Hexaaqua­zinc(II) dichloride bis­(hexa­methyl­enetetramine) tetra­hydrate

Yao, X.J.; Xuan, Y.W.; Wu, W.

doi:10.1107/S1600536808024793

metal-organic compounds

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ISSN: 2056-9890

Volume 64| Part 9| September 2008| Page m1132

doi:10.1107/S1600536808024793

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Hexaaquazinc(II) dichloride bis(hexamethylenetetramine) tetrahydrate

Xin Jian Yao,^a Ya Wen Xuan ^a ^* and Wen Wu ^a

^aDepartment of Chemistry, Zhokou Normal University, Henan 466001, People's Republic of China
^*Correspondence e-mail: bookw@126.com

(Received 19 July 2008; accepted 1 August 2008; online 6 August 2008)

The title compound, [Zn(H₂O)₆]Cl₂·2C₆H₁₂N₄·4H₂O, has been prepared under mild hydrothermal conditions. The Zn^II atom, located on a centre of symmetry, is coordinated by six water molecules in a distorted octahedral coordination geometry. The hexamethylenetetramine molecule is not coordinated to Zn^II but links the Zn complexes via three O—H⋯N hydrogen bonds. The remaining N atom of the hexamethylenetetramine molecule is hydrogen-bonded to a solvent water molecule. In the crystal structure, intermolecular O—H⋯O, O—H⋯N and O—H⋯Cl hydrogen bonds link the components into a three-dimensional network.

Related literature

For related compounds, see: Zhang et al. (2000 ).

Experimental

Crystal data

[Zn(H₂O)₆]Cl₂·2C₆H₁₂N₄·4H₂O
M_r = 596.84
Triclinic, $[P \overline 1]$
a = 9.345 (3) Å
b = 9.4176 (15) Å
c = 9.4535 (15) Å
α = 119.521 (1)°
β = 94.218 (2)°
γ = 100.969 (2)°
V = 697.0 (3) Å³
Z = 1
Mo Kα radiation
μ = 1.13 mm⁻¹
T = 291 (2) K
0.36 × 0.29 × 0.15 mm

Data collection

Bruker SMART CCD area-detector diffractometer
Absorption correction: multi-scan (SADABS; Sheldrick, 1996 ) T_min = 0.690, T_max = 0.849
5184 measured reflections
2576 independent reflections
2466 reflections with I > 2σ(I)
R_int = 0.018

Refinement

R[F² > 2σ(F²)] = 0.038
wR(F²) = 0.113
S = 1.05
2576 reflections
151 parameters
H-atom parameters constrained
Δρ_max = 0.51 e Å⁻³
Δρ_min = −0.68 e Å⁻³

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
O1—H1W⋯N3	0.82	2.05	2.827 (3)	158
O1—H2W⋯O5ⁱ	0.83	1.94	2.743 (3)	162
O2—H3W⋯N2ⁱⁱ	0.83	1.99	2.804 (3)	167
O2—H4W⋯O4ⁱⁱ	0.84	1.90	2.711 (3)	165
O3—H5W⋯Cl1	0.82	2.55	3.197 (2)	137
O3—H6W⋯N1ⁱⁱⁱ	0.83	2.01	2.813 (3)	165
O4—H7W⋯Cl1	0.83	2.36	3.175 (2)	168
O4—H8W⋯N4^iv	0.84	2.00	2.835 (3)	174
O5—H9W⋯Cl1	0.83	2.43	3.255 (3)	168
O5—H10W⋯Cl1^v	0.83	2.38	3.213 (3)	175
Symmetry codes: (i) x, y, z-1; (ii) -x+1, -y+1, -z+1; (iii) x, y-1, z; (iv) -x, -y+1, -z+1; (v) -x, -y, -z+1.

Data collection: SMART (Bruker, 2002 ); cell refinement: SAINT (Bruker, 2002); data reduction: SAINT; 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.

Supporting information

Crystal structure: contains datablocks I, global. DOI: 10.1107/S1600536808024793/kp2186sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536808024793/kp2186Isup2.hkl

checkCIF report

Comment top

The asymmetric unit (Fig.1) consists of one half of hexaaua Zn^II octahedron, one chloride ion, one uncoordinated neutral hexamethylenetetramine and two molecules of water of crystallization. The hexamethylenetetramine molecule is linked to the [Zn(H₂O)₆]²⁺ via three O—H···N hydrogen bonds, while atom N3 of hexamethylenetetramine is hydrogen-bonded to O5 of the solvent water molecule. The Cl^- anions link to the [Zn(H₂O)₆]²⁺ and water of crystallization via O—H···Cl hydrogen bonding, Hydrogen bonding of these anionic and cationic frameworks results in the formation of a three-dimensional network (Table 1, Fig. 2).

Related literature top

For related compounds, see: Zhang et al. (2000).

Experimental top

All the reagents were of AR grade and used without further purification. C₆H₁₂N₄ (1.401 g, 10 mmol) were dissolved in 50 mL H₂O solution, then the resultant solution was added in 10 mL double-distilled water containing ZnCl₂ (0.273 g, 2 mmol). The resulting solution was heated at 423 K for 96 h. After cooling to room temperature, block crystals were obtained in a yield up to 21.1%.

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

	Fig. 1. Plot of an asymmetric unit with the 30% probability ellipsoids.
	Fig. 2. Three-dimensional network of hydrogen-bonding pattern with the motif R₄⁴(16) linking the cationic moieties with hexamine which are in turn interwoven with anionic moieties via water molecules.

Hexaaquazinc(II) dichloride bis(hexamethylenetetramine) tetrahydrate top

Crystal data top

[Zn(H₂O)₆]Cl₂·2C₆H₁₂N₄·4H₂O	Z = 1
M_r = 596.84	F(000) = 316
Triclinic, P1	D_x = 1.422 Mg m⁻³
Hall symbol: -P 1	Mo Kα radiation, λ = 0.71073 Å
a = 9.345 (3) Å	Cell parameters from 2143 reflections
b = 9.4176 (15) Å	θ = 2.5–25.5°
c = 9.4535 (15) Å	µ = 1.13 mm⁻¹
α = 119.521 (1)°	T = 291 K
β = 94.218 (2)°	Block, colorless
γ = 100.969 (2)°	0.36 × 0.29 × 0.15 mm
V = 697.0 (3) Å³

Data collection top

Bruker SMART CCD area-detector diffractometer	2576 independent reflections
Radiation source: fine-focus sealed tube	2466 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.018
Detector resolution: 0 pixels mm^-1	θ_max = 25.5°, θ_min = 2.5°
ϕ and ω scans	h = −11→11
Absorption correction: multi-scan (SADABS; Sheldrick, 1996)	k = −11→11
T_min = 0.690, T_max = 0.849	l = −11→11
5184 measured reflections

Refinement top

Refinement on F²	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F² > 2σ(F²)] = 0.038	Hydrogen site location: inferred from neighbouring sites
wR(F²) = 0.113	H-atom parameters constrained
S = 1.05	w = 1/[σ²(F_o²) + (0.0645P)² + 0.6988P] where P = (F_o² + 2F_c²)/3
2576 reflections	(Δ/σ)_max < 0.001
151 parameters	Δρ_max = 0.51 e Å⁻³
0 restraints	Δρ_min = −0.68 e Å⁻³

Crystal data top

[Zn(H₂O)₆]Cl₂·2C₆H₁₂N₄·4H₂O	γ = 100.969 (2)°
M_r = 596.84	V = 697.0 (3) Å³
Triclinic, P1	Z = 1
a = 9.345 (3) Å	Mo Kα radiation
b = 9.4176 (15) Å	µ = 1.13 mm⁻¹
c = 9.4535 (15) Å	T = 291 K
α = 119.521 (1)°	0.36 × 0.29 × 0.15 mm
β = 94.218 (2)°

Data collection top

Bruker SMART CCD area-detector diffractometer	2576 independent reflections
Absorption correction: multi-scan (SADABS; Sheldrick, 1996)	2466 reflections with I > 2σ(I)
T_min = 0.690, T_max = 0.849	R_int = 0.018
5184 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.038	0 restraints
wR(F²) = 0.113	H-atom parameters constrained
S = 1.05	Δρ_max = 0.51 e Å⁻³
2576 reflections	Δρ_min = −0.68 e Å⁻³
151 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

	x	y	z	U_iso*/U_eq
Zn1	0.5000	0.0000	0.0000	0.02942 (16)
Cl1	0.18939 (10)	0.17479 (10)	0.43496 (10)	0.0530 (2)
O1	0.3835 (2)	0.1347 (2)	−0.0573 (2)	0.0385 (5)
H1W	0.3888	0.2257	0.0269	0.058*
H2W	0.3035	0.0961	−0.1235	0.058*
O2	0.6192 (2)	0.2238 (2)	0.1987 (2)	0.0445 (5)
H3W	0.6176	0.2521	0.2964	0.067*
H4W	0.6833	0.2949	0.1930	0.067*
O3	0.3580 (2)	−0.0281 (3)	0.1463 (3)	0.0449 (5)
H5W	0.3405	0.0629	0.2078	0.067*
H6W	0.3658	−0.0839	0.1907	0.067*
O4	0.1961 (2)	0.5028 (3)	0.7778 (3)	0.0453 (5)
H7W	0.2082	0.4201	0.6931	0.068*
H8W	0.1053	0.4939	0.7784	0.068*
O5	0.1487 (3)	0.0521 (4)	0.7002 (4)	0.0734 (8)
H9W	0.1699	0.0950	0.6431	0.110*
H10W	0.0600	−0.0024	0.6715	0.110*
N1	0.3345 (3)	0.7407 (3)	0.2554 (3)	0.0330 (5)
N2	0.3362 (3)	0.6543 (3)	0.4602 (3)	0.0333 (5)
N3	0.3418 (3)	0.4524 (3)	0.1728 (3)	0.0321 (5)
N4	0.1152 (2)	0.5427 (3)	0.2441 (3)	0.0340 (5)
C1	0.3865 (3)	0.7935 (3)	0.4289 (3)	0.0356 (6)
H1A	0.3492	0.8886	0.5010	0.043*
H1B	0.4942	0.8302	0.4552	0.043*
C2	0.3944 (3)	0.5116 (3)	0.3488 (3)	0.0342 (6)
H2A	0.3628	0.4191	0.3677	0.041*
H2B	0.5022	0.5466	0.3743	0.041*
C3	0.1782 (3)	0.4021 (3)	0.1376 (3)	0.0381 (6)
H3A	0.1440	0.3088	0.1546	0.046*
H3B	0.1422	0.3632	0.0223	0.046*
C4	0.1706 (3)	0.6838 (4)	0.2176 (4)	0.0375 (6)
H4A	0.1348	0.6480	0.1032	0.045*
H4B	0.1311	0.7778	0.2876	0.045*
C5	0.3925 (3)	0.5955 (4)	0.1482 (3)	0.0353 (6)
H5A	0.3595	0.5592	0.0331	0.042*
H5B	0.5003	0.6306	0.1729	0.042*
C6	0.1728 (3)	0.5995 (4)	0.4186 (3)	0.0369 (6)
H6A	0.1333	0.6925	0.4910	0.044*
H6B	0.1386	0.5076	0.4377	0.044*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Zn1	0.0395 (3)	0.0233 (2)	0.0250 (2)	0.00991 (17)	0.00711 (17)	0.01148 (18)
Cl1	0.0591 (5)	0.0399 (4)	0.0534 (5)	0.0135 (3)	0.0246 (4)	0.0173 (4)
O1	0.0604 (11)	0.0278 (9)	0.0329 (10)	0.0192 (8)	−0.0009 (8)	0.0105 (8)
O2	0.0710 (13)	0.0280 (10)	0.0220 (9)	−0.0076 (9)	0.0009 (9)	0.0081 (8)
O3	0.0675 (14)	0.0422 (11)	0.0548 (12)	0.0327 (10)	0.0401 (11)	0.0367 (10)
O4	0.0398 (11)	0.0386 (11)	0.0417 (11)	0.0003 (9)	0.0055 (9)	0.0130 (9)
O5	0.0604 (16)	0.086 (2)	0.0720 (18)	0.0027 (14)	−0.0156 (13)	0.0485 (16)
N1	0.0423 (12)	0.0298 (11)	0.0371 (12)	0.0160 (10)	0.0161 (10)	0.0214 (10)
N2	0.0426 (12)	0.0285 (11)	0.0242 (10)	0.0036 (9)	0.0050 (9)	0.0125 (9)
N3	0.0407 (12)	0.0255 (11)	0.0275 (11)	0.0144 (9)	0.0046 (9)	0.0102 (9)
N4	0.0345 (11)	0.0289 (11)	0.0335 (11)	0.0088 (9)	0.0063 (9)	0.0124 (9)
C1	0.0433 (15)	0.0221 (12)	0.0348 (14)	0.0048 (10)	0.0100 (11)	0.0110 (11)
C2	0.0432 (15)	0.0284 (13)	0.0326 (13)	0.0090 (11)	0.0007 (11)	0.0178 (11)
C3	0.0417 (15)	0.0264 (13)	0.0327 (14)	0.0086 (11)	−0.0005 (11)	0.0066 (11)
C4	0.0443 (15)	0.0376 (15)	0.0404 (15)	0.0219 (12)	0.0136 (12)	0.0226 (12)
C5	0.0452 (15)	0.0399 (15)	0.0314 (13)	0.0217 (12)	0.0167 (11)	0.0214 (12)
C6	0.0427 (15)	0.0337 (14)	0.0326 (14)	0.0055 (11)	0.0128 (11)	0.0167 (11)

Geometric parameters (Å, º) top

Zn1—O2ⁱ	2.0269 (18)	N2—C2	1.476 (3)
Zn1—O2	2.0269 (18)	N2—C1	1.481 (3)
Zn1—O1	2.0507 (17)	N3—C3	1.472 (4)
Zn1—O1ⁱ	2.0507 (17)	N3—C5	1.476 (3)
Zn1—O3ⁱ	2.0595 (18)	N3—C2	1.477 (3)
Zn1—O3	2.0595 (18)	N4—C4	1.477 (4)
O1—H1W	0.8223	N4—C3	1.478 (3)
O1—H2W	0.8281	N4—C6	1.479 (4)
O2—H3W	0.8279	C1—H1A	0.9700
O2—H4W	0.8360	C1—H1B	0.9700
O3—H5W	0.8221	C2—H2A	0.9700
O3—H6W	0.8267	C2—H2B	0.9700
O4—H7W	0.8326	C3—H3A	0.9700
O4—H8W	0.8374	C3—H3B	0.9700
O5—H9W	0.8338	C4—H4A	0.9700
O5—H10W	0.8316	C4—H4B	0.9700
N1—C1	1.470 (4)	C5—H5A	0.9700
N1—C4	1.476 (4)	C5—H5B	0.9700
N1—C5	1.479 (3)	C6—H6A	0.9700
N2—C6	1.471 (4)	C6—H6B	0.9700

O2ⁱ—Zn1—O2	180.00 (11)	C3—N4—C6	107.8 (2)
O2ⁱ—Zn1—O1	92.66 (8)	N1—C1—N2	111.7 (2)
O2—Zn1—O1	87.34 (8)	N1—C1—H1A	109.3
O2ⁱ—Zn1—O1ⁱ	87.34 (8)	N2—C1—H1A	109.3
O2—Zn1—O1ⁱ	92.66 (8)	N1—C1—H1B	109.3
O1—Zn1—O1ⁱ	180.00 (12)	N2—C1—H1B	109.3
O2ⁱ—Zn1—O3ⁱ	89.59 (9)	H1A—C1—H1B	107.9
O2—Zn1—O3ⁱ	90.41 (9)	N2—C2—N3	111.6 (2)
O1—Zn1—O3ⁱ	86.57 (8)	N2—C2—H2A	109.3
O1ⁱ—Zn1—O3ⁱ	93.43 (8)	N3—C2—H2A	109.3
O2ⁱ—Zn1—O3	90.41 (9)	N2—C2—H2B	109.3
O2—Zn1—O3	89.59 (9)	N3—C2—H2B	109.3
O1—Zn1—O3	93.43 (8)	H2A—C2—H2B	108.0
O1ⁱ—Zn1—O3	86.57 (8)	N3—C3—N4	112.2 (2)
O3ⁱ—Zn1—O3	180.00 (19)	N3—C3—H3A	109.2
Zn1—O1—H1W	109.5	N4—C3—H3A	109.2
Zn1—O1—H2W	126.6	N3—C3—H3B	109.2
H1W—O1—H2W	113.2	N4—C3—H3B	109.2
Zn1—O2—H3W	124.5	H3A—C3—H3B	107.9
Zn1—O2—H4W	124.2	N1—C4—N4	112.2 (2)
H3W—O2—H4W	111.0	N1—C4—H4A	109.2
Zn1—O3—H5W	109.6	N4—C4—H4A	109.2
Zn1—O3—H6W	123.5	N1—C4—H4B	109.2
H5W—O3—H6W	113.5	N4—C4—H4B	109.2
H7W—O4—H8W	110.1	H4A—C4—H4B	107.9
H9W—O5—H10W	111.5	N3—C5—N1	111.7 (2)
C1—N1—C4	108.4 (2)	N3—C5—H5A	109.3
C1—N1—C5	108.2 (2)	N1—C5—H5A	109.3
C4—N1—C5	108.3 (2)	N3—C5—H5B	109.3
C6—N2—C2	108.5 (2)	N1—C5—H5B	109.3
C6—N2—C1	108.5 (2)	H5A—C5—H5B	108.0
C2—N2—C1	108.0 (2)	N2—C6—N4	112.0 (2)
C3—N3—C5	108.7 (2)	N2—C6—H6A	109.2
C3—N3—C2	108.4 (2)	N4—C6—H6A	109.2
C5—N3—C2	107.9 (2)	N2—C6—H6B	109.2
C4—N4—C3	108.0 (2)	N4—C6—H6B	109.2
C4—N4—C6	108.1 (2)	H6A—C6—H6B	107.9

C4—N1—C1—N2	58.4 (3)	C1—N1—C4—N4	−58.6 (3)
C5—N1—C1—N2	−58.8 (3)	C5—N1—C4—N4	58.6 (3)
C6—N2—C1—N1	−58.6 (3)	C3—N4—C4—N1	−58.3 (3)
C2—N2—C1—N1	58.9 (3)	C6—N4—C4—N1	58.1 (3)
C6—N2—C2—N3	58.3 (3)	C3—N3—C5—N1	58.1 (3)
C1—N2—C2—N3	−59.1 (3)	C2—N3—C5—N1	−59.2 (3)
C3—N3—C2—N2	−58.2 (3)	C1—N1—C5—N3	59.2 (3)
C5—N3—C2—N2	59.4 (3)	C4—N1—C5—N3	−58.1 (3)
C5—N3—C3—N4	−58.3 (3)	C2—N2—C6—N4	−58.8 (3)
C2—N3—C3—N4	58.7 (3)	C1—N2—C6—N4	58.4 (3)
C4—N4—C3—N3	58.0 (3)	C4—N4—C6—N2	−58.0 (3)
C6—N4—C3—N3	−58.6 (3)	C3—N4—C6—N2	58.5 (3)

Symmetry code: (i) −x+1, −y, −z.

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O1—H1W···N3	0.82	2.05	2.827 (3)	158
O1—H2W···O5ⁱⁱ	0.83	1.94	2.743 (3)	162
O2—H3W···N2ⁱⁱⁱ	0.83	1.99	2.804 (3)	167
O2—H4W···O4ⁱⁱⁱ	0.84	1.90	2.711 (3)	165
O3—H5W···Cl1	0.82	2.55	3.197 (2)	137
O3—H6W···N1^iv	0.83	2.01	2.813 (3)	165
O4—H7W···Cl1	0.83	2.36	3.175 (2)	168
O4—H8W···N4^v	0.84	2.00	2.835 (3)	174
O5—H9W···Cl1	0.83	2.43	3.255 (3)	168
O5—H10W···Cl1^vi	0.83	2.38	3.213 (3)	175

Symmetry codes: (ii) x, y, z−1; (iii) −x+1, −y+1, −z+1; (iv) x, y−1, z; (v) −x, −y+1, −z+1; (vi) −x, −y, −z+1.

Experimental details

Crystal data
Chemical formula	[Zn(H₂O)₆]Cl₂·2C₆H₁₂N₄·4H₂O
M_r	596.84
Crystal system, space group	Triclinic, P1
Temperature (K)	291
a, b, c (Å)	9.345 (3), 9.4176 (15), 9.4535 (15)
α, β, γ (°)	119.521 (1), 94.218 (2), 100.969 (2)
V (Å³)	697.0 (3)
Z	1
Radiation type	Mo Kα
µ (mm⁻¹)	1.13
Crystal size (mm)	0.36 × 0.29 × 0.15

Data collection
Diffractometer	Bruker SMART CCD area-detector diffractometer
Absorption correction	Multi-scan (SADABS; Sheldrick, 1996)
T_min, T_max	0.690, 0.849
No. of measured, independent and observed [I > 2σ(I)] reflections	5184, 2576, 2466
R_int	0.018
(sin θ/λ)_max (Å⁻¹)	0.606

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.038, 0.113, 1.05
No. of reflections	2576
No. of parameters	151
H-atom treatment	H-atom parameters constrained
Δρ_max, Δρ_min (e Å⁻³)	0.51, −0.68

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···A	D—H	H···A	D···A	D—H···A
O1—H1W···N3	0.82	2.05	2.827 (3)	157.6
O1—H2W···O5ⁱ	0.83	1.94	2.743 (3)	162.0
O2—H3W···N2ⁱⁱ	0.83	1.99	2.804 (3)	166.6
O2—H4W···O4ⁱⁱ	0.84	1.90	2.711 (3)	164.6
O3—H5W···Cl1	0.82	2.55	3.197 (2)	136.6
O3—H6W···N1ⁱⁱⁱ	0.83	2.01	2.813 (3)	164.9
O4—H7W···Cl1	0.83	2.36	3.175 (2)	168.2
O4—H8W···N4^iv	0.84	2.00	2.835 (3)	174.4
O5—H9W···Cl1	0.83	2.43	3.255 (3)	168.4
O5—H10W···Cl1^v	0.83	2.38	3.213 (3)	174.8

Symmetry codes: (i) x, y, z−1; (ii) −x+1, −y+1, −z+1; (iii) x, y−1, z; (iv) −x, −y+1, −z+1; (v) −x, −y, −z+1.

Acknowledgements

We thank the Natural Science Foundation of Henan Province and the Key Discipline Foundation of Zhoukou Normal University for financial support of this research.

References

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Sheldrick, G. M. (1996). SADABS. University of Göttingen, Germany. Google Scholar
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