metal-organic compounds\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

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ISSN: 2056-9890

Di­aqua­bis­(5-carb­­oxy-2-methyl-1H-imidazole-4-carboxyl­ato-κ2N3,O4)cobalt(II) di­methyl­formamide disolvate

aDepartment of Chemistry and Material Science, Hengyang Normal University, Hengyang, Hunan 421008, People's Republic of China.
*Correspondence e-mail: sptang88@163.com

(Received 4 April 2009; accepted 5 April 2009; online 10 April 2009)

In the title compound, [Co(C6H5N2O4)2(H2O)2]·2C3H7NO, the CoII ion lies on an inversion center and adopts a slightly distorted CoN2O4 octa­hedral geometry binding two bidentate chelating 5-carb­oxy-2-methyl-1H-imidazole-4-carboxyl­ate (H2MIDA) monoanionic ligands and two axial aqua ligands. In the crystal structure, inter­molecular O—H⋯O hydrogen bonds link neighboring mol­ecules, generating a two-dimensional framework containing eight-membered H4O4 rings. In addition, the dimethyl­formamide solvent mol­ecules are hydrogen bonded to the two-dimensional framework via the NH groups of the H2MIDA ligands.

Related literature

For background to N-heterocyclic carboxylic acids as ligands in coordination complexes, see: Gao et al. (2004[Gao, S., Zhang, X.-F., Huo, L.-H. & Zhao, H. (2004). Acta Cryst. E60, m1790-m1792.]); Shimizu et al. (2004[Shimizu, E., Kondo, M., Fuwa, Y., Sarker, R. P., Miyazawa, M., Ueno, M., Naito, T., Maeda, K. & Uchida, F. (2004). Inorg. Chem. Commun. 7, 1191-1194.]); Zhang et al. (2006[Zhang, M.-B., Chen, Y.-M., Zheng, S.-T. & Yang, G.-Y. (2006). Eur. J. Inorg. Chem. pp. 1423-1428.]). For related structures, see: Liu et al. (2007[Liu, C.-B., Nie, X.-L. & Wen, H.-L. (2007). Acta Cryst. E63, m2244.]); Nie et al. (2007[Nie, X.-L., Wen, H.-L., Wu, Z.-S., Liu, D.-B. & Liu, C.-B. (2007). Acta Cryst. E63, m753-m755.]); Zeng et al. (2008[Zeng, J.-Z., Yi, X.-G., Lin, J.-Y., Ying, S.-M. & Huang, G.-S. (2008). Acta Cryst. E64, m476.]).

[Scheme 1]

Experimental

Crystal data
  • [Co(C6H5N2O4)2(H2O)2]·2C3H7NO

  • Mr = 579.39

  • Triclinic, [P \overline 1]

  • a = 7.1979 (11) Å

  • b = 9.2180 (15) Å

  • c = 10.8659 (17) Å

  • α = 65.173 (2)°

  • β = 83.459 (2)°

  • γ = 68.254 (2)°

  • V = 607.02 (17) Å3

  • Z = 1

  • Mo Kα radiation

  • μ = 0.78 mm−1

  • T = 123 K

  • 0.20 × 0.15 × 0.14 mm

Data collection
  • Bruker SMART APEX area-detector diffractometer

  • Absorption correction: multi-scan (SADABS; Sheldrick, 1996[Sheldrick, G. M. (1996). SADABS. University of Göttingen, Germany.]) Tmin = 0.859, Tmax = 0.899

  • 4588 measured reflections

  • 2335 independent reflections

  • 2176 reflections with I > 2σ(I)

  • Rint = 0.026

Refinement
  • R[F2 > 2σ(F2)] = 0.039

  • wR(F2) = 0.106

  • S = 1.06

  • 2335 reflections

  • 172 parameters

  • 1 restraint

  • H atoms treated by a mixture of independent and constrained refinement

  • Δρmax = 0.65 e Å−3

  • Δρmin = −0.49 e Å−3

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A D—H H⋯A DA D—H⋯A
N2—H2⋯O5i 0.86 1.83 2.670 (2) 166
O2—H2A⋯O3 0.863 (10) 1.593 (11) 2.452 (2) 173 (3)
O1W—H1W2⋯O1ii 0.85 1.91 2.7579 (19) 178
O1W—H1W1⋯O1iii 0.85 2.03 2.847 (2) 160
Symmetry codes: (i) -x+1, -y, -z+1; (ii) -x+1, -y+1, -z+1; (iii) x, y+1, z.

Data collection: SMART (Bruker, 2002[Bruker (2002). SMART and SAINT. Bruker AXS Inc., Madison, Wisconsin, USA.]); cell refinement: SAINT (Bruker, 2002[Bruker (2002). SMART and SAINT. Bruker AXS Inc., Madison, Wisconsin, USA.]); data reduction: SAINT; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008[Sheldrick, G. M. (2008). Acta Cryst. A64, 112-122.]); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008[Sheldrick, G. M. (2008). Acta Cryst. A64, 112-122.]); molecular graphics: SHELXTL (Sheldrick, 2008[Sheldrick, G. M. (2008). Acta Cryst. A64, 112-122.]); software used to prepare material for publication: SHELXTL.

Supporting information


Comment top

N-heterocyclic carboxylic acids have attracted considerable interests as ligands in metal complexes because of their structural diversity as multidentate chelating or bridging ligands (Gao et al., 2004; Shimizu et al., 2004; Zhang et al., 2006). Recently, 2-methyl-1H-imidazole-4,5- dicarboxylic acid (H3MIDA) has been used as a chelating ligand to generate mononuclear complexes with cadmium (II), cobalt (II) and manganese (II) ions (Liu et al., 2007; Nie et al., 2007; Zeng et al., 2008). We report here the synthesis and structure of a new cobalt (II) complex incorporating H3MIDA.

The title compound is composed of one Co (II) ion, two mono-deprotonated H2MIDA ligands, two aqua ligands and two DMF solvent molecules, Fig 1. The Co (II) cation lies on a crystallographic inversion center and has a distorted octahedral geometry with the basal plane occupied by two carboxylate O atoms and two N atoms from two chelating H2MIDA- ligands. There are two axial aqua ligands. In the H2MIDA ligand, the carboxyl and carboxylate groups form an intramolecular hydrogen bond with an O···O distance of 2.452 (2) Å.

In the crystal packing, each aqua ligand is involved in two intermolecular O—H···O hydrogen bonds with two carboxyl O atoms from two neighboring molecules to genetrate a two-dimensional supramolecular structure (Fig. 2), in which two aqua ligands and two carboxyl O atoms form a H4O4 eight-membered ring. In addition, the DMF solvates are hydrogen-bonded to the two-dimensional framework via the –NH groups of the H2MIDA- ligands.

Related literature top

For background to N-heterocyclic carboxylic acids as ligands in coordination complexes, see: Gao et al. (2004); Shimizu et al. (2004); Zhang et al. (2006). For related structures, see: Liu et al. (2007); Nie et al. (2007); Zeng et al. (2008).

Experimental top

A solution of cobalt perchlorate hexahydrate (73.2 mg, 0.2 mmol) and H3MIDA (15.8 mg, 0.1 mmol) in DMF (6 ml) and methanol (1 ml) was stirred for 5 h. After filtering, the filtrate was left for about two months and pink, block-like crystals of the title compound appeared. Yield: 11 mg (38%).

Refinement top

The carboxyl and water H atoms were located in a difference Fourier map and refined with Uiso=1.5Ueq (O). The O—H distances of water were refined with idealized values of 0.85 Å, however, that of carboxyl is refined freely. All other H-atoms were positioned geometrically and refined using a riding model with C—H (methyl) = 0.96 Å, C—H (aldehyde) = 0.93 Å, N—H = 0.86 Å, and Uiso = 1.2Ueq (C, N).

Computing details top

Data collection: SMART (Bruker, 2002); cell refinement: SAINT (Bruker, 2002); data reduction: SAINT (Bruker, 2002); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXTL (Sheldrick, 2008).

Figures top
[Figure 1] Fig. 1. The title molecule with displacement ellipsoids drawn at the 50% probability level, and with the H atoms shown as spheres of arbitrary radii.
[Figure 2] Fig. 2. Packing diagram of the title compound showing the intra/intermolecular O—H···O hydrogen bonds as dashed lines. The H atoms not involved in hydrogen bonds and the DMF solvate molecules have been omitted for clarity.
Diaquabis(5-carboxy-2-methyl-1H-imidazole-4-carboxylato- κ2N3,O4)cobalt(II) dimethylformamide disolvate top
Crystal data top
[Co(C6H5N2O4)2(H2O)2]·2C3H7NOZ = 1
Mr = 579.39F(000) = 301
Triclinic, P1Dx = 1.585 Mg m3
Hall symbol: -P 1Mo Kα radiation, λ = 0.71073 Å
a = 7.1979 (11) ÅCell parameters from 3423 reflections
b = 9.2180 (15) Åθ = 2.6–27.8°
c = 10.8659 (17) ŵ = 0.78 mm1
α = 65.173 (2)°T = 123 K
β = 83.459 (2)°Block, pink
γ = 68.254 (2)°0.20 × 0.15 × 0.14 mm
V = 607.02 (17) Å3
Data collection top
Bruker SMART APEX area-detector
diffractometer
2335 independent reflections
Radiation source: fine-focus sealed tube2176 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.026
ϕ and ω scansθmax = 26.0°, θmin = 2.6°
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)
h = 87
Tmin = 0.859, Tmax = 0.899k = 1111
4588 measured reflectionsl = 1313
Refinement top
Refinement on F2Primary atom site location: structure-invariant direct methods
Least-squares matrix: fullSecondary atom site location: difference Fourier map
R[F2 > 2σ(F2)] = 0.039Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.106H atoms treated by a mixture of independent and constrained refinement
S = 1.06 w = 1/[σ2(Fo2) + (0.0631P)2 + 0.2477P]
where P = (Fo2 + 2Fc2)/3
2335 reflections(Δ/σ)max < 0.001
172 parametersΔρmax = 0.65 e Å3
1 restraintΔρmin = 0.49 e Å3
Crystal data top
[Co(C6H5N2O4)2(H2O)2]·2C3H7NOγ = 68.254 (2)°
Mr = 579.39V = 607.02 (17) Å3
Triclinic, P1Z = 1
a = 7.1979 (11) ÅMo Kα radiation
b = 9.2180 (15) ŵ = 0.78 mm1
c = 10.8659 (17) ÅT = 123 K
α = 65.173 (2)°0.20 × 0.15 × 0.14 mm
β = 83.459 (2)°
Data collection top
Bruker SMART APEX area-detector
diffractometer
2335 independent reflections
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)
2176 reflections with I > 2σ(I)
Tmin = 0.859, Tmax = 0.899Rint = 0.026
4588 measured reflections
Refinement top
R[F2 > 2σ(F2)] = 0.0391 restraint
wR(F2) = 0.106H atoms treated by a mixture of independent and constrained refinement
S = 1.06Δρmax = 0.65 e Å3
2335 reflectionsΔρmin = 0.49 e Å3
172 parameters
Special details top

Geometry. All e.s.d.'s (except the e.s.d. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell e.s.d.'s are taken into account individually in the estimation of e.s.d.'s in distances, angles and torsion angles; correlations between e.s.d.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell e.s.d.'s is used for estimating e.s.d.'s involving l.s. planes.

Refinement. Refinement of F2 against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F2, conventional R-factors R are based on F, with F set to zero for negative F2. The threshold expression of F2 > σ(F2) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F2 are statistically about twice as large as those based on F, and R- factors based on ALL data will be even larger.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/Ueq
Co10.00001.00000.50000.01688 (15)
O40.1583 (2)0.97632 (18)0.32480 (14)0.0200 (3)
O1W0.2666 (2)0.96647 (18)0.58550 (14)0.0214 (3)
H1W20.37160.90170.56430.032*
H1W10.28571.05850.56850.032*
O30.3261 (2)0.78687 (18)0.23787 (14)0.0220 (3)
O20.4205 (2)0.47831 (19)0.30617 (15)0.0233 (3)
H2A0.391 (4)0.5850 (15)0.288 (3)0.035*
O10.3990 (2)0.24920 (18)0.48355 (16)0.0241 (3)
O50.8297 (3)0.1528 (2)0.20333 (17)0.0343 (4)
N10.0922 (2)0.7355 (2)0.55599 (16)0.0171 (4)
N20.1798 (2)0.4572 (2)0.62169 (17)0.0181 (4)
H20.19320.35230.67130.022*
N30.6518 (3)0.1293 (2)0.08740 (18)0.0296 (4)
C60.2260 (3)0.8282 (3)0.33158 (19)0.0179 (4)
C50.1936 (3)0.6909 (2)0.4550 (2)0.0174 (4)
C40.2493 (3)0.5176 (2)0.4945 (2)0.0173 (4)
C30.3636 (3)0.4049 (3)0.4250 (2)0.0194 (4)
C20.0862 (3)0.5915 (2)0.6556 (2)0.0181 (4)
C10.0105 (3)0.5776 (3)0.7860 (2)0.0247 (5)
H1A0.08820.50480.85970.037*
H1B0.11210.52970.79610.037*
H1C0.07030.68920.78660.037*
C70.4556 (5)0.2630 (4)0.0431 (3)0.0559 (8)
H7A0.35390.21360.07540.084*
H7B0.44500.31760.05430.084*
H7C0.43840.34610.07880.084*
C80.8227 (5)0.1820 (4)0.0586 (3)0.0480 (7)
H8A0.82130.25000.03690.072*
H8B0.94340.08270.08540.072*
H8C0.81750.24840.10810.072*
C90.6694 (4)0.0318 (3)0.1589 (2)0.0309 (5)
H90.55240.05630.17730.037*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Co10.0189 (2)0.0105 (2)0.0213 (2)0.00437 (16)0.00295 (15)0.00778 (16)
O40.0233 (8)0.0124 (7)0.0237 (7)0.0055 (6)0.0032 (5)0.0082 (6)
O1W0.0208 (7)0.0153 (7)0.0301 (8)0.0054 (6)0.0030 (6)0.0126 (6)
O30.0273 (8)0.0180 (7)0.0221 (7)0.0078 (6)0.0063 (6)0.0110 (6)
O20.0279 (8)0.0168 (7)0.0297 (8)0.0076 (6)0.0065 (6)0.0151 (7)
O10.0227 (8)0.0146 (8)0.0375 (8)0.0059 (6)0.0044 (6)0.0144 (6)
O50.0393 (10)0.0179 (8)0.0395 (9)0.0122 (7)0.0049 (7)0.0051 (7)
N10.0172 (9)0.0136 (8)0.0209 (8)0.0043 (7)0.0028 (6)0.0087 (7)
N20.0200 (8)0.0098 (8)0.0240 (8)0.0056 (7)0.0010 (6)0.0063 (7)
N30.0419 (11)0.0206 (9)0.0236 (9)0.0106 (8)0.0034 (8)0.0076 (7)
C60.0152 (9)0.0167 (10)0.0228 (10)0.0060 (8)0.0005 (7)0.0086 (8)
C50.0161 (9)0.0155 (10)0.0218 (9)0.0049 (8)0.0017 (7)0.0094 (8)
C40.0163 (9)0.0147 (10)0.0227 (9)0.0057 (8)0.0003 (7)0.0091 (8)
C30.0152 (9)0.0160 (10)0.0303 (10)0.0045 (8)0.0002 (8)0.0130 (9)
C20.0172 (10)0.0128 (10)0.0243 (10)0.0054 (8)0.0001 (7)0.0074 (8)
C10.0299 (11)0.0205 (11)0.0245 (10)0.0105 (9)0.0045 (8)0.0096 (9)
C70.0534 (15)0.0351 (15)0.0471 (15)0.0034 (13)0.0177 (13)0.0078 (12)
C80.0650 (16)0.0353 (14)0.0426 (14)0.0331 (13)0.0161 (13)0.0028 (12)
C90.0366 (13)0.0268 (12)0.0336 (12)0.0164 (11)0.0115 (10)0.0143 (10)
Geometric parameters (Å, º) top
Co1—O1Wi2.0895 (14)N2—H20.8600
Co1—O1W2.0895 (14)N3—C91.319 (3)
Co1—N1i2.0982 (16)N3—C81.440 (3)
Co1—N12.0982 (16)N3—C71.453 (3)
Co1—O42.1543 (14)C6—C51.476 (3)
Co1—O4i2.1543 (14)C5—C41.374 (3)
O4—C61.240 (2)C4—C31.486 (3)
O1W—H1W20.8500C2—C11.482 (3)
O1W—H1W10.8500C1—H1A0.9600
O3—C61.287 (2)C1—H1B0.9600
O2—C31.282 (3)C1—H1C0.9600
O2—H2A0.863 (10)C7—H7A0.9600
O1—C31.237 (2)C7—H7B0.9600
O5—C91.237 (3)C7—H7C0.9600
N1—C21.327 (3)C8—H8A0.9600
N1—C51.371 (2)C8—H8B0.9600
N2—C21.353 (3)C8—H8C0.9600
N2—C41.368 (3)C9—H90.9300
O1Wi—Co1—O1W180.00 (8)N1—C5—C6117.57 (17)
O1Wi—Co1—N1i90.58 (6)C4—C5—C6132.78 (18)
O1W—Co1—N1i89.42 (6)N2—C4—C5105.63 (17)
O1Wi—Co1—N189.42 (6)N2—C4—C3123.03 (18)
O1W—Co1—N190.58 (6)C5—C4—C3131.33 (19)
N1i—Co1—N1180.0O1—C3—O2124.18 (18)
O1Wi—Co1—O490.84 (5)O1—C3—C4119.29 (19)
O1W—Co1—O489.16 (5)O2—C3—C4116.53 (17)
N1i—Co1—O4101.17 (6)N1—C2—N2110.61 (17)
N1—Co1—O478.83 (6)N1—C2—C1125.33 (18)
O1Wi—Co1—O4i89.16 (5)N2—C2—C1124.06 (18)
O1W—Co1—O4i90.84 (5)C2—C1—H1A109.5
N1i—Co1—O4i78.83 (6)C2—C1—H1B109.5
N1—Co1—O4i101.17 (6)H1A—C1—H1B109.5
O4—Co1—O4i180.0C2—C1—H1C109.5
C6—O4—Co1113.98 (12)H1A—C1—H1C109.5
Co1—O1W—H1W2114.5H1B—C1—H1C109.5
Co1—O1W—H1W1114.9N3—C7—H7A109.5
H1W2—O1W—H1W1106.7N3—C7—H7B109.5
C3—O2—H2A108.9 (18)H7A—C7—H7B109.5
C2—N1—C5106.13 (15)N3—C7—H7C109.5
C2—N1—Co1142.57 (14)H7A—C7—H7C109.5
C5—N1—Co1111.29 (12)H7B—C7—H7C109.5
C2—N2—C4107.99 (16)N3—C8—H8A109.5
C2—N2—H2126.0N3—C8—H8B109.5
C4—N2—H2126.0H8A—C8—H8B109.5
C9—N3—C8121.9 (2)N3—C8—H8C109.5
C9—N3—C7120.7 (2)H8A—C8—H8C109.5
C8—N3—C7117.2 (2)H8B—C8—H8C109.5
O4—C6—O3123.66 (18)O5—C9—N3124.9 (2)
O4—C6—C5118.29 (17)O5—C9—H9117.5
O3—C6—C5118.04 (17)N3—C9—H9117.5
N1—C5—C4109.64 (17)
O1Wi—Co1—O4—C687.79 (14)O4—C6—C5—C4179.21 (19)
O1W—Co1—O4—C692.21 (14)O3—C6—C5—C40.4 (3)
N1i—Co1—O4—C6178.54 (13)C2—N2—C4—C50.0 (2)
N1—Co1—O4—C61.46 (13)C2—N2—C4—C3178.91 (17)
O1Wi—Co1—N1—C289.6 (2)N1—C5—C4—N20.0 (2)
O1W—Co1—N1—C290.4 (2)C6—C5—C4—N2178.5 (2)
O4—Co1—N1—C2179.4 (2)N1—C5—C4—C3178.80 (19)
O4i—Co1—N1—C20.6 (2)C6—C5—C4—C30.3 (4)
O1Wi—Co1—N1—C589.19 (13)N2—C4—C3—O10.1 (3)
O1W—Co1—N1—C590.81 (13)C5—C4—C3—O1178.5 (2)
O4—Co1—N1—C51.78 (12)N2—C4—C3—O2179.72 (17)
O4i—Co1—N1—C5178.22 (12)C5—C4—C3—O21.1 (3)
Co1—O4—C6—O3179.57 (15)C5—N1—C2—N20.0 (2)
Co1—O4—C6—C50.8 (2)Co1—N1—C2—N2178.85 (15)
C2—N1—C5—C40.0 (2)C5—N1—C2—C1179.54 (18)
Co1—N1—C5—C4179.25 (13)Co1—N1—C2—C10.7 (4)
C2—N1—C5—C6178.74 (17)C4—N2—C2—N10.0 (2)
Co1—N1—C5—C62.0 (2)C4—N2—C2—C1179.55 (18)
O4—C6—C5—N10.8 (3)C8—N3—C9—O52.9 (4)
O3—C6—C5—N1178.00 (16)C7—N3—C9—O5178.2 (2)
Symmetry code: (i) x, y+2, z+1.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
N2—H2···O5ii0.861.832.670 (2)166
O2—H2A···O30.86 (1)1.59 (1)2.452 (2)173 (3)
O1W—H1W2···O1iii0.851.912.7579 (19)178
O1W—H1W1···O1iv0.852.032.847 (2)160
Symmetry codes: (ii) x+1, y, z+1; (iii) x+1, y+1, z+1; (iv) x, y+1, z.

Experimental details

Crystal data
Chemical formula[Co(C6H5N2O4)2(H2O)2]·2C3H7NO
Mr579.39
Crystal system, space groupTriclinic, P1
Temperature (K)123
a, b, c (Å)7.1979 (11), 9.2180 (15), 10.8659 (17)
α, β, γ (°)65.173 (2), 83.459 (2), 68.254 (2)
V3)607.02 (17)
Z1
Radiation typeMo Kα
µ (mm1)0.78
Crystal size (mm)0.20 × 0.15 × 0.14
Data collection
DiffractometerBruker SMART APEX area-detector
diffractometer
Absorption correctionMulti-scan
(SADABS; Sheldrick, 1996)
Tmin, Tmax0.859, 0.899
No. of measured, independent and
observed [I > 2σ(I)] reflections
4588, 2335, 2176
Rint0.026
(sin θ/λ)max1)0.617
Refinement
R[F2 > 2σ(F2)], wR(F2), S 0.039, 0.106, 1.06
No. of reflections2335
No. of parameters172
No. of restraints1
H-atom treatmentH atoms treated by a mixture of independent and constrained refinement
Δρmax, Δρmin (e Å3)0.65, 0.49

Computer programs: SMART (Bruker, 2002), SAINT (Bruker, 2002), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), SHELXTL (Sheldrick, 2008).

Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
N2—H2···O5i0.861.832.670 (2)166.2
O2—H2A···O30.863 (10)1.593 (11)2.452 (2)173 (3)
O1W—H1W2···O1ii0.851.912.7579 (19)177.6
O1W—H1W1···O1iii0.852.032.847 (2)160.0
Symmetry codes: (i) x+1, y, z+1; (ii) x+1, y+1, z+1; (iii) x, y+1, z.
 

Acknowledgements

We thank Hengyang Normal University for supporting this study.

References

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