catena-Poly[[tri­aqua­copper(II)]-μ-5-carb­oxy­benzene-1,3-di­carboxyl­ato-κ2O1:O3]

Ma, Y.-H.; Ma, P.-Z.; Yao, T.; Hao, J.-T.

doi:10.1107/S1600536813024781

metal-organic compounds

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ISSN: 2056-9890

Volume 69| Part 10| October 2013| Page m538

doi:10.1107/S1600536813024781

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catena-Poly[[triaquacopper(II)]-μ-5-carboxybenzene-1,3-dicarboxylato-κ²O¹:O³]

Yu-Hong Ma,^a ^* Pi-Zhuang Ma,^b Ting Yao ^a and Jing-Tuan Hao ^a

^aAir Force Service College, Xuzhou 221000, People's Republic of China, and ^bLogistics College, Beijing 100858, People's Republic of China
^*Correspondence e-mail: myhmayuhong@163.com

(Received 1 September 2013; accepted 5 September 2013; online 12 September 2013)

In the title complex, [Cu(C₉H₄O₆)(H₂O)₃]_n, the Cu^II cation exhibits a distorted square-pyramidal coordination geometry involving five O atoms from two monodentate 5-carboxybenzene-1,3-dicarboxylate anions and three water molecules. The 5-carboxybenzene-1,3-dicarboxylate anions bridge Cu^II cations into zigzag polymeric chains running along the b-axis direction. These chains are further linked by O—H⋯O hydrogen bonds between coordinating water molecules or carboxyl groups and carboxylate groups into a three-dimensional supramolecular architecture. In the crystal, π–π stacking is observed between parallel benzene rings of adjacent chains, the centroid–centroid distances being 3.584 (3) and 3.684 (3) Å.

Related literature

For background to complexes derived from 1,3,5-benzenetricarboxylic acid and for related structures, see: Lei et al. (2012 ); Liu (2012 ); Yao & Yuan (2011 ).

Experimental

Crystal data

[Cu(C₉H₄O₆)(H₂O)₃]
M_r = 325.71
Monoclinic, P 2₁ /c
a = 6.8551 (14) Å
b = 18.892 (4) Å
c = 10.716 (3) Å
β = 126.87 (2)°
V = 1110.2 (5) Å³
Z = 4
Mo Kα radiation
μ = 2.01 mm⁻¹
T = 293 K
0.24 × 0.21 × 0.21 mm

Data collection

Rigaku SCXmini diffractometer
Absorption correction: multi-scan (ABSCOR; Higashi, 1995 ) T_min = 0.644, T_max = 0.677
9530 measured reflections
1957 independent reflections
1744 reflections with I > 2σ(I)
R_int = 0.047

Refinement

R[F² > 2σ(F²)] = 0.040
wR(F²) = 0.083
S = 1.01
1957 reflections
193 parameters
10 restraints
H atoms treated by a mixture of independent and constrained refinement
Δρ_max = 0.42 e Å⁻³
Δρ_min = −0.47 e Å⁻³

Table 1
Selected bond lengths (Å)

Cu1—O2	1.934 (2)
Cu1—O5ⁱ	1.917 (2)
Cu1—O1W	2.258 (3)
Cu1—O2W	1.987 (3)
Cu1—O3W	1.984 (3)
Symmetry code: (i) $[-x+2, y-{\script{1\over 2}}, -z+{\script{3\over 2}}]$ .

Table 2
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
O3—H1⋯O1ⁱⁱ	0.83 (1)	1.80 (2)	2.570 (4)	153 (4)
O1W—H1WA⋯O1ⁱⁱⁱ	0.84 (1)	2.37 (2)	3.077 (4)	142 (3)
O1W—H1WB⋯O4^iv	0.83 (1)	1.97 (1)	2.801 (4)	171 (4)
O2W—H2WA⋯O6^v	0.84 (1)	1.91 (2)	2.697 (4)	155 (4)
O2W—H2WB⋯O3^vi	0.84 (1)	2.07 (2)	2.875 (4)	162 (3)
O3W—H3WA⋯O6^vii	0.83 (1)	1.92 (2)	2.717 (4)	161 (4)
O3W—H3WB⋯O2^viii	0.83 (1)	2.33 (2)	3.130 (4)	161 (3)
Symmetry codes: (ii) x-1, y, z-1; (iii) $[x-1, -y+{\script{3\over 2}}, z-{\script{1\over 2}}]$ ; (iv) $[-x+1, y-{\script{1\over 2}}, -z+{\script{1\over 2}}]$ ; (v) -x+2, -y+2, -z+1; (vi) $[x+1, -y+{\script{3\over 2}}, z+{\script{1\over 2}}]$ ; (vii) -x+1, -y+2, -z+1; (viii) $[x, -y+{\script{3\over 2}}, z+{\script{1\over 2}}]$ .

Data collection: CrystalClear (Rigaku, 2005 ); cell refinement: CrystalClear; data reduction: CrystalClear; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008 ); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEPII (Johnson, 1976 ) and DIAMOND (Brandenburg, 1999 ); software used to prepare material for publication: SHELXL97.

Supporting information

Crystal structure: contains datablocks I, global. DOI: 10.1107/S1600536813024781/xu5736sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536813024781/xu5736Isup2.hkl

checkCIF report

Comment top

The design and synthesis of novel coordination polymer are currently of great interest due to their potential application. Aromatic polycarboxylate compounds, such as 1,3,5-benzenetricarboxylic acid, have been proved to useful ligands to synthesis coordination polymers due to the versatile binding modes (Lei et al., (2012); Liu (2012); Yao & Yuan, (2011). Herein we report the synthesis and structures of the title compound.

As illustrated in Figure 1, there is one Cu(II) ion in the asymmetry unit. Cu(II) atom exhibits a distorted square-pyramidal coordination sphere, defined by three O atoms from three water molecules and two O atoms from two different carboxylate ligands. O3w is axially positioned, and the other four O atoms are formed the basal plane. The 5-carboxybenzene-1,3-dicarboxylate ligands connect two Cu(II) ions and build a one-dimensional zigzag chain (Fig. 2). The chains are further self-assembled into a three-dimensional supramolecular network through hydrogen bonds between the water molecules and carboxylate groups (Fig. 3). In the crystal, π-π stacking is observed between parallel benzene rings of adjacent chains, centroids distances are 3.584 (3) and 3.684 (3) Å.

Related literature top

For background to 1,3,5-benzenetricarboxylic acid complexes and related structures, see: Lei et al. (2012); Liu (2012); Yao & Yuan (2011).

Experimental top

A mixture of copper nitrate (0.2 mmol), 1,3,5-benzenetricarboxylic acid (0.2 mmol), NaOH (0.2 mmol) and H₂O (5 ml) was sealed in a 23 ml Tefon-lined stainless-steel reactor and then heated to 413 K for three days under autogenous pressure, then the mixture was slowly cooled to room temperature. The crystals were obtained from the mixture.

Computing details top

Data collection: CrystalClear (Rigaku, 2005); cell refinement: CrystalClear (Rigaku, 2005); data reduction: CrystalClear (Rigaku, 2005); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEPII (Johnson, 1976) and DIAMOND (Brandenburg, 1999); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Figures top

	Fig. 1. The structure of the title compound, showing the atomic numbering scheme with 30% probability displacement ellipsoids. [Symmetry codes: (i) 2 - x, -0.5 + y, 1.5 - z]
	Fig. 2. A view of the chain structure of title compound.
	Fig. 3. A view of the three-dimensional constructed by hydrogen bonds.

catena-Poly[[triaquacopper(II)]-µ-5-carboxybenzene-1,3-dicarboxylato-κ²O¹:O³] top

Crystal data top

[Cu(C₉H₄O₆)(H₂O)₃]	F(000) = 660
M_r = 325.71	D_x = 1.949 Mg m⁻³
Monoclinic, P2₁/c	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybc	Cell parameters from 10073 reflections
a = 6.8551 (14) Å	θ = 3.2–27.5°
b = 18.892 (4) Å	µ = 2.01 mm⁻¹
c = 10.716 (3) Å	T = 293 K
β = 126.87 (2)°	Block, blue
V = 1110.2 (5) Å³	0.24 × 0.21 × 0.21 mm
Z = 4

Data collection top

Rigaku SCXmini diffractometer	1957 independent reflections
Radiation source: fine-focus sealed tube	1744 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.047
ω scans	θ_max = 25.0°, θ_min = 3.2°
Absorption correction: multi-scan (ABSCOR; Higashi, 1995)	h = −8→8
T_min = 0.644, T_max = 0.677	k = −22→22
9530 measured reflections	l = −12→12

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)] = 0.040	Hydrogen site location: inferred from neighbouring sites
wR(F²) = 0.083	H atoms treated by a mixture of independent and constrained refinement
S = 1.01	w = 1/[σ²(F_o²) + (0.0257P)² + 3.7798P] where P = (F_o² + 2F_c²)/3
1957 reflections	(Δ/σ)_max = 0.001
193 parameters	Δρ_max = 0.42 e Å⁻³
10 restraints	Δρ_min = −0.47 e Å⁻³

Crystal data top

[Cu(C₉H₄O₆)(H₂O)₃]	V = 1110.2 (5) Å³
M_r = 325.71	Z = 4
Monoclinic, P2₁/c	Mo Kα radiation
a = 6.8551 (14) Å	µ = 2.01 mm⁻¹
b = 18.892 (4) Å	T = 293 K
c = 10.716 (3) Å	0.24 × 0.21 × 0.21 mm
β = 126.87 (2)°

Data collection top

Rigaku SCXmini diffractometer	1957 independent reflections
Absorption correction: multi-scan (ABSCOR; Higashi, 1995)	1744 reflections with I > 2σ(I)
T_min = 0.644, T_max = 0.677	R_int = 0.047
9530 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.040	10 restraints
wR(F²) = 0.083	H atoms treated by a mixture of independent and constrained refinement
S = 1.01	Δρ_max = 0.42 e Å⁻³
1957 reflections	Δρ_min = −0.47 e Å⁻³
193 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
Cu1	0.88938 (8)	0.72640 (2)	0.67913 (5)	0.01706 (15)
O1	1.0451 (6)	0.87364 (14)	0.7671 (3)	0.0342 (7)
O2	0.7879 (5)	0.81134 (12)	0.5539 (3)	0.0218 (6)
O3	0.2513 (5)	0.91554 (14)	0.0486 (3)	0.0310 (7)
O4	0.2713 (6)	1.02917 (15)	0.0054 (3)	0.0460 (9)
O5	1.0069 (5)	1.13796 (13)	0.7111 (3)	0.0235 (6)
O6	0.7413 (5)	1.18913 (13)	0.4807 (3)	0.0258 (6)
O2W	1.1364 (6)	0.71108 (15)	0.6410 (4)	0.0398 (8)
O3W	0.6921 (6)	0.75211 (17)	0.7518 (3)	0.0367 (7)
O1W	0.5840 (5)	0.67089 (15)	0.4599 (3)	0.0337 (7)
C2	0.7775 (6)	0.93545 (17)	0.5254 (4)	0.0144 (7)
C1	0.8790 (7)	0.86999 (18)	0.6244 (4)	0.0171 (8)
C9	0.8375 (7)	1.13575 (18)	0.5656 (4)	0.0174 (8)
C7	0.8509 (6)	1.00280 (18)	0.5912 (4)	0.0148 (7)
H7	0.9642	1.0078	0.6986	0.018*
C5	0.5841 (7)	1.05491 (18)	0.3357 (4)	0.0169 (8)
H5	0.5191	1.0949	0.2726	0.020*
C3	0.6077 (6)	0.92832 (18)	0.3648 (4)	0.0160 (8)
H3	0.5587	0.8834	0.3208	0.019*
C4	0.5101 (7)	0.98796 (18)	0.2688 (4)	0.0165 (8)
C8	0.3348 (7)	0.98117 (19)	0.0959 (4)	0.0223 (8)
C6	0.7546 (6)	1.06249 (18)	0.4962 (4)	0.0150 (7)
H1	0.167 (6)	0.915 (2)	−0.0485 (13)	0.023*
H2WA	1.132 (7)	0.7397 (13)	0.579 (4)	0.023*
H3WB	0.715 (6)	0.746 (2)	0.836 (2)	0.023*
H2WB	1.166 (7)	0.6699 (7)	0.628 (4)	0.023*
H1WA	0.439 (3)	0.6782 (16)	0.424 (4)	0.023*
H3WA	0.575 (5)	0.7788 (17)	0.691 (3)	0.023*
H1WB	0.619 (6)	0.6279 (7)	0.475 (4)	0.023*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Cu1	0.0235 (3)	0.0090 (2)	0.0143 (2)	0.00247 (19)	0.00896 (19)	0.00336 (18)
O1	0.0463 (19)	0.0200 (15)	0.0137 (14)	0.0008 (13)	0.0059 (13)	0.0007 (11)
O2	0.0280 (14)	0.0083 (12)	0.0176 (13)	−0.0011 (11)	0.0074 (12)	0.0014 (10)
O3	0.0441 (18)	0.0208 (14)	0.0097 (13)	−0.0087 (13)	0.0063 (13)	−0.0049 (11)
O4	0.068 (2)	0.0180 (15)	0.0167 (16)	−0.0039 (15)	0.0064 (16)	0.0040 (13)
O5	0.0303 (15)	0.0119 (13)	0.0153 (14)	−0.0014 (11)	0.0068 (12)	−0.0034 (10)
O6	0.0296 (15)	0.0119 (13)	0.0257 (15)	−0.0008 (11)	0.0112 (13)	0.0036 (11)
O2W	0.056 (2)	0.0206 (16)	0.066 (2)	0.0127 (15)	0.0494 (19)	0.0162 (15)
O3W	0.0430 (19)	0.0456 (19)	0.0305 (17)	0.0199 (15)	0.0270 (16)	0.0153 (14)
O1W	0.0320 (17)	0.0236 (15)	0.0294 (16)	−0.0021 (13)	0.0098 (14)	−0.0048 (13)
C2	0.0169 (18)	0.0100 (17)	0.0149 (18)	−0.0020 (14)	0.0088 (15)	−0.0020 (14)
C1	0.023 (2)	0.0138 (18)	0.0140 (19)	0.0014 (15)	0.0103 (17)	0.0002 (14)
C9	0.0220 (19)	0.0128 (18)	0.022 (2)	−0.0017 (15)	0.0152 (17)	−0.0031 (15)
C7	0.0182 (19)	0.0148 (18)	0.0100 (17)	0.0004 (14)	0.0078 (15)	0.0002 (14)
C5	0.025 (2)	0.0094 (17)	0.0167 (19)	0.0022 (15)	0.0124 (16)	0.0014 (14)
C3	0.0208 (19)	0.0088 (17)	0.0180 (19)	−0.0018 (14)	0.0114 (16)	−0.0021 (14)
C4	0.0184 (18)	0.016 (2)	0.0131 (18)	−0.0013 (15)	0.0084 (15)	−0.0009 (14)
C8	0.028 (2)	0.0137 (19)	0.017 (2)	−0.0002 (16)	0.0089 (17)	−0.0029 (16)
C6	0.0161 (18)	0.0126 (17)	0.0148 (18)	−0.0025 (14)	0.0085 (15)	−0.0024 (14)

Geometric parameters (Å, º) top

Cu1—O2	1.934 (2)	O3W—H3WA	0.832 (10)
Cu1—O5ⁱ	1.917 (2)	O1W—H1WA	0.836 (10)
Cu1—O1W	2.258 (3)	O1W—H1WB	0.834 (10)
Cu1—O2W	1.987 (3)	C2—C3	1.390 (5)
Cu1—O3W	1.984 (3)	C2—C7	1.394 (5)
O1—C1	1.245 (4)	C2—C1	1.501 (5)
O2—C1	1.273 (4)	C9—C6	1.511 (5)
O3—C8	1.332 (4)	C7—C6	1.392 (5)
O3—H1	0.833 (10)	C7—H7	0.9300
O4—C8	1.202 (5)	C5—C6	1.391 (5)
O5—C9	1.268 (4)	C5—C4	1.391 (5)
O5—Cu1ⁱⁱ	1.917 (2)	C5—H5	0.9300
O6—C9	1.249 (4)	C3—C4	1.396 (5)
O2W—H2WA	0.840 (10)	C3—H3	0.9300
O2W—H2WB	0.838 (10)	C4—C8	1.491 (5)
O3W—H3WB	0.832 (10)

O5ⁱ—Cu1—O2	174.44 (11)	O1—C1—O2	122.6 (3)
O5ⁱ—Cu1—O3W	93.54 (12)	O1—C1—C2	121.1 (3)
O2—Cu1—O3W	91.33 (12)	O2—C1—C2	116.3 (3)
O5ⁱ—Cu1—O2W	87.19 (12)	O6—C9—O5	124.2 (3)
O2—Cu1—O2W	88.53 (12)	O6—C9—C6	120.2 (3)
O3W—Cu1—O2W	169.17 (15)	O5—C9—C6	115.6 (3)
O5ⁱ—Cu1—O1W	90.26 (11)	C6—C7—C2	120.0 (3)
O2—Cu1—O1W	86.59 (11)	C6—C7—H7	120.0
O3W—Cu1—O1W	95.58 (13)	C2—C7—H7	120.0
O2W—Cu1—O1W	95.22 (14)	C6—C5—C4	120.5 (3)
C1—O2—Cu1	117.8 (2)	C6—C5—H5	119.8
C8—O3—H1	108 (3)	C4—C5—H5	119.8
C9—O5—Cu1ⁱⁱ	120.8 (2)	C2—C3—C4	120.6 (3)
Cu1—O2W—H2WA	116 (2)	C2—C3—H3	119.7
Cu1—O2W—H2WB	120 (3)	C4—C3—H3	119.7
H2WA—O2W—H2WB	111.3 (17)	C5—C4—C3	119.3 (3)
Cu1—O3W—H3WB	132 (2)	C5—C4—C8	119.4 (3)
Cu1—O3W—H3WA	114 (2)	C3—C4—C8	121.3 (3)
H3WB—O3W—H3WA	113.6 (18)	O4—C8—O3	122.0 (3)
Cu1—O1W—H1WA	120 (3)	O4—C8—C4	124.7 (3)
Cu1—O1W—H1WB	106 (3)	O3—C8—C4	113.3 (3)
H1WA—O1W—H1WB	112.3 (18)	C5—C6—C7	119.9 (3)
C3—C2—C7	119.7 (3)	C5—C6—C9	119.5 (3)
C3—C2—C1	119.0 (3)	C7—C6—C9	120.6 (3)
C7—C2—C1	121.4 (3)

O5ⁱ—Cu1—O2—C1	−135.3 (11)	C6—C5—C4—C3	0.4 (5)
O3W—Cu1—O2—C1	73.6 (3)	C6—C5—C4—C8	−177.5 (3)
O2W—Cu1—O2—C1	−95.6 (3)	C2—C3—C4—C5	−0.2 (5)
O1W—Cu1—O2—C1	169.1 (3)	C2—C3—C4—C8	177.8 (3)
Cu1—O2—C1—O1	6.9 (5)	C5—C4—C8—O4	9.3 (6)
Cu1—O2—C1—C2	−174.0 (2)	C3—C4—C8—O4	−168.6 (4)
C3—C2—C1—O1	174.3 (4)	C5—C4—C8—O3	−170.6 (3)
C7—C2—C1—O1	−5.3 (5)	C3—C4—C8—O3	11.5 (5)
C3—C2—C1—O2	−4.8 (5)	C4—C5—C6—C7	−0.5 (5)
C7—C2—C1—O2	175.5 (3)	C4—C5—C6—C9	178.2 (3)
Cu1ⁱⁱ—O5—C9—O6	6.7 (5)	C2—C7—C6—C5	0.3 (5)
Cu1ⁱⁱ—O5—C9—C6	−173.7 (2)	C2—C7—C6—C9	−178.4 (3)
C3—C2—C7—C6	−0.1 (5)	O6—C9—C6—C5	4.9 (5)
C1—C2—C7—C6	179.6 (3)	O5—C9—C6—C5	−174.7 (3)
C7—C2—C3—C4	0.0 (5)	O6—C9—C6—C7	−176.3 (3)
C1—C2—C3—C4	−179.7 (3)	O5—C9—C6—C7	4.0 (5)

Symmetry codes: (i) −x+2, y−1/2, −z+3/2; (ii) −x+2, y+1/2, −z+3/2.

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O3—H1···O1ⁱⁱⁱ	0.83 (1)	1.80 (2)	2.570 (4)	153 (4)
O1W—H1WA···O1^iv	0.84 (1)	2.37 (2)	3.077 (4)	142 (3)
O1W—H1WB···O4^v	0.83 (1)	1.97 (1)	2.801 (4)	171 (4)
O2W—H2WA···O6^vi	0.84 (1)	1.91 (2)	2.697 (4)	155 (4)
O2W—H2WB···O3^vii	0.84 (1)	2.07 (2)	2.875 (4)	162 (3)
O3W—H3WA···O6^viii	0.83 (1)	1.92 (2)	2.717 (4)	161 (4)
O3W—H3WB···O2^ix	0.83 (1)	2.33 (2)	3.130 (4)	161 (3)

Symmetry codes: (iii) x−1, y, z−1; (iv) x−1, −y+3/2, z−1/2; (v) −x+1, y−1/2, −z+1/2; (vi) −x+2, −y+2, −z+1; (vii) x+1, −y+3/2, z+1/2; (viii) −x+1, −y+2, −z+1; (ix) x, −y+3/2, z+1/2.

Selected bond lengths (Å) top

Cu1—O2	1.934 (2)	Cu1—O2W	1.987 (3)
Cu1—O5ⁱ	1.917 (2)	Cu1—O3W	1.984 (3)
Cu1—O1W	2.258 (3)

Symmetry code: (i) −x+2, y−1/2, −z+3/2.

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O3—H1···O1ⁱⁱ	0.833 (10)	1.80 (2)	2.570 (4)	153 (4)
O1W—H1WA···O1ⁱⁱⁱ	0.836 (10)	2.37 (2)	3.077 (4)	142 (3)
O1W—H1WB···O4^iv	0.834 (10)	1.974 (11)	2.801 (4)	171 (4)
O2W—H2WA···O6^v	0.840 (10)	1.910 (19)	2.697 (4)	155 (4)
O2W—H2WB···O3^vi	0.838 (10)	2.065 (15)	2.875 (4)	162 (3)
O3W—H3WA···O6^vii	0.832 (10)	1.918 (16)	2.717 (4)	161 (4)
O3W—H3WB···O2^viii	0.832 (10)	2.332 (15)	3.130 (4)	161 (3)

Symmetry codes: (ii) x−1, y, z−1; (iii) x−1, −y+3/2, z−1/2; (iv) −x+1, y−1/2, −z+1/2; (v) −x+2, −y+2, −z+1; (vi) x+1, −y+3/2, z+1/2; (vii) −x+1, −y+2, −z+1; (viii) x, −y+3/2, z+1/2.

Acknowledgements

The authors acknowledge the Air Force Service College for supporting this work.

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