2) | 0.026 (2) | 0.0030 (16) | 0.0054 (16) | 0.0068 (16) |
C5 | 0.0193 (19) | 0.021 (2) | 0.029 (2) | 0.0039 (15) | 0.0043 (16) | 0.0036 (17) |
C6 | 0.023 (2) | 0.025 (2) | 0.027 (2) | 0.0074 (17) | 0.0017 (17) | −0.0025 (17) |
C7A | 0.0157 (18) | 0.0229 (19) | 0.0178 (19) | 0.0077 (15) | 0.0057 (14) | 0.0051 (15) |
C7 | 0.022 (2) | 0.023 (2) | 0.020 (2) | 0.0074 (15) | 0.0052 (16) | 0.0010 (15) |
Sn1—Cl1i | 2.4255 (9) | C2—H2C | 0.9800 |
Sn1—Cl1 | 2.4255 (9) | N3A—C3 | 1.345 (6) |
Sn1—Cl3 | 2.4777 (8) | N3A—C4 | 1.395 (5) |
Sn1—Cl3i | 2.4778 (8) | N3A—C7A | 1.401 (5) |
Sn1—Cl2i | 2.4881 (8) | C3—H3 | 0.9500 |
Sn1—Cl2 | 2.4881 (8) | C4—C5 | 1.350 (6) |
C1—N2 | 1.369 (5) | C4—H4 | 0.9500 |
C1—C7A | 1.373 (6) | C5—C6 | 1.433 (6) |
C1—H1 | 0.9500 | C5—H5 | 0.9500 |
N2—C3 | 1.337 (5) | C6—C7 | 1.352 (6) |
N2—C2 | 1.465 (5) | C6—H6 | 0.9500 |
C2—H2A | 0.9800 | C7A—C7 | 1.416 (6) |
C2—H2B | 0.9800 | C7—H7 | 0.9500 |
Cl1i—Sn1—Cl1 | 180.0 | N2—C2—H2C | 109.5 |
Cl1i—Sn1—Cl3 | 89.72 (3) | H2A—C2—H2C | 109.5 |
Cl1—Sn1—Cl3 | 90.28 (3) | H2B—C2—H2C | 109.5 |
Cl1i—Sn1—Cl3i | 90.28 (3) | C3—N3A—C4 | 129.1 (4) |
Cl1—Sn1—Cl3i | 89.72 (3) | C3—N3A—C7A | 109.1 (3) |
Cl3—Sn1—Cl3i | 180.00 (2) | C4—N3A—C7A | 121.8 (4) |
Cl1i—Sn1—Cl2i | 89.80 (3) | N2—C3—N3A | 107.6 (4) |
Cl1—Sn1—Cl2i | 90.20 (3) | N2—C3—H3 | 126.2 |
Cl3—Sn1—Cl2i | 90.59 (3) | N3A—C3—H3 | 126.2 |
Cl3i—Sn1—Cl2i | 89.41 (3) | C5—C4—N3A | 118.6 (4) |
Cl1i—Sn1—Cl2 | 90.20 (3) | C5—C4—H4 | 120.7 |
Cl1—Sn1—Cl2 | 89.80 (3) | N3A—C4—H4 | 120.7 |
Cl3—Sn1—Cl2 | 89.41 (3) | C4—C5—C6 | 120.6 (4) |
Cl3i—Sn1—Cl2 | 90.59 (3) | C4—C5—H5 | 119.7 |
Cl2i—Sn1—Cl2 | 180.0 | C6—C5—H5 | 119.7 |
N2—C1—C7A | 107.2 (4) | C7—C6—C5 | 121.1 (4) |
N2—C1—H1 | 126.4 | C7—C6—H6 | 119.4 |
C7A—C1—H1 | 126.4 | C5—C6—H6 | 119.4 |
C3—N2—C1 | 110.1 (4) | C1—C7A—N3A | 105.9 (4) |
C3—N2—C2 | 124.2 (4) | C1—C7A—C7 | 135.3 (4) |
C1—N2—C2 | 125.7 (4) | N3A—C7A—C7 | 118.8 (4) |
N2—C2—H2A | 109.5 | C6—C7—C7A | 119.0 (4) |
N2—C2—H2B | 109.5 | C6—C7—H7 | 120.5 |
H2A—C2—H2B | 109.5 | C7A—C7—H7 | 120.5 |
C7A—C1—N2—C3 | −0.2 (4) | N2—C1—C7A—N3A | 0.5 (4) |
C7A—C1—N2—C2 | 178.1 (4) | N2—C1—C7A—C7 | −178.1 (4) |
C1—N2—C3—N3A | −0.2 (4) | C3—N3A—C7A—C1 | −0.6 (4) |
C2—N2—C3—N3A | −178.5 (3) | C4—N3A—C7A—C1 | 177.7 (3) |
C4—N3A—C3—N2 | −177.7 (4) | C3—N3A—C7A—C7 | 178.2 (3) |
C7A—N3A—C3—N2 | 0.5 (4) | C4—N3A—C7A—C7 | −3.4 (6) |
C3—N3A—C4—C5 | −179.8 (4) | C5—C6—C7—C7A | −0.1 (6) |
C7A—N3A—C4—C5 | 2.2 (6) | C1—C7A—C7—C6 | −179.2 (4) |
N3A—C4—C5—C6 | 0.1 (6) | N3A—C7A—C7—C6 | 2.3 (6) |
C4—C5—C6—C7 | −1.1 (6) |
Symmetry code: (i) −x+1, −y+1, −z+1. |
D—H···A | D—H | H···A | D···A | D—H···A |
C5—H5···Cl3ii | 0.95 | 2.97 | 3.614 (5) | 126 |
C3—H3···Cl2iii | 0.95 | 2.65 | 3.549 (4) | 158 |
C3—H3···Cl1iii | 0.95 | 2.91 | 3.483 (4) | 120 |
C4—H4···Cl3iii | 0.95 | 2.96 | 3.473 (4) | 115 |
C7—H7···Cl3iv | 0.95 | 2.96 | 3.746 (5) | 141 |
C7—H7···Cl1iv | 0.95 | 2.91 | 3.605 (4) | 131 |
C2—H2A···Cl1iii | 0.98 | 2.86 | 3.668 (5) | 140 |
C2—H2B···Cl2v | 0.98 | 2.91 | 3.658 (5) | 134 |
C2—H2C···Cl2i | 0.98 | 2.87 | 3.804 (4) | 161 |
C2—H2C···Cl1 | 0.98 | 2.96 | 3.615 (4) | 125 |
Symmetry codes: (i) −x+1, −y+1, −z+1; (ii) x, y+1, z; (iii) x−1/2, −y+3/2, z−1/2; (iv) −x+1/2, y+1/2, −z+3/2; (v) x−1, y, z. |
Funding information
Funding for this research was provided by: Ministry of Education and Science of Ukraine (project No. 22BP037-13; grant for the perspective development of the scientific direction `Mathematical sciences and natural sciences' at the Taras Shevchenko National University of Kyiv).
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