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Journal logoCRYSTALLOGRAPHIC
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ISSN: 2056-9890
Volume 70| Part 12| December 2014| Pages o1290-o1291

Crystal structure of bis­­(1-benzyl-1H-1,2,4-triazole) perchloric acid monosolvate

aLaboratory of Medicinal Chemistry, Lvliang University, Lvliang, Shanxi 033001, People's Republic of China, and bDepartment of Chemistry and Chemical Engineering, Lvliang University, Lvliang, Shanxi 033001, People's Republic of China
*Correspondence e-mail: qinyq2003@163.com

Edited by V. Rybakov, Moscow State University, Russia (Received 23 March 2014; accepted 12 November 2014; online 26 November 2014)

The title compound, 2C9H9N3·HClO4, was prepared by reaction of 1-benzyl-1H-1,2,4-triazole and HClO4 in ethanol at room temperature. The asymmetric unit consists of two mol­ecules of 1-benzyl-1H-1,2,4-triazole and one of HClO4 mol­ecule. The benzene and triazole rings make dihedral angles of 85.45 (8) and 84.76 (8)° in the two mol­ecules. The H-atom position of the perchloric acid mol­ecule is split over two O atoms (real peaks on difference map), with site-occupation factors of 0.5. These H atoms form two classical hydrogen bonds [2.546 (5) and 2.620 (4) Å] with the same N atoms in both mol­ecules. Five inter­molecular non-classical C—H⋯O inter­actions, with C⋯O distances in the range 3.147 (5)–3.483 (5) Å, are found in the crystal structure.

1. Related literature

For the anti­viral activity of triazole derivatives, see: Madan & Taneja (1991[Madan, V. K. & Taneja, A. D. (1991). J. Indian Chem. Soc. 68, 162-163.]); Borisova et al. (2007[Borisova, N. E., Reshetova, M. D. & Ustynyuk, Yu. A. (2007). Chem. Rev. 107, 46-79.]) and of polyligand complexes with metals, see: Xu et al. (2004[Xu, L.-Zh., Jian, F.-F., Qin, Y.-Q., Yu, G.-P. & Jiao, K. (2004). Chem. Res. Chin. Univ. 20, 305-307.]). For a related structure, see: Ji et al. (2002[Ji, B.-M., Du, C.-X., Zhu, Y. & Wang, Y. (2002). Chin. J. Struct. Chem. 21, 252-255.]).

[Scheme 1]

2. Experimental

2.1. Crystal data

  • 2C9H9N3·HClO4

  • Mr = 418.84

  • Monoclinic, P 21

  • a = 5.7173 (10) Å

  • b = 7.7671 (12) Å

  • c = 21.955 (4) Å

  • β = 95.136 (4)°

  • V = 971.0 (3) Å3

  • Z = 2

  • Mo Kα radiation

  • μ = 0.24 mm−1

  • T = 296 K

  • 0.25 × 0.23 × 0.20 mm

2.2. Data collection

  • Bruker SMART CCD diffractometer

  • 9373 measured reflections

  • 3981 independent reflections

  • 2680 reflections with I > 2σ(I)

  • Rint = 0.050

2.3. Refinement

  • R[F2 > 2σ(F2)] = 0.053

  • wR(F2) = 0.137

  • S = 1.05

  • 3981 reflections

  • 263 parameters

  • 1 restraint

  • H-atom parameters constrained

  • Δρmax = 0.35 e Å−3

  • Δρmin = −0.39 e Å−3

  • Absolute structure: Flack (1983[Flack, H. D. (1983). Acta Cryst. A39, 876-881.]), 873 Friedel pairs

  • Absolute structure parameter: 0.27 (9)

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A D—H H⋯A DA D—H⋯A
O3—H3A⋯N6i 0.82 2.04 2.620 (4) 127
O1—H1⋯N3ii 0.82 1.78 2.546 (5) 154
C10—H10B⋯O4 0.97 2.35 3.282 (5) 160
C9—H9⋯O2iii 0.93 2.52 3.352 (6) 150
C1—H1A⋯O2iv 0.97 2.56 3.483 (5) 158
C17—H17⋯O4v 0.93 2.43 3.147 (5) 134
C18—H18⋯O3vi 0.93 2.44 3.185 (5) 137
Symmetry codes: (i) x-1, y-1, z; (ii) [-x+1, y+{\script{1\over 2}}, -z+1]; (iii) [-x, y-{\script{1\over 2}}, -z+1]; (iv) [-x, y+{\script{1\over 2}}, -z+1]; (v) x+1, y, z; (vi) x, y+1, z.

Data collection: SMART (Bruker, 1997[Bruker (1997). SMART and SAINT. Bruker AXS Inc., Madison, Wisconsin, USA.]); cell refinement: SAINT (Bruker, 1997[Bruker (1997). 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

It is well known, that triazole derivatives possess anti-viral activities (Madan & Taneja, 1991; Borisova et al., 2007). A materials about anti-viral activities of the such polyligand complexes with metals were published Xu et al., 2004. These facts were of a reason for structural study of 1-benzyl-1H-1,2,4-triazole HClO4 solvate.

In the molecule of the title compound (Fig. 1), bond lengths and angles are generally normal in benzene and 1,2,4-triazole ring (Ji et al., 2002). The torsion angles of C2—C1—N1—C8 and N1—C1—C2—C3 are -108.2 (4)° and 60.3 (5)°, respectively; C11—C10—N4—C17 and N4—C10—C11—C12 are 115.2 (4)° and 124.3 (4)° respectively. The dihedral angle formed by the benzene and the triazole ring is 85.45 (8)° and 84.76 (8)° for first and second molecules respectively.

There are some classical O—H···N and non-classical C—H···O intermolecular interactions stabilize the title crystal structure (Fig. 2).

Related literature top

For the antiviral activity of triazole derivatives, see: Madan & Taneja (1991); Borisova et al. (2007) and of polyligand complexes with metals, see: Xu et al. (2004). For a related structure, see: Ji et al. (2002).

Experimental top

The title compound was prepared by reaction of benzyl chloride (0.05 mol), triazole (0.05 mol), potassium carbonate (0.06 mol) and potassium iodide (0.5 g) in the acetone solution (40 ml) at 333 K for 8 h, filtering, evaporating the solvent, affording the tile compound (6.2 g, yield 78%) by solidification in the etanol solution of HClO4 (5%, 15 ml). In the reaction, potassium carbonate can powerfully adsorb HCl to promote the reaction and potassium iodide is catalyst. Single crystals of the title compound suitable for X-ray measurements was obtained by recrystallization from ethanol/acetone (v/v = 1:1) at room temperature.

Refinement top

The H atoms were fixed geometrically and allowed to ride on their attached atoms, with C—H distances = 0.93–0.97 Å; O—H = 0.82 Å and Uiso(H) = 1.2–1.5Ueq(C, O).

Computing details top

Data collection: SMART (Bruker, 1997); cell refinement: SAINT (Bruker, 1997); data reduction: SAINT (Bruker, 1997); 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. The molecular structure of the title compound with the atom numbering scheme. The displacement ellipsoids are drawn at the 40% probability level H atoms are presented as a small spheres of arbitrary radius.

Fig. 2. The packing of the title compound, viewed down the b axis, showing short contact (dashed lines).
Bis(1-benzyl-1H-1,2,4-triazole) perchloric acid monosolvate top
Crystal data top
2C9H9N3·HClO4F(000) = 436
Mr = 418.84Dx = 1.433 Mg m3
Monoclinic, P21Mo Kα radiation, λ = 0.71073 Å
a = 5.7173 (10) ÅCell parameters from 637 reflections
b = 7.7671 (12) Åθ = 3.2–19.6°
c = 21.955 (4) ŵ = 0.24 mm1
β = 95.136 (4)°T = 296 K
V = 971.0 (3) Å3Rectangle, colourless
Z = 20.25 × 0.23 × 0.20 mm
Data collection top
Bruker SMART CCD
diffractometer
2680 reflections with I > 2σ(I)
Radiation source: fine-focus sealed tubeRint = 0.050
Graphite monochromatorθmax = 28.1°, θmin = 2.8°
ϕ and ω scansh = 77
9373 measured reflectionsk = 99
3981 independent reflectionsl = 2828
Refinement top
Refinement on F2Hydrogen site location: mixed
Least-squares matrix: fullH-atom parameters constrained
R[F2 > 2σ(F2)] = 0.053 w = 1/[σ2(Fo2) + (0.0604P)2 + 0.021P]
where P = (Fo2 + 2Fc2)/3
wR(F2) = 0.137(Δ/σ)max < 0.001
S = 1.05Δρmax = 0.35 e Å3
3981 reflectionsΔρmin = 0.39 e Å3
263 parametersExtinction correction: SHELXL97 (Sheldrick, 2008), Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4
1 restraintExtinction coefficient: 0.007 (2)
Primary atom site location: structure-invariant direct methodsAbsolute structure: Flack (1983) parameter determined using 873 quotients
Secondary atom site location: difference Fourier mapAbsolute structure parameter: 0.27 (9)
Crystal data top
2C9H9N3·HClO4V = 971.0 (3) Å3
Mr = 418.84Z = 2
Monoclinic, P21Mo Kα radiation
a = 5.7173 (10) ŵ = 0.24 mm1
b = 7.7671 (12) ÅT = 296 K
c = 21.955 (4) Å0.25 × 0.23 × 0.20 mm
β = 95.136 (4)°
Data collection top
Bruker SMART CCD
diffractometer
2680 reflections with I > 2σ(I)
9373 measured reflectionsRint = 0.050
3981 independent reflections
Refinement top
R[F2 > 2σ(F2)] = 0.053H-atom parameters constrained
wR(F2) = 0.137Δρmax = 0.35 e Å3
S = 1.05Δρmin = 0.39 e Å3
3981 reflectionsAbsolute structure: Flack (1983) parameter determined using 873 quotients
263 parametersAbsolute structure parameter: 0.27 (9)
1 restraint
Special details top

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

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/UeqOcc. (<1)
Cl10.22466 (15)0.25879 (13)0.75737 (4)0.0376 (3)
O20.0268 (4)0.2501 (5)0.75956 (13)0.0501 (7)
O30.3257 (5)0.0851 (3)0.75971 (14)0.0506 (8)
H3A0.29750.03800.79170.076*0.50
O10.2712 (5)0.3307 (4)0.69478 (12)0.0520 (8)
H10.34400.41930.70390.078*0.50
O40.3397 (5)0.3708 (4)0.80363 (14)0.0579 (9)
N20.1586 (5)0.2466 (5)0.33197 (15)0.0446 (8)
C50.1887 (8)0.5483 (7)0.5465 (2)0.0544 (12)
H50.15520.55180.58710.065*
N30.4745 (6)0.0982 (5)0.31116 (16)0.0459 (9)
C90.2415 (8)0.0996 (6)0.3156 (2)0.0474 (11)
H90.14730.00320.30760.057*
N10.3519 (6)0.3489 (4)0.33810 (14)0.0351 (8)
C10.3406 (8)0.5273 (5)0.35882 (18)0.0443 (11)
H1A0.21890.58770.33360.053*
H1B0.48910.58380.35390.053*
C60.0396 (8)0.6210 (7)0.5020 (2)0.0549 (13)
H60.09670.67440.51250.066*
C20.2893 (7)0.5377 (5)0.42460 (18)0.0364 (9)
C80.5381 (6)0.2598 (7)0.32578 (17)0.0417 (9)
H80.69030.30260.32710.050*
C40.3899 (8)0.4693 (6)0.53045 (19)0.0507 (12)
H40.49270.41910.56050.061*
C70.0878 (7)0.6168 (6)0.4414 (2)0.0460 (11)
H70.01610.66750.41170.055*
C30.4401 (7)0.4640 (6)0.4706 (2)0.0454 (11)
H30.57700.41040.46060.054*
N50.7031 (5)0.7708 (5)0.83783 (15)0.0417 (8)
N40.8811 (6)0.6541 (4)0.84658 (14)0.0366 (8)
C180.7984 (7)0.9014 (6)0.81290 (17)0.0395 (10)
H180.71881.00270.80190.047*
C160.9431 (7)0.5756 (6)0.98327 (19)0.0420 (10)
H161.08210.62470.97260.050*
N61.0286 (6)0.8733 (4)0.80462 (14)0.0356 (8)
C110.7924 (6)0.4980 (5)0.93922 (17)0.0317 (9)
C120.5874 (7)0.4219 (6)0.95742 (18)0.0405 (10)
H120.48340.36810.92840.049*
C100.8470 (8)0.4849 (5)0.87376 (19)0.0441 (11)
H10A0.98820.41670.87160.053*
H10B0.71920.42590.85030.053*
C140.6861 (8)0.5066 (6)1.0598 (2)0.0485 (12)
H140.64930.51131.10020.058*
C171.0749 (7)0.7164 (5)0.82694 (18)0.0357 (10)
H171.21920.66050.82840.043*
C150.8882 (8)0.5809 (6)1.0437 (2)0.0507 (12)
H150.98940.63521.07320.061*
C130.5389 (8)0.4258 (6)1.01696 (19)0.0482 (12)
H130.40380.37271.02840.058*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Cl10.0384 (5)0.0346 (5)0.0404 (5)0.0025 (5)0.0060 (4)0.0025 (5)
O20.0300 (13)0.0547 (19)0.0666 (19)0.0054 (16)0.0088 (12)0.0001 (19)
O30.0517 (17)0.0285 (16)0.075 (2)0.0139 (14)0.0271 (15)0.0174 (15)
O10.078 (2)0.0441 (18)0.0338 (16)0.0108 (16)0.0065 (15)0.0126 (13)
O40.0575 (19)0.065 (2)0.0497 (19)0.0129 (17)0.0052 (15)0.0157 (17)
N20.0429 (17)0.039 (2)0.052 (2)0.000 (2)0.0037 (15)0.003 (2)
C50.063 (3)0.053 (3)0.048 (3)0.005 (3)0.010 (2)0.010 (2)
N30.057 (2)0.045 (2)0.035 (2)0.0044 (19)0.0028 (17)0.0077 (17)
C90.050 (3)0.044 (3)0.047 (3)0.010 (2)0.005 (2)0.008 (2)
N10.0440 (19)0.033 (2)0.0289 (17)0.0003 (16)0.0082 (15)0.0037 (15)
C10.057 (3)0.031 (3)0.045 (3)0.001 (2)0.007 (2)0.007 (2)
C60.042 (3)0.066 (4)0.059 (3)0.001 (2)0.014 (2)0.014 (3)
C20.040 (2)0.029 (2)0.040 (2)0.0019 (18)0.0032 (18)0.0044 (19)
C80.041 (2)0.045 (3)0.040 (2)0.009 (3)0.0093 (17)0.002 (3)
C40.065 (3)0.046 (3)0.039 (3)0.003 (2)0.011 (2)0.003 (2)
C70.048 (3)0.041 (3)0.048 (3)0.008 (2)0.001 (2)0.001 (2)
C30.043 (2)0.043 (3)0.050 (3)0.009 (2)0.004 (2)0.010 (2)
N50.0370 (16)0.040 (2)0.049 (2)0.0052 (18)0.0065 (14)0.003 (2)
N40.042 (2)0.033 (2)0.0354 (18)0.0029 (15)0.0076 (16)0.0000 (16)
C180.044 (2)0.040 (3)0.034 (2)0.007 (2)0.0028 (18)0.005 (2)
C160.040 (2)0.039 (2)0.047 (3)0.0060 (19)0.0015 (19)0.000 (2)
N60.0432 (19)0.033 (2)0.0307 (18)0.0002 (16)0.0060 (15)0.0007 (16)
C110.039 (2)0.022 (2)0.034 (2)0.0028 (17)0.0034 (17)0.0023 (17)
C120.040 (2)0.045 (3)0.036 (2)0.0062 (19)0.0009 (18)0.007 (2)
C100.053 (3)0.033 (2)0.049 (3)0.002 (2)0.016 (2)0.008 (2)
C140.060 (3)0.047 (3)0.040 (2)0.014 (2)0.008 (2)0.014 (2)
C170.037 (2)0.031 (3)0.039 (2)0.0017 (18)0.0049 (17)0.0010 (17)
C150.062 (3)0.043 (3)0.044 (3)0.001 (2)0.012 (2)0.002 (2)
C130.046 (3)0.058 (3)0.043 (3)0.005 (2)0.012 (2)0.013 (2)
Geometric parameters (Å, º) top
Cl1—O21.444 (2)C4—H40.9300
Cl1—O41.449 (3)C7—H70.9300
Cl1—O31.466 (3)C3—H30.9300
Cl1—O11.529 (3)N5—C181.295 (5)
O3—H3A0.8200N5—N41.363 (4)
O1—H10.8200N4—C171.317 (5)
N2—C91.299 (6)N4—C101.464 (5)
N2—N11.358 (4)C18—N61.362 (5)
C5—C61.359 (7)C18—H180.9300
C5—C41.377 (6)C16—C111.375 (5)
C5—H50.9300C16—C151.390 (6)
N3—C81.338 (6)C16—H160.9300
N3—C91.345 (5)N6—C171.331 (5)
C9—H90.9300C11—C121.402 (5)
N1—C81.318 (5)C11—C101.501 (5)
N1—C11.462 (5)C12—C131.361 (5)
C1—C21.501 (5)C12—H120.9300
C1—H1A0.9700C10—H10A0.9700
C1—H1B0.9700C10—H10B0.9700
C6—C71.384 (6)C14—C131.359 (6)
C6—H60.9300C14—C151.367 (6)
C2—C71.384 (5)C14—H140.9300
C2—C31.391 (6)C17—H170.9300
C8—H80.9300C15—H150.9300
C4—C31.370 (6)C13—H130.9300
O2—Cl1—O4113.29 (19)C2—C7—H7119.7
O2—Cl1—O3110.2 (2)C4—C3—C2121.0 (4)
O4—Cl1—O3112.0 (2)C4—C3—H3119.5
O2—Cl1—O1107.50 (18)C2—C3—H3119.5
O4—Cl1—O1107.82 (19)C18—N5—N4104.1 (3)
O3—Cl1—O1105.54 (17)C17—N4—N5110.4 (3)
Cl1—O3—H3A109.5C17—N4—C10127.6 (3)
Cl1—O1—H1102.2N5—N4—C10122.0 (3)
C9—N2—N1103.2 (3)N5—C18—N6112.3 (4)
C6—C5—C4119.1 (4)N5—C18—H18123.9
C6—C5—H5120.4N6—C18—H18123.9
C4—C5—H5120.4C11—C16—C15120.2 (4)
C8—N3—C9103.0 (4)C11—C16—H16119.9
N2—C9—N3114.6 (4)C15—C16—H16119.9
N2—C9—H9122.7C17—N6—C18105.1 (4)
N3—C9—H9122.7C16—C11—C12118.1 (4)
C8—N1—N2109.7 (3)C16—C11—C10122.2 (4)
C8—N1—C1128.4 (4)C12—C11—C10119.7 (4)
N2—N1—C1121.8 (3)C13—C12—C11120.8 (4)
N1—C1—C2111.6 (3)C13—C12—H12119.6
N1—C1—H1A109.3C11—C12—H12119.6
C2—C1—H1A109.3N4—C10—C11112.1 (3)
N1—C1—H1B109.3N4—C10—H10A109.2
C2—C1—H1B109.3C11—C10—H10A109.2
H1A—C1—H1B108.0N4—C10—H10B109.2
C5—C6—C7120.9 (4)C11—C10—H10B109.2
C5—C6—H6119.5H10A—C10—H10B107.9
C7—C6—H6119.5C13—C14—C15120.1 (4)
C7—C2—C3117.7 (4)C13—C14—H14120.0
C7—C2—C1121.3 (4)C15—C14—H14120.0
C3—C2—C1121.0 (4)N4—C17—N6108.1 (3)
N1—C8—N3109.6 (3)N4—C17—H17125.9
N1—C8—H8125.2N6—C17—H17125.9
N3—C8—H8125.2C14—C15—C16120.2 (4)
C3—C4—C5120.6 (4)C14—C15—H15119.9
C3—C4—H4119.7C16—C15—H15119.9
C5—C4—H4119.7C14—C13—C12120.6 (4)
C6—C7—C2120.6 (4)C14—C13—H13119.7
C6—C7—H7119.7C12—C13—H13119.7
N1—N2—C9—N30.6 (5)C18—N5—N4—C170.1 (4)
C8—N3—C9—N20.5 (5)C18—N5—N4—C10179.5 (3)
C9—N2—N1—C80.4 (4)N4—N5—C18—N60.7 (4)
C9—N2—N1—C1177.4 (4)N5—C18—N6—C171.0 (4)
C8—N1—C1—C2108.2 (4)C15—C16—C11—C121.5 (6)
N2—N1—C1—C268.1 (5)C15—C16—C11—C10178.3 (4)
C4—C5—C6—C70.1 (8)C16—C11—C12—C130.4 (6)
N1—C1—C2—C7118.1 (4)C10—C11—C12—C13177.2 (4)
N1—C1—C2—C360.3 (5)C17—N4—C10—C11115.2 (4)
N2—N1—C8—N30.2 (4)N5—N4—C10—C1165.5 (5)
C1—N1—C8—N3176.9 (4)C16—C11—C10—N458.9 (5)
C9—N3—C8—N10.2 (5)C12—C11—C10—N4124.3 (4)
C6—C5—C4—C30.1 (8)N5—N4—C17—N60.5 (4)
C5—C6—C7—C20.2 (7)C10—N4—C17—N6178.9 (4)
C3—C2—C7—C60.2 (6)C18—N6—C17—N40.8 (4)
C1—C2—C7—C6178.2 (4)C13—C14—C15—C160.6 (7)
C5—C4—C3—C20.2 (7)C11—C16—C15—C141.0 (6)
C7—C2—C3—C40.2 (6)C15—C14—C13—C121.8 (7)
C1—C2—C3—C4178.2 (4)C11—C12—C13—C141.3 (6)
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
O3—H3A···N6i0.822.042.620 (4)127
O1—H1···N3ii0.821.782.546 (5)154
C10—H10B···O40.972.353.282 (5)160
C9—H9···O2iii0.932.523.352 (6)150
C1—H1A···O2iv0.972.563.483 (5)158
C17—H17···O4v0.932.433.147 (5)134
C18—H18···O3vi0.932.443.185 (5)137
Symmetry codes: (i) x1, y1, z; (ii) x+1, y+1/2, z+1; (iii) x, y1/2, z+1; (iv) x, y+1/2, z+1; (v) x+1, y, z; (vi) x, y+1, z.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
O3—H3A···N6i0.822.042.620 (4)127.4
O1—H1···N3ii0.821.782.546 (5)154.1
C10—H10B···O40.972.353.282 (5)159.7
C9—H9···O2iii0.932.523.352 (6)149.7
C1—H1A···O2iv0.972.563.483 (5)157.9
C17—H17···O4v0.932.433.147 (5)133.9
C18—H18···O3vi0.932.443.185 (5)137.2
Symmetry codes: (i) x1, y1, z; (ii) x+1, y+1/2, z+1; (iii) x, y1/2, z+1; (iv) x, y+1/2, z+1; (v) x+1, y, z; (vi) x, y+1, z.
 

Acknowledgements

The authors thank the Colleges and Universities Technology Project of Shanxi Province (20121033), and the Natural Science Fund of Lvliang University (ZRXN201206 and ZRXN201210).

References

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Volume 70| Part 12| December 2014| Pages o1290-o1291
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