Poly[diaqua-μ-oxalato-μ-pyrazine-2-carbox­yl­ato-lanthanum(III)]

Han, L.; Meng, Q.-H.; Hao, J.-D.; Luo, Y.-F.; Zeng, R.-H.

doi:10.1107/S1600536808041664

metal-organic compounds

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Volume 65| Part 1| January 2009| Page m76

doi:10.1107/S1600536808041664

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Poly[diaqua-μ-oxalato-μ-pyrazine-2-carboxylato-lanthanum(III)]

Lu Han,^a Qun-Hui Meng,^a Jian-Dong Hao,^a Yi-Fan Luo ^a ^* and Rong-Hua Zeng ^a,^b

^aSchool of Chemistry and the Environment, South China Normal University, Guangzhou 510006, People's Republic of China, and ^bKey Laboratory of Technology on Electrochemical Energy Storage and Power Generation in Guangdong Universities, Guangzhou 510631, People's Republic of China
^*Correspondence e-mail: luoyf2004@yahoo.com.cn

(Received 1 December 2008; accepted 9 December 2008; online 17 December 2008)

In the title complex, [La(C₅H₃N₂O₂)(C₂O₄)(H₂O)₂]_n, the La^III ion is coordinated by one N and three O atoms from two pyrazine-2-carboxylate ligands, by four O atoms from two oxalate ligands and by two O atoms of two water molecules, displaying a distorted bicapped square-antiprismatic geometry. The carboxylate groups of pyrazine-2-carboxylate and oxalate ligands link the lanthanum metal centres, forming layers parallel to (10 $[\overline{1}]$ ). The layers are further connected by intermolecular O—H⋯O and N—H⋯O hydrogen-bonding interactions, forming a three-dimensional supramolecular network.

Related literature

For general background, see: Eddaoudi et al. (2001 ); Rizk et al. (2005 ); Zeng et al. (2007 ).

Experimental

Crystal data

[La(C₅H₃N₂O₂)(C₂O₄)(H₂O)₂]
M_r = 386.06
Triclinic, $[P \overline 1]$
a = 8.040 (3) Å
b = 8.7343 (18) Å
c = 8.8329 (18) Å
α = 115.552 (2)°
β = 101.447 (3)°
γ = 95.789 (3)°
V = 536.1 (3) Å³
Z = 2
Mo Kα radiation
μ = 4.02 mm⁻¹
T = 296 (2) K
0.17 × 0.16 × 0.14 mm

Data collection

Bruker APEXII area-detector diffractometer
Absorption correction: multi-scan (APEX2; Bruker, 2004 ) T_min = 0.548, T_max = 0.603 (expected range = 0.518–0.569)
2761 measured reflections
1898 independent reflections
1787 reflections with I > 2σ(I)
R_int = 0.020

Refinement

R[F² > 2σ(F²)] = 0.031
wR(F²) = 0.078
S = 1.07
1898 reflections
163 parameters
6 restraints
H-atom parameters constrained
Δρ_max = 1.61 e Å⁻³
Δρ_min = −1.22 e Å⁻³

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
O1W—H1W⋯N1ⁱ	0.84	1.97	2.796 (6)	170
O2W—H3W⋯O2ⁱⁱ	0.84	1.94	2.737 (5)	157
O1W—H2W⋯O3ⁱⁱⁱ	0.84	2.05	2.874 (5)	167
O2W—H4W⋯O6^iv	0.84	2.09	2.825 (5)	146
Symmetry codes: (i) x, y-1, z-1; (ii) x, y, z-1; (iii) -x+1, -y, -z+1; (iv) -x+1, -y, -z.

Data collection: APEX2 (Bruker, 2004); cell refinement: SAINT (Bruker, 2004); data reduction: SAINT; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008 ); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEPIII (Burnett & Johnson, 1996 ), PLATON (Spek, 2003 ) and SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXL97.

Supporting information

Crystal structure: contains datablocks I, global. DOI: 10.1107/S1600536808041664/dn2413sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536808041664/dn2413Isup2.hkl

checkCIF report

Comment top

The design, synthesis, characterization, and properties of supramolecular networks formed by using functionalized organic molecules as bridges between metal centers are of great interest(Eddaoudi et al., 2001; Rizk et al., 2005; Zeng et al.,2007). As a building block, pyrazine-2-carboxylic acid and oxalic acid are excellent candidates for the construction of supramolecular complexes. Herein, we reported the new coordination polymer, (I).

In (I), each La^III centre is coordinated by seven oxygen atoms and one nitrogen atom from two pyrazine-2-carboxylate ligands, two oxalate ligands and two water molecules (Fig. 1), and represents a distorted bicapped square antiprismatic geometry. The La^III ions are linked by pyrazine-2-carboxylate ligands and oxalate ligands to form layers parallel to the (1 0 -1) plane (Fig.2), and the adjacent La···La separations are 6.570 (4) and 4.506 (5) Å, respectively. O—H···O and N—H···O hydrogen bonds (Table 1), involving the pyrazine-2-carboxylate ligands, coordinating water molecules and oxalate ligands assemble neighboring layers into a three-dimensional supramolecular network motif .

Related literature top

For general background, see: Eddaoudi et al. (2001); Rizk et al. (2005); Zeng et al. (2007).

Experimental top

A mixture of La₂O₃ (0.245 g; 0.75 mmol), pyrazine-2-carboxylic acid (0.186 g; 1.5 mmol), oxalic acid(0.135 g; 1.5 mmol), water (10 mL) in the presence of HNO₃ (0.024 g; 0.385 mmol) was stirred vigorously for 20 min and then sealed in a Teflon-lined stainless-steel autoclave (20 mL, capacity). The autoclave was heated and maintained at 433K for 3 days, and then cooled to room temperature at 5 K h^-1 and obtained the colorless block crystals.

Computing details top

Data collection: APEX2 (Bruker, 2004); cell refinement: SAINT (Bruker, 2004); data reduction: SAINT (Bruker, 2004); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEPIII (Burnett & Johnson, 1996), PLATON (Spek, 2003) and SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Figures top

	Fig. 1. ORTEP view showing the atomic-numbering scheme. Displacement ellipsoids are drawn at the 30% probability level. [Symmetry codes: (i)1-x, 1-y, 1-z; (ii)1-x, -y, -z; (iii)1-x, -y, 1-z]
	Fig. 2. View of the layered network of the title structure.

Poly[diaqua-µ-oxalato-µ-pyrazine-2-carboxylato-lanthanum(III)] top

Crystal data top

[La(C₅H₃N₂O₂)(C₂O₄)(H₂O)₂]	Z = 2
M_r = 386.06	F(000) = 368
Triclinic, P1	D_x = 2.391 Mg m⁻³
Hall symbol: -P 1	Mo Kα radiation, λ = 0.71073 Å
a = 8.040 (3) Å	Cell parameters from 6377 reflections
b = 8.7343 (18) Å	θ = 1.7–28.0°
c = 8.8329 (18) Å	µ = 4.02 mm⁻¹
α = 115.552 (2)°	T = 296 K
β = 101.447 (3)°	Block, colourless
γ = 95.789 (3)°	0.17 × 0.16 × 0.14 mm
V = 536.1 (3) Å³

Data collection top

Bruker APEXII area-detector diffractometer	1898 independent reflections
Radiation source: fine-focus sealed tube	1787 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.020
ϕ and ω scans	θ_max = 25.2°, θ_min = 2.6°
Absorption correction: multi-scan (APEX2; Bruker, 2004)	h = −5→9
T_min = 0.548, T_max = 0.603	k = −10→10
2761 measured reflections	l = −10→10

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)] = 0.031	Hydrogen site location: inferred from neighbouring sites
wR(F²) = 0.078	H-atom parameters constrained
S = 1.07	w = 1/[σ²(F_o²) + (0.0533P)²] where P = (F_o² + 2F_c²)/3
1898 reflections	(Δ/σ)_max < 0.001
163 parameters	Δρ_max = 1.61 e Å⁻³
6 restraints	Δρ_min = −1.22 e Å⁻³

Crystal data top

[La(C₅H₃N₂O₂)(C₂O₄)(H₂O)₂]	γ = 95.789 (3)°
M_r = 386.06	V = 536.1 (3) Å³
Triclinic, P1	Z = 2
a = 8.040 (3) Å	Mo Kα radiation
b = 8.7343 (18) Å	µ = 4.02 mm⁻¹
c = 8.8329 (18) Å	T = 296 K
α = 115.552 (2)°	0.17 × 0.16 × 0.14 mm
β = 101.447 (3)°

Data collection top

Bruker APEXII area-detector diffractometer	1898 independent reflections
Absorption correction: multi-scan (APEX2; Bruker, 2004)	1787 reflections with I > 2σ(I)
T_min = 0.548, T_max = 0.603	R_int = 0.020
2761 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.031	6 restraints
wR(F²) = 0.078	H-atom parameters constrained
S = 1.07	Δρ_max = 1.61 e Å⁻³
1898 reflections	Δρ_min = −1.22 e Å⁻³
163 parameters

Special details top

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.

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 > 2sigma(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
C1	0.2750 (6)	0.2794 (6)	0.7523 (6)	0.0218 (10)
C2	0.2577 (7)	0.4053 (7)	0.9066 (7)	0.0285 (11)
H2	0.3191	0.4122	1.0111	0.034*
C3	0.0676 (7)	0.4974 (7)	0.7564 (7)	0.0304 (12)
H3	−0.0079	0.5706	0.7529	0.036*
C4	0.0849 (7)	0.3716 (7)	0.6014 (7)	0.0289 (11)
H4	0.0202	0.3621	0.4968	0.035*
C5	0.5718 (6)	0.4928 (6)	0.5701 (6)	0.0204 (10)
C6	0.0553 (6)	−0.0282 (6)	−0.0653 (6)	0.0218 (10)
C7	0.3936 (6)	0.1564 (7)	0.7458 (6)	0.0238 (10)
La1	0.33941 (3)	0.08521 (3)	0.32247 (3)	0.01745 (13)
N1	0.1560 (6)	0.5173 (6)	0.9109 (6)	0.0284 (10)
N8	0.1914 (5)	0.2634 (5)	0.5974 (5)	0.0235 (9)
O1	0.3920 (5)	0.0398 (4)	0.5980 (4)	0.0248 (8)
O2	0.4933 (5)	0.1760 (5)	0.8841 (5)	0.0314 (8)
O3	0.5571 (5)	0.3495 (4)	0.5715 (4)	0.0246 (7)
O4	0.6879 (4)	0.6241 (4)	0.6678 (4)	0.0264 (8)
O5	−0.0155 (4)	−0.0721 (5)	−0.2201 (4)	0.0299 (8)
O6	0.2119 (4)	−0.0240 (5)	−0.0025 (4)	0.0292 (8)
O1W	0.2022 (5)	−0.2208 (4)	0.2499 (4)	0.0271 (8)
O2W	0.5746 (5)	0.1980 (5)	0.2097 (4)	0.0300 (8)
H1W	0.1827	−0.2913	0.1441	0.045*
H3W	0.5293	0.2079	0.1210	0.045*
H2W	0.2583	−0.2630	0.3071	0.045*
H4W	0.6611	0.1534	0.1900	0.045*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
C1	0.018 (2)	0.027 (2)	0.021 (2)	0.007 (2)	0.005 (2)	0.012 (2)
C2	0.024 (3)	0.033 (3)	0.026 (3)	0.008 (2)	0.004 (2)	0.012 (2)
C3	0.030 (3)	0.030 (3)	0.033 (3)	0.012 (2)	0.010 (2)	0.014 (2)
C4	0.031 (3)	0.033 (3)	0.029 (3)	0.017 (2)	0.007 (2)	0.018 (2)
C5	0.015 (2)	0.025 (2)	0.021 (2)	0.0083 (19)	0.005 (2)	0.009 (2)
C6	0.017 (2)	0.023 (2)	0.024 (2)	0.0043 (19)	0.004 (2)	0.010 (2)
C7	0.018 (2)	0.033 (3)	0.025 (3)	0.005 (2)	0.007 (2)	0.016 (2)
La1	0.01352 (18)	0.01999 (18)	0.01806 (18)	0.00464 (11)	0.00163 (12)	0.00898 (13)
N1	0.022 (2)	0.027 (2)	0.031 (2)	0.0072 (18)	0.0067 (19)	0.0093 (19)
N8	0.022 (2)	0.027 (2)	0.023 (2)	0.0084 (17)	0.0035 (17)	0.0132 (18)
O1	0.0267 (19)	0.0270 (18)	0.0238 (18)	0.0104 (15)	0.0089 (15)	0.0126 (15)
O2	0.028 (2)	0.046 (2)	0.0244 (18)	0.0153 (17)	0.0049 (16)	0.0195 (17)
O3	0.0261 (19)	0.0225 (17)	0.0260 (18)	0.0047 (14)	0.0020 (15)	0.0143 (15)
O4	0.0196 (18)	0.0241 (18)	0.0316 (19)	0.0037 (15)	−0.0015 (15)	0.0131 (16)
O5	0.0172 (18)	0.049 (2)	0.0197 (18)	0.0097 (16)	0.0016 (15)	0.0132 (16)
O6	0.0147 (18)	0.046 (2)	0.0220 (17)	0.0100 (16)	0.0020 (14)	0.0123 (16)
O1W	0.0262 (19)	0.0239 (18)	0.0263 (18)	0.0031 (15)	0.0015 (15)	0.0100 (15)
O2W	0.027 (2)	0.039 (2)	0.0272 (19)	0.0109 (17)	0.0114 (16)	0.0166 (17)

Geometric parameters (Å, º) top

C1—N8	1.340 (6)	C7—O1	1.259 (6)
C1—C2	1.378 (7)	La1—O1W	2.533 (3)
C1—C7	1.497 (7)	La1—O4ⁱ	2.536 (3)
C2—N1	1.330 (7)	La1—O6	2.544 (3)
C2—H2	0.9300	La1—O3	2.551 (3)
C3—N1	1.336 (7)	La1—O5ⁱⁱ	2.555 (4)
C3—C4	1.382 (7)	La1—O1	2.592 (3)
C3—H3	0.9300	La1—O2W	2.600 (4)
C4—N8	1.332 (7)	La1—O1ⁱⁱⁱ	2.623 (3)
C4—H4	0.9300	La1—N8	2.828 (4)
C5—O4	1.242 (6)	La1—O2ⁱⁱⁱ	2.889 (4)
C5—O3	1.250 (6)	La1—C7ⁱⁱⁱ	3.124 (5)
C5—C5ⁱ	1.574 (9)	O1W—H1W	0.8385
C6—O5	1.239 (6)	O1W—H2W	0.8353
C6—O6	1.261 (6)	O2W—H3W	0.8400
C6—C6ⁱⁱ	1.539 (9)	O2W—H4W	0.8421
C7—O2	1.252 (6)

N8—C1—C2	122.0 (5)	O5ⁱⁱ—La1—O1ⁱⁱⁱ	154.21 (12)
N8—C1—C7	115.3 (4)	O1—La1—O1ⁱⁱⁱ	60.45 (13)
C2—C1—C7	122.7 (5)	O2W—La1—O1ⁱⁱⁱ	75.58 (11)
N1—C2—C1	122.1 (5)	O1W—La1—N8	97.55 (12)
N1—C2—H2	118.9	O4ⁱ—La1—N8	72.37 (12)
C1—C2—H2	118.9	O6—La1—N8	128.17 (11)
N1—C3—C4	122.0 (5)	O3—La1—N8	68.57 (12)
N1—C3—H3	119.0	O5ⁱⁱ—La1—N8	66.44 (11)
C4—C3—H3	119.0	O1—La1—N8	58.52 (11)
N8—C4—C3	121.9 (5)	O2W—La1—N8	131.02 (12)
N8—C4—H4	119.0	O1ⁱⁱⁱ—La1—N8	115.86 (11)
C3—C4—H4	119.0	O1W—La1—O2ⁱⁱⁱ	66.41 (11)
O4—C5—O3	126.6 (4)	O4ⁱ—La1—O2ⁱⁱⁱ	129.49 (11)
O4—C5—C5ⁱ	117.0 (5)	O6—La1—O2ⁱⁱⁱ	68.05 (11)
O3—C5—C5ⁱ	116.4 (5)	O3—La1—O2ⁱⁱⁱ	112.40 (11)
O5—C6—O6	126.1 (4)	O5ⁱⁱ—La1—O2ⁱⁱⁱ	121.50 (11)
O5—C6—C6ⁱⁱ	118.0 (5)	O1—La1—O2ⁱⁱⁱ	99.45 (10)
O6—C6—C6ⁱⁱ	115.9 (5)	O2W—La1—O2ⁱⁱⁱ	65.10 (11)
O2—C7—O1	122.7 (5)	O1ⁱⁱⁱ—La1—O2ⁱⁱⁱ	46.83 (10)
O2—C7—C1	119.7 (5)	N8—La1—O2ⁱⁱⁱ	157.23 (11)
O1—C7—C1	117.5 (4)	O1W—La1—C7ⁱⁱⁱ	68.77 (12)
O2—C7—La1ⁱⁱⁱ	67.6 (3)	O4ⁱ—La1—C7ⁱⁱⁱ	137.10 (12)
O1—C7—La1ⁱⁱⁱ	55.5 (3)	O6—La1—C7ⁱⁱⁱ	91.66 (12)
C1—C7—La1ⁱⁱⁱ	169.0 (3)	O3—La1—C7ⁱⁱⁱ	95.00 (12)
O1W—La1—O4ⁱ	150.33 (12)	O5ⁱⁱ—La1—C7ⁱⁱⁱ	139.98 (12)
O1W—La1—O6	82.82 (11)	O1—La1—C7ⁱⁱⁱ	78.74 (11)
O4ⁱ—La1—O6	82.26 (12)	O2W—La1—C7ⁱⁱⁱ	70.04 (12)
O1W—La1—O3	139.37 (10)	O1ⁱⁱⁱ—La1—C7ⁱⁱⁱ	23.29 (11)
O4ⁱ—La1—O3	63.95 (11)	N8—La1—C7ⁱⁱⁱ	136.97 (12)
O6—La1—O3	136.33 (11)	O2ⁱⁱⁱ—La1—C7ⁱⁱⁱ	23.62 (11)
O1W—La1—O5ⁱⁱ	77.14 (12)	C2—N1—C3	116.0 (5)
O4ⁱ—La1—O5ⁱⁱ	73.25 (12)	C4—N8—C1	115.9 (4)
O6—La1—O5ⁱⁱ	63.19 (11)	C4—N8—La1	126.3 (3)
O3—La1—O5ⁱⁱ	124.85 (11)	C1—N8—La1	114.8 (3)
O1W—La1—O1	68.69 (11)	C7—O1—La1	122.9 (3)
O4ⁱ—La1—O1	122.78 (11)	C7—O1—La1ⁱⁱⁱ	101.2 (3)
O6—La1—O1	151.51 (12)	La1—O1—La1ⁱⁱⁱ	119.55 (13)
O3—La1—O1	71.74 (11)	C7—O2—La1ⁱⁱⁱ	88.8 (3)
O5ⁱⁱ—La1—O1	108.11 (11)	C5—O3—La1	120.1 (3)
O1W—La1—O2W	130.93 (11)	C5—O4—La1ⁱ	120.6 (3)
O4ⁱ—La1—O2W	67.57 (12)	C6—O5—La1ⁱⁱ	121.0 (3)
O6—La1—O2W	72.86 (11)	C6—O6—La1	121.8 (3)
O3—La1—O2W	69.18 (11)	La1—O1W—H1W	113.1
O5ⁱⁱ—La1—O2W	123.82 (12)	La1—O1W—H2W	115.0
O1—La1—O2W	126.58 (11)	H1W—O1W—H2W	107.3
O1W—La1—O1ⁱⁱⁱ	77.11 (11)	La1—O2W—H3W	111.1
O4ⁱ—La1—O1ⁱⁱⁱ	132.53 (11)	La1—O2W—H4W	123.8
O6—La1—O1ⁱⁱⁱ	114.65 (11)	H3W—O2W—H4W	106.5
O3—La1—O1ⁱⁱⁱ	75.79 (11)

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

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O1W—H1W···N1^iv	0.84	1.97	2.796 (6)	170
O2W—H3W···O2^v	0.84	1.94	2.737 (5)	157
O1W—H2W···O3ⁱⁱⁱ	0.84	2.05	2.874 (5)	167
O2W—H4W···O6^vi	0.84	2.09	2.825 (5)	146

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

Experimental details

Crystal data
Chemical formula	[La(C₅H₃N₂O₂)(C₂O₄)(H₂O)₂]
M_r	386.06
Crystal system, space group	Triclinic, P1
Temperature (K)	296
a, b, c (Å)	8.040 (3), 8.7343 (18), 8.8329 (18)
α, β, γ (°)	115.552 (2), 101.447 (3), 95.789 (3)
V (Å³)	536.1 (3)
Z	2
Radiation type	Mo Kα
µ (mm⁻¹)	4.02
Crystal size (mm)	0.17 × 0.16 × 0.14

Data collection
Diffractometer	Bruker APEXII area-detector diffractometer
Absorption correction	Multi-scan (APEX2; Bruker, 2004)
T_min, T_max	0.548, 0.603
No. of measured, independent and observed [I > 2σ(I)] reflections	2761, 1898, 1787
R_int	0.020
(sin θ/λ)_max (Å⁻¹)	0.599

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.031, 0.078, 1.07
No. of reflections	1898
No. of parameters	163
No. of restraints	6
H-atom treatment	H-atom parameters constrained
Δρ_max, Δρ_min (e Å⁻³)	1.61, −1.22

Computer programs: APEX2 (Bruker, 2004), SAINT (Bruker, 2004), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), ORTEPIII (Burnett & Johnson, 1996), PLATON (Spek, 2003) and SHELXTL (Sheldrick, 2008).

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O1W—H1W···N1ⁱ	0.84	1.97	2.796 (6)	170.0
O2W—H3W···O2ⁱⁱ	0.84	1.94	2.737 (5)	156.8
O1W—H2W···O3ⁱⁱⁱ	0.84	2.05	2.874 (5)	166.6
O2W—H4W···O6^iv	0.84	2.09	2.825 (5)	146.1

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

Acknowledgements

The authors acknowledge South China Normal University for supporting this work.

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