catena-Poly[[tetra­aqua­zinc(II)]-μ-1,3,4-thia­diazol-2,5-diyldithio­diacetato-κ2O:O′]

Yu, Y.-H.; He, C.; Hou, G.-F.; Gao, J.-S.; Zhang, H.-K.

doi:10.1107/S1600536808013251

metal-organic compounds

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ISSN: 2056-9890

Volume 64| Part 6| June 2008| Page m794

doi:10.1107/S1600536808013251

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catena-Poly[[tetraaquazinc(II)]-μ-1,3,4-thiadiazol-2,5-diyldithiodiacetato-κ²O:O′]

Ying-Hui Yu,^a Chuan He,^b Guang-Feng Hou,^a Jin-Sheng Gao ^a ^* and Hong-Kun Zhang ^c

^aCollege of Chemistry and Materials Science, Heilongjiang University, Harbin 150080, People's Republic of China, ^bSchool of Resources and Safety Engineering, China University of Mining and Techology (Beijing Campus), Beijing 100083, People's Republic of China, and ^cDepartment of Food and Environmental Engineering, Heilongjiang East College, Harbin 150086, People's Republic of China
^*Correspondence e-mail: hgf1000@163.com

(Received 12 April 2008; accepted 5 May 2008; online 10 May 2008)

In the title linear coordination polymer, [Zn(C₆H₄N₂O₄S₃)(H₂O)₄]_n, the Zn^II atom is coordinated by four O atoms from four water molecules and two O atoms from two [5-(carboxylatomethylsulfanyl)-1,3,4-thiadiazol-2-ylsulfanyl]acetate units in an octahedral coordination environment. The chains are linked into a three-dimensional supramolecular network via O—H⋯O and O—H⋯N hydrogen bonds.

Related literature

For the structure of other metal 1,3,4-thiadiazolyl-2,5-dithioacetates, see Gao et al. (2005 , 2006 ); Zhang et al. (2006 ).

Experimental

Crystal data

[Zn(C₆H₄N₂O₄S₃)(H₂O)₄]
M_r = 401.73
Monoclinic, P 2₁
a = 5.1554 (10) Å
b = 9.5043 (19) Å
c = 13.627 (3) Å
β = 94.82 (3)°
V = 665.3 (2) Å³
Z = 2
Mo Kα radiation
μ = 2.35 mm⁻¹
T = 291 (2) K
0.42 × 0.18 × 0.18 mm

Data collection

Rigaku R-AXIS RAPID diffractometer
Absorption correction: multi-scan (ABSCOR; Higashi, 1995 ) T_min = 0.439, T_max = 0.675
6408 measured reflections
2774 independent reflections
2624 reflections with I > 2σ(I)
R_int = 0.022

Refinement

R[F² > 2σ(F²)] = 0.021
wR(F²) = 0.046
S = 1.06
2774 reflections
181 parameters
1 restraint
H-atom parameters constrained
Δρ_max = 0.26 e Å⁻³
Δρ_min = −0.33 e Å⁻³
Absolute structure: Flack (1983 ), 1151 Friedel pairs
Flack parameter: 0.014 (8)

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
O5—H6⋯O3	0.85	2.53	2.971 (3)	113
O5—H6⋯O6ⁱ	0.85	2.23	3.053 (3)	165
O5—H5⋯N2ⁱ	0.85	2.05	2.897 (3)	172
O6—H8⋯O4ⁱⁱ	0.85	2.02	2.762 (2)	145
O6—H7⋯O7ⁱⁱ	0.85	2.36	3.116 (3)	148
O7—H10⋯O3ⁱⁱⁱ	0.85	1.94	2.770 (3)	166
O7—H9⋯O1^iv	0.85	2.61	2.992 (2)	109
O7—H9⋯O2^iv	0.85	1.85	2.680 (3)	166
O8—H12⋯N1ⁱ	0.85	1.98	2.819 (3)	172
O8—H11⋯O1^v	0.85	1.87	2.713 (2)	175
Symmetry codes: (i) $[-x, y-{\script{1\over 2}}, -z+1]$ ; (ii) x-1, y, z; (iii) $[-x, y+{\script{1\over 2}}, -z+1]$ ; (iv) x, y, z-1; (v) x+1, y, z-1.

Data collection: RAPID-AUTO (Rigaku, 1998 ); cell refinement: RAPID-AUTO; data reduction: CrystalStructure (Rigaku/MSC, 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: SHELXL97.

Supporting information

Crystal structure: contains datablocks global, I. DOI: 10.1107/S1600536808013251/ng2446sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536808013251/ng2446Isup2.hkl

checkCIF report

Comment top

1,3,4-Thiadiazolyl-2,5-dithioacetic acid is a multidentate flexible aromatic carboxylic acid having two —S—CH₂CO₂H arms, and its N atoms can be considered as a potential coordinate candidate which also would coordinate with metal atoms fomed a supramolecular complexes. The structure of 1,3,4-Thiadiazolyl-2,5-dithioacetic acid was reported by Gao et al. (2005) and Zhang et al., (2006). In this paper, we reporte a new one-dimensional title compound crystal structure, synthesized by the reaction of 1,3,4-Thiadiazolyl-2,5-dithioacetic acid and zinc dichloride in aqueous solution.

Complex (I) consists of molecules of tetraaquazinc(II)-1,3,4-thiadiazol-2,5-diyldithiodiacetato. The zinc atom is six-coordinated in an octahedron environment (Figure 1), each zinc atom connect with two 1,3,4-Thiadiazolyl-2,5-dithioacetic acid ligand and four water molecules formed a one-dimensional chain structure along c axis (Figure 2).

There are eight symmetry-independent 'active' H atoms in the crystal structure; all of them participate in hydrogen bonds, which link the one-dimensional chain structure into an infinite three-dimensional network (Table 1, Figure 3).

Related literature top

For the structure of other metal 1,3,4-thiadiazolyl-2,5-dithioacetates, see Gao et al. (2005, 2006); Zhang et al. (2006).

Experimental top

1,3,4-Thiadiazolyl-2,5-dithioacetic acid was prepared from 2,5-dimercapto-1,3,4-thiadiazole, using the method for synthesis of benzene-1,2-dioxyacetic acid reported by us (Gao et al., 2006). The colorless zinc complex was obtained from the reaction of zinc dichloride hexahydrate (0.244 g, 1 mmol) and 1,3,4-Thiadiazolyl-2,5-dithioacetic acid (0.532 g, 2 mmol) in hot water (20 ml), and then the pH was adjusted to about 6 with 0.2 M sodium hydroxide. The resulting solution was filtered and allowed to stand in a desiccator at room temperature for several days.

Computing details top

Data collection: RAPID-AUTO (Rigaku, 1998); cell refinement: RAPID-AUTO (Rigaku, 1998); data reduction: CrystalStructure (Rigaku/MSC, 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: SHELXL97 (Sheldrick, 2008).

Figures top

	Fig. 1. The molecular structure of (I), showing displacement ellipsoids at the 30% probability level for non-H atoms.
	Fig. 2. One-dimensional chain structure of the title complex. H atoms have been omitted for clarity.
	Fig. 3. A partial packing view, showing the three-dimensional hydrogen-bonding network. Dashed lines indicate the hydrogen-bonding interactions. H atoms not involved in hydrogen bonds have been omitted for clarity.

catena-Poly[[tetraaquazinc(II)]-µ-1,3,4-thiadiazol-2,5-diyldithiodiacetato-κ²O:O'] top

Crystal data top

[Zn(C₆H₄N₂O₄S₃)(H₂O)₄]	F(000) = 408
M_r = 401.73	D_x = 2.005 Mg m⁻³
Monoclinic, P2₁	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: P 2yb	Cell parameters from 6179 reflections
a = 5.1554 (10) Å	θ = 3.0–27.5°
b = 9.5043 (19) Å	µ = 2.35 mm⁻¹
c = 13.627 (3) Å	T = 291 K
β = 94.82 (3)°	Block, colorless
V = 665.3 (2) Å³	0.42 × 0.18 × 0.18 mm
Z = 2

Data collection top

Rigaku R-AXIS RAPID diffractometer	2774 independent reflections
Radiation source: fine-focus sealed tube	2624 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.022
ω scans	θ_max = 27.5°, θ_min = 3.0°
Absorption correction: multi-scan (ABSCOR; Higashi, 1995)	h = −6→6
T_min = 0.439, T_max = 0.675	k = −11→12
6408 measured reflections	l = −17→17

Refinement top

Refinement on F²	Secondary atom site location: difference Fourier map
Least-squares matrix: full	Hydrogen site location: inferred from neighbouring sites
R[F² > 2σ(F²)] = 0.021	H-atom parameters constrained
wR(F²) = 0.046	w = 1/[σ²(F_o²) + (0.0189P)²] where P = (F_o² + 2F_c²)/3
S = 1.06	(Δ/σ)_max = 0.001
2774 reflections	Δρ_max = 0.26 e Å⁻³
181 parameters	Δρ_min = −0.33 e Å⁻³
1 restraint	Absolute structure: Flack (1983), 1151 Friedel pairs
Primary atom site location: structure-invariant direct methods	Absolute structure parameter: 0.014 (8)

Crystal data top

[Zn(C₆H₄N₂O₄S₃)(H₂O)₄]	V = 665.3 (2) Å³
M_r = 401.73	Z = 2
Monoclinic, P2₁	Mo Kα radiation
a = 5.1554 (10) Å	µ = 2.35 mm⁻¹
b = 9.5043 (19) Å	T = 291 K
c = 13.627 (3) Å	0.42 × 0.18 × 0.18 mm
β = 94.82 (3)°

Data collection top

Rigaku R-AXIS RAPID diffractometer	2774 independent reflections
Absorption correction: multi-scan (ABSCOR; Higashi, 1995)	2624 reflections with I > 2σ(I)
T_min = 0.439, T_max = 0.675	R_int = 0.022
6408 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.021	H-atom parameters constrained
wR(F²) = 0.046	Δρ_max = 0.26 e Å⁻³
S = 1.06	Δρ_min = −0.33 e Å⁻³
2774 reflections	Absolute structure: Flack (1983), 1151 Friedel pairs
181 parameters	Absolute structure parameter: 0.014 (8)
1 restraint

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
C1	−0.1447 (4)	0.4252 (3)	1.17156 (18)	0.0201 (5)
C2	−0.3790 (4)	0.4114 (3)	1.09577 (18)	0.0218 (5)
H1	−0.5359	0.4172	1.1301	0.026*
H2	−0.3749	0.3184	1.0665	0.026*
C3	−0.1709 (4)	0.4813 (2)	0.92174 (18)	0.0199 (5)
C4	0.1357 (4)	0.3689 (2)	0.82727 (18)	0.0213 (5)
C5	0.4372 (4)	0.3491 (3)	0.67199 (16)	0.0211 (4)
H3	0.4337	0.4499	0.6833	0.025*
H4	0.6113	0.3248	0.6558	0.025*
C6	0.2476 (4)	0.3147 (2)	0.58475 (18)	0.0207 (5)
N1	−0.1311 (4)	0.5509 (2)	0.84268 (16)	0.0278 (5)
N2	0.0486 (4)	0.4848 (2)	0.78684 (16)	0.0280 (5)
O1	−0.1686 (3)	0.3552 (2)	1.24944 (12)	0.0262 (4)
O2	0.0419 (3)	0.5006 (2)	1.15424 (13)	0.0272 (4)
O3	0.0336 (3)	0.2622 (2)	0.59581 (14)	0.0328 (4)
O4	0.3282 (3)	0.3463 (3)	0.50187 (11)	0.0275 (3)
O5	−0.0626 (3)	0.14243 (19)	0.39464 (14)	0.0290 (4)
H5	−0.0598	0.0891	0.3447	0.044*
H6	0.0049	0.1028	0.4466	0.044*
O6	−0.2103 (3)	0.45593 (19)	0.44603 (14)	0.0279 (4)
H7	−0.3137	0.5040	0.4074	0.042*
H8	−0.3032	0.3988	0.4759	0.042*
O7	0.2564 (3)	0.54165 (19)	0.33802 (14)	0.0274 (4)
H9	0.2034	0.5402	0.2772	0.041*
H10	0.1894	0.6112	0.3658	0.041*
O8	0.3690 (3)	0.2453 (2)	0.29289 (14)	0.0291 (4)
H11	0.5185	0.2761	0.2817	0.044*
H12	0.3081	0.1896	0.2479	0.044*
S1	−0.40220 (11)	0.53897 (6)	0.99806 (5)	0.02380 (13)
S2	0.00595 (10)	0.32704 (7)	0.93689 (4)	0.02359 (12)
S3	0.37405 (12)	0.26074 (7)	0.78411 (5)	0.02603 (14)
Zn1	0.09545 (4)	0.34105 (3)	0.374965 (19)	0.02112 (7)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
C1	0.0183 (10)	0.0232 (12)	0.0194 (14)	0.0027 (9)	0.0046 (9)	−0.0024 (10)
C2	0.0213 (10)	0.0259 (12)	0.0183 (13)	−0.0012 (9)	0.0025 (9)	0.0012 (10)
C3	0.0216 (11)	0.0202 (12)	0.0177 (13)	0.0025 (9)	0.0008 (8)	−0.0002 (9)
C4	0.0243 (10)	0.0253 (14)	0.0143 (12)	0.0034 (9)	0.0026 (8)	0.0016 (9)
C5	0.0212 (9)	0.0246 (11)	0.0180 (11)	0.0037 (11)	0.0039 (8)	0.0002 (12)
C6	0.0219 (10)	0.0218 (13)	0.0187 (12)	0.0012 (9)	0.0043 (8)	−0.0012 (9)
N1	0.0349 (11)	0.0273 (11)	0.0222 (12)	0.0108 (9)	0.0086 (9)	0.0074 (9)
N2	0.0352 (11)	0.0306 (12)	0.0193 (12)	0.0100 (9)	0.0089 (9)	0.0079 (9)
O1	0.0229 (7)	0.0355 (10)	0.0199 (9)	−0.0069 (8)	−0.0007 (6)	0.0066 (9)
O2	0.0226 (8)	0.0360 (10)	0.0233 (10)	−0.0072 (7)	0.0047 (7)	0.0047 (8)
O3	0.0296 (9)	0.0432 (11)	0.0261 (11)	−0.0132 (8)	0.0053 (8)	0.0036 (9)
O4	0.0249 (7)	0.0422 (9)	0.0159 (8)	−0.0072 (9)	0.0044 (6)	−0.0006 (10)
O5	0.0394 (10)	0.0293 (9)	0.0182 (10)	−0.0110 (8)	0.0016 (7)	−0.0012 (8)
O6	0.0249 (8)	0.0308 (10)	0.0291 (11)	−0.0037 (7)	0.0088 (7)	0.0034 (8)
O7	0.0337 (9)	0.0244 (9)	0.0250 (10)	−0.0053 (7)	0.0079 (7)	−0.0032 (8)
O8	0.0223 (8)	0.0364 (11)	0.0296 (11)	−0.0093 (7)	0.0075 (7)	−0.0132 (8)
S1	0.0246 (3)	0.0267 (3)	0.0207 (3)	0.0070 (2)	0.0053 (2)	0.0032 (2)
S2	0.0284 (2)	0.0245 (3)	0.0185 (3)	0.0081 (3)	0.0058 (2)	0.0065 (3)
S3	0.0302 (3)	0.0303 (3)	0.0183 (3)	0.0106 (3)	0.0058 (2)	0.0043 (3)
Zn1	0.02055 (11)	0.02492 (14)	0.01799 (14)	−0.00523 (11)	0.00214 (9)	−0.00268 (13)

Geometric parameters (Å, º) top

C1—O2	1.238 (3)	C6—O4	1.272 (3)
C1—O1	1.267 (3)	N1—N2	1.397 (3)
C1—C2	1.527 (3)	O1—Zn1ⁱ	2.0989 (17)
C2—S1	1.797 (2)	O4—Zn1	2.0209 (17)
C2—H1	0.9700	O5—Zn1	2.0824 (18)
C2—H2	0.9700	O5—H5	0.8500
C3—N1	1.295 (3)	O5—H6	0.8500
C3—S2	1.730 (2)	O6—Zn1	2.2072 (17)
C3—S1	1.736 (2)	O6—H7	0.8500
C4—N2	1.295 (3)	O6—H8	0.8500
C4—S2	1.734 (2)	O7—Zn1	2.1557 (18)
C4—S3	1.742 (2)	O7—H9	0.8501
C5—C6	1.510 (3)	O7—H10	0.8500
C5—S3	1.797 (2)	O8—Zn1	2.0815 (17)
C5—H3	0.9700	O8—H11	0.8500
C5—H4	0.9700	O8—H12	0.8500
C6—O3	1.231 (3)	Zn1—O1ⁱⁱ	2.0989 (17)

O2—C1—O1	126.5 (2)	Zn1—O5—H6	111.8
O2—C1—C2	120.2 (2)	H5—O5—H6	111.7
O1—C1—C2	113.2 (2)	Zn1—O6—H7	115.4
C1—C2—S1	116.27 (17)	Zn1—O6—H8	110.2
C1—C2—H1	108.2	H7—O6—H8	106.9
S1—C2—H1	108.2	Zn1—O7—H9	96.6
C1—C2—H2	108.2	Zn1—O7—H10	113.9
S1—C2—H2	108.2	H9—O7—H10	109.7
H1—C2—H2	107.4	Zn1—O8—H11	127.7
N1—C3—S2	114.44 (18)	Zn1—O8—H12	115.7
N1—C3—S1	120.11 (17)	H11—O8—H12	111.7
S2—C3—S1	125.31 (14)	C3—S1—C2	102.97 (11)
N2—C4—S2	114.60 (17)	C3—S2—C4	86.59 (11)
N2—C4—S3	125.98 (19)	C4—S3—C5	101.23 (12)
S2—C4—S3	119.35 (13)	O4—Zn1—O8	95.17 (7)
C6—C5—S3	114.60 (17)	O4—Zn1—O5	97.06 (8)
C6—C5—H3	108.6	O8—Zn1—O5	87.88 (7)
S3—C5—H3	108.6	O4—Zn1—O1ⁱⁱ	173.48 (9)
C6—C5—H4	108.6	O8—Zn1—O1ⁱⁱ	90.71 (7)
S3—C5—H4	108.6	O5—Zn1—O1ⁱⁱ	85.94 (7)
H3—C5—H4	107.6	O4—Zn1—O7	88.02 (8)
O3—C6—O4	124.6 (2)	O8—Zn1—O7	88.26 (7)
O3—C6—C5	121.2 (2)	O5—Zn1—O7	173.88 (8)
O4—C6—C5	114.17 (19)	O1ⁱⁱ—Zn1—O7	89.36 (7)
C3—N1—N2	112.4 (2)	O4—Zn1—O6	90.40 (7)
C4—N2—N1	111.9 (2)	O8—Zn1—O6	173.27 (8)
C1—O1—Zn1ⁱ	127.90 (15)	O5—Zn1—O6	95.18 (7)
C6—O4—Zn1	122.62 (14)	O1ⁱⁱ—Zn1—O6	83.55 (7)
Zn1—O5—H5	113.8	O7—Zn1—O6	88.17 (6)

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

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O5—H6···O3	0.85	2.53	2.971 (3)	113
O5—H6···O6ⁱⁱⁱ	0.85	2.23	3.053 (3)	165
O5—H5···N2ⁱⁱⁱ	0.85	2.05	2.897 (3)	172
O6—H8···O4^iv	0.85	2.02	2.762 (2)	145
O6—H7···O7^iv	0.85	2.36	3.116 (3)	148
O7—H10···O3^v	0.85	1.94	2.770 (3)	166
O7—H9···O1ⁱⁱ	0.85	2.61	2.992 (2)	109
O7—H9···O2ⁱⁱ	0.85	1.85	2.680 (3)	166
O8—H12···N1ⁱⁱⁱ	0.85	1.98	2.819 (3)	172
O8—H11···O1^vi	0.85	1.87	2.713 (2)	175

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

Experimental details

Crystal data
Chemical formula	[Zn(C₆H₄N₂O₄S₃)(H₂O)₄]
M_r	401.73
Crystal system, space group	Monoclinic, P2₁
Temperature (K)	291
a, b, c (Å)	5.1554 (10), 9.5043 (19), 13.627 (3)
β (°)	94.82 (3)
V (Å³)	665.3 (2)
Z	2
Radiation type	Mo Kα
µ (mm⁻¹)	2.35
Crystal size (mm)	0.42 × 0.18 × 0.18

Data collection
Diffractometer	Rigaku R-AXIS RAPID diffractometer
Absorption correction	Multi-scan (ABSCOR; Higashi, 1995)
T_min, T_max	0.439, 0.675
No. of measured, independent and observed [I > 2σ(I)] reflections	6408, 2774, 2624
R_int	0.022
(sin θ/λ)_max (Å⁻¹)	0.649

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.021, 0.046, 1.06
No. of reflections	2774
No. of parameters	181
No. of restraints	1
H-atom treatment	H-atom parameters constrained
Δρ_max, Δρ_min (e Å⁻³)	0.26, −0.33
Absolute structure	Flack (1983), 1151 Friedel pairs
Absolute structure parameter	0.014 (8)

Computer programs: RAPID-AUTO (Rigaku, 1998), CrystalStructure (Rigaku/MSC, 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
O5—H6···O3	0.85	2.53	2.971 (3)	113.3
O5—H6···O6ⁱ	0.85	2.23	3.053 (3)	164.7
O5—H5···N2ⁱ	0.85	2.05	2.897 (3)	172.3
O6—H8···O4ⁱⁱ	0.85	2.02	2.762 (2)	144.7
O6—H7···O7ⁱⁱ	0.85	2.36	3.116 (3)	147.9
O7—H10···O3ⁱⁱⁱ	0.85	1.94	2.770 (3)	165.7
O7—H9···O1^iv	0.85	2.61	2.992 (2)	109.0
O7—H9···O2^iv	0.85	1.85	2.680 (3)	166.2
O8—H12···N1ⁱ	0.85	1.98	2.819 (3)	171.6
O8—H11···O1^v	0.85	1.87	2.713 (2)	174.9

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

Acknowledgements

The authors thank Heilongjiang University for supporting this study.

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