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Poly[[aqua­[μ3-(2,6-2H2)-isonicotinato-κ3N:O:O′][μ2-(2,6-2H2)-isonicotinato-κ2N:O]manganese(II)] ethanol solvate]

aOrdered Matter Science Research Center, College of Chemistry and Chemical Engineering, Southeast University, Nanjing 210096, People's Republic of China
*Correspondence e-mail: cherrydai01@yahoo.com.cn

(Received 8 May 2008; accepted 28 May 2008; online 16 July 2008)

In the title compound, {[Mn(C6H2D2NO2)2(H2O)]·C2H6O}n, the MnII metal centre displays a slightly distorted octa­hedral coordination geometry, provided by three O and two N atoms of five isonicotinate ligands and one O atom of a water mol­ecule. There are two types of isonicotinate anions, one acting as a bridging tridentate group and the other in a bridging bidentate fashion, to form a polymeric three-dimensional network. The structure is stabilized by intra- and inter­molecular O—H⋯O and C—H⋯O hydrogen-bond inter­actions.

Related literature

For related literature, see: Akutagawa et al. (2004[Akutagawa, T., Takeda, S., Hasegawa, T. & Nakamura, T. (2004). J. Am. Chem. Soc. 126, 291-294.]); Cova et al. (2001[Cova, B., Briceno, A. & Atencio, R. (2001). New J. Chem. 25, 1516-1519.]); Pavlik & Laohhasurayotin (2005[Pavlik, J. W. & Laohhasurayotin, S. (2005). J. Heterocycl. Chem. 42, 73-76.]); Sekiya & Nishikiori (2001[Sekiya, R. & Nishikiori, S. (2001). Chem. Commun. pp. 2612-2613.]).

[Scheme 1]

Experimental

Crystal data
  • [Mn(C6H2D2NO2)2(H2O)]·C2H6O

  • Mr = 367.24

  • Monoclinic, P 21 /n

  • a = 10.903 (2) Å

  • b = 12.180 (2) Å

  • c = 13.015 (3) Å

  • β = 110.02 (3)°

  • V = 1623.9 (6) Å3

  • Z = 4

  • Mo Kα radiation

  • μ = 0.84 mm−1

  • T = 293 (2) K

  • 0.20 × 0.20 × 0.20 mm

Data collection
  • Rigaku Mercury2 diffractometer

  • Absorption correction: multi-scan (CrystalClear; Rigaku, 2005[Rigaku (2005). CrystalClear. Rigaku Corporation, Tokyo, Japan.]) Tmin = 0.795, Tmax = 0.841

  • 16339 measured reflections

  • 3701 independent reflections

  • 3087 reflections with I > 2σ(I)

  • Rint = 0.042

Refinement
  • R[F2 > 2σ(F2)] = 0.040

  • wR(F2) = 0.142

  • S = 1.06

  • 3701 reflections

  • 248 parameters

  • H atoms treated by a mixture of independent and constrained refinement

  • Δρmax = 0.52 e Å−3

  • Δρmin = −0.52 e Å−3

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A D—H H⋯A DA D—H⋯A
C11—H11A⋯O5 0.98 (4) 2.44 (3) 2.784 (3) 100 (2)
O4—H2W⋯O6 0.79 (4) 1.90 (4) 2.680 (5) 166 (4)
O6—H6A⋯O3i 0.85 1.90 2.729 (3) 165
C11—H11A⋯O3ii 0.98 (4) 2.50 (4) 3.404 (4) 153 (3)
O4—H1W⋯O3ii 0.92 (4) 1.89 (4) 2.793 (3) 166 (3)
Symmetry codes: (i) [x+{\script{1\over 2}}, -y+{\script{3\over 2}}, z+{\script{1\over 2}}]; (ii) -x+1, -y+1, -z.

Data collection: CrystalClear (Rigaku, 2005[Rigaku (2005). CrystalClear. Rigaku Corporation, Tokyo, Japan.]); cell refinement: CrystalClear; data reduction: CrystalClear; 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: SHELXL97.

Supporting information


Comment top

Isonicotinic acid is a good mono- or bidentate ligand for the construction of supramolecular complexes with versatile binding modes (Cova et al., 2001; Sekiya & Nishikiori, 2001). Until now, a large number of metal-organic framework structures containing isonicotinic acid ligands have been reported. Investigations on the effect of deuteration onto the physical properties like permittivity has become of increasing interest (Akutagawa et al., 2004).

In the crystal structure of the title compound the manganese atom displays a slightly distorted octahedral geometry provided by two N atoms and three carboxylate O atoms of five different isonicotinato anions and one O atom of a water molecule (Fig. 1). The structure contains two types of isonicotinato ligands, one acting as a bridging trichelate group, the other as bridging bidentate group to form a polymeric three-dimensional network (Fig. 2). The shortest interatomic Mn···Mn separation is 4.9182 (12) Å. The structure is stabilized by intra- and intermolecular O—H···O and C—H···O hydrogen bonds (Table 1).

Related literature top

For related literature, see: Akutagawa et al. (2004); Cova et al. (2001); Pavlik & Laohhasurayotin (2005); Sekiya & Nishikiori (2001).

Experimental top

A mixture of isonicotinic acid N-oxide (21.6 mmol) in deuterium oxide (10.0 mL) and sodium deuteroxide (31.0 mmol) was acidified with concentrated hydrochloric acid. After 4 h isonicotinic acid N-oxide-2,6-D2 was separated as a white solid. A solution of this compound (9.0 mmol) in dichloromethane (60 mL) was added dropwise to phosphorus trichloride (1.2 mL). The mixture was refluxed for 1 h, mixed with ice–water (30 mL), made alkaline with aqueous NaOH (10 N) and extracted with dichloromethane (5 x 20mL) according to the method reported by Pavlik & Laohhasurayotin (2005). The organic phase was dried over anhydrous sodium sulfate to give isonicotinic acid-2,6-D2. A mixture of isonicotinic acid-2,6-D2 (0.1 mmol), manganese(II) acetate (0.2 mmol), ethanol (1 ml) and water (0.1 ml) was then transferred into a sealed Pyrex tube and heated at 100°C for 2 d. Yellow crystals of the title compound suitable for X-ray analysis were obtained on slow cooling to room temperature.

Refinement top

H and D atoms associated with the pyridine rings and water molecule were located in a difference Fourier map and refined freely. All other H atoms were placed in calculated positions, with C—H = 0.86–0.96 Å, O—H = 0.85 Å, and with Uiso(H) = 1.5 Ueq(C, O) or 1.5 Ueq(C) for methylene H atoms.

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: SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Figures top
[Figure 1] Fig. 1. A view of the title compound showing the coordination around the manganese(II) atom. Displacement ellipsoids are drawn at the 30% probability level. [Symmetry codes: (A) 1/2+x, 1/2-y, 1/2+z; (B) 1/2+x, 3/2-y; (C) 1/2+z and 2-x, 1-y, -z]
[Figure 2] Fig. 2. Packing diagram of the title compound viewed along the a axis. Ethanol solvent molecules are omitted for clarity.
Poly[[aqua[µ3-(2,6-2H2)-isonicotinato-κ3N:O:O'][µ2-(2,6-2H2)- isonicotinato-κ2N:O]manganese(II)] ethanol solvate] top
Crystal data top
[Mn(C6H2D2NO2)2(H2O)]·C2H6OF(000) = 748
Mr = 367.24Dx = 1.502 Mg m3
Monoclinic, P21/nMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ynCell parameters from 19580 reflections
a = 10.903 (2) Åθ = 3.0–27.5°
b = 12.180 (2) ŵ = 0.84 mm1
c = 13.015 (3) ÅT = 293 K
β = 110.02 (3)°Block, yellow
V = 1623.9 (6) Å30.20 × 0.20 × 0.20 mm
Z = 4
Data collection top
Rigaku Mercury2 (2x2 bin mode)
diffractometer
3701 independent reflections
Radiation source: fine-focus sealed tube3087 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.042
Detector resolution: 13.6612 pixels mm-1θmax = 27.5°, θmin = 3.0°
CCD profile fitting scansh = 1414
Absorption correction: multi-scan
(CrystalClear; Rigaku, 2005)
k = 1515
Tmin = 0.795, Tmax = 0.841l = 1616
16339 measured reflections
Refinement top
Refinement on F2Primary atom site location: structure-invariant direct methods
Least-squares matrix: fullSecondary atom site location: difference Fourier map
R[F2 > 2σ(F2)] = 0.040Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.142H atoms treated by a mixture of independent and constrained refinement
S = 1.06 w = 1/[σ2(Fo2) + (0.1P)2]
where P = (Fo2 + 2Fc2)/3
3701 reflections(Δ/σ)max < 0.001
248 parametersΔρmax = 0.52 e Å3
0 restraintsΔρmin = 0.52 e Å3
Crystal data top
[Mn(C6H2D2NO2)2(H2O)]·C2H6OV = 1623.9 (6) Å3
Mr = 367.24Z = 4
Monoclinic, P21/nMo Kα radiation
a = 10.903 (2) ŵ = 0.84 mm1
b = 12.180 (2) ÅT = 293 K
c = 13.015 (3) Å0.20 × 0.20 × 0.20 mm
β = 110.02 (3)°
Data collection top
Rigaku Mercury2 (2x2 bin mode)
diffractometer
3701 independent reflections
Absorption correction: multi-scan
(CrystalClear; Rigaku, 2005)
3087 reflections with I > 2σ(I)
Tmin = 0.795, Tmax = 0.841Rint = 0.042
16339 measured reflections
Refinement top
R[F2 > 2σ(F2)] = 0.0400 restraints
wR(F2) = 0.142H atoms treated by a mixture of independent and constrained refinement
S = 1.06Δρmax = 0.52 e Å3
3701 reflectionsΔρmin = 0.52 e Å3
248 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
xyzUiso*/Ueq
Mn10.93671 (3)0.50175 (2)0.16410 (2)0.02182 (15)
O50.87201 (17)0.38008 (13)0.03681 (14)0.0391 (4)
O40.79806 (18)0.41661 (16)0.22877 (17)0.0411 (4)
N10.77113 (19)0.61593 (15)0.06381 (16)0.0334 (4)
C70.7734 (2)0.27522 (17)0.12181 (17)0.0260 (4)
C120.8682 (2)0.36318 (16)0.05830 (18)0.0268 (4)
C110.6930 (3)0.2208 (2)0.0751 (2)0.0395 (6)
C50.7818 (2)0.72543 (19)0.0584 (2)0.0376 (5)
C80.7617 (2)0.2480 (2)0.2283 (2)0.0353 (5)
C10.6582 (2)0.5717 (2)0.0011 (2)0.0404 (6)
C20.5536 (2)0.63237 (19)0.0664 (2)0.0395 (6)
C30.5668 (2)0.74530 (17)0.07336 (16)0.0273 (4)
C40.6847 (2)0.79184 (19)0.00909 (19)0.0358 (5)
C60.4577 (2)0.81308 (17)0.15183 (17)0.0274 (4)
O30.34326 (15)0.77890 (14)0.17448 (14)0.0386 (4)
O20.49329 (16)0.89829 (13)0.18801 (14)0.0385 (4)
O10.9325 (2)0.40991 (17)0.10687 (17)0.0554 (5)
C100.6056 (3)0.1433 (2)0.1362 (2)0.0409 (6)
C90.6720 (2)0.1688 (2)0.2837 (2)0.0355 (5)
N20.59483 (18)0.11618 (15)0.23923 (15)0.0311 (4)
O60.6716 (3)0.5555 (2)0.3174 (4)0.1366 (18)
H6A0.7151 (3)0.6101 (2)0.3078 (4)0.205*
C140.5578 (4)0.5893 (4)0.3214 (6)0.118 (2)
H14A0.5176 (4)0.6222 (4)0.2618 (6)0.142*
H14B0.5716 (4)0.6358 (4)0.3729 (6)0.142*
C130.4751 (7)0.5076 (5)0.3359 (7)0.132 (3)
H13A0.3950 (7)0.5394 (5)0.3370 (7)0.197*
H13B0.4566 (7)0.4556 (5)0.2771 (7)0.197*
H13C0.5183 (7)0.4711 (5)0.4041 (7)0.197*
H8A0.804 (3)0.287 (2)0.269 (2)0.038 (7)*
D30.662 (2)0.1472 (19)0.355 (2)0.028 (6)*
D10.648 (4)0.495 (3)0.008 (3)0.063 (12)*
D40.549 (3)0.109 (2)0.105 (3)0.044 (7)*
H4A0.701 (3)0.872 (2)0.015 (2)0.044 (7)*
D20.864 (3)0.758 (2)0.109 (2)0.044 (7)*
H11A0.708 (3)0.236 (3)0.002 (3)0.065 (10)*
H2A0.472 (3)0.591 (3)0.105 (3)0.066 (10)*
H1W0.758 (4)0.351 (3)0.202 (3)0.069 (10)*
H2W0.751 (4)0.450 (3)0.252 (3)0.066 (11)*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Mn10.0206 (2)0.0197 (2)0.0215 (2)0.00130 (10)0.00247 (15)0.00019 (10)
O50.0464 (10)0.0376 (9)0.0314 (9)0.0144 (7)0.0108 (7)0.0118 (7)
O40.0420 (10)0.0345 (9)0.0554 (11)0.0090 (8)0.0275 (9)0.0037 (8)
N10.0309 (10)0.0305 (9)0.0331 (10)0.0022 (8)0.0037 (8)0.0083 (7)
C70.0240 (10)0.0263 (10)0.0262 (10)0.0052 (8)0.0067 (8)0.0016 (8)
C120.0233 (10)0.0227 (10)0.0313 (11)0.0033 (8)0.0054 (8)0.0014 (8)
C110.0475 (14)0.0470 (13)0.0266 (12)0.0224 (11)0.0159 (10)0.0072 (10)
C50.0368 (13)0.0316 (11)0.0339 (12)0.0019 (10)0.0014 (10)0.0003 (9)
C80.0331 (12)0.0427 (14)0.0344 (12)0.0145 (10)0.0173 (10)0.0070 (10)
C10.0309 (12)0.0290 (12)0.0532 (15)0.0014 (9)0.0042 (10)0.0125 (10)
C20.0299 (12)0.0307 (12)0.0488 (15)0.0029 (9)0.0018 (10)0.0105 (10)
C30.0280 (10)0.0297 (11)0.0237 (10)0.0024 (8)0.0083 (8)0.0076 (8)
C40.0389 (13)0.0268 (11)0.0337 (12)0.0038 (10)0.0022 (10)0.0015 (9)
C60.0308 (11)0.0272 (10)0.0228 (10)0.0021 (8)0.0074 (8)0.0046 (8)
O30.0279 (8)0.0365 (9)0.0472 (10)0.0006 (7)0.0074 (7)0.0112 (7)
O20.0381 (9)0.0329 (8)0.0362 (9)0.0031 (7)0.0021 (7)0.0171 (7)
O10.0586 (12)0.0619 (12)0.0511 (12)0.0404 (10)0.0255 (10)0.0110 (9)
C100.0474 (14)0.0483 (14)0.0303 (12)0.0261 (12)0.0175 (11)0.0060 (10)
C90.0390 (13)0.0431 (13)0.0277 (12)0.0136 (10)0.0157 (10)0.0124 (9)
N20.0320 (10)0.0329 (10)0.0261 (9)0.0122 (8)0.0071 (7)0.0055 (7)
O60.103 (2)0.0659 (18)0.295 (5)0.0334 (17)0.138 (3)0.077 (3)
C140.064 (3)0.085 (3)0.216 (7)0.019 (2)0.062 (3)0.044 (3)
C130.098 (5)0.143 (6)0.184 (8)0.016 (3)0.087 (5)0.011 (4)
Geometric parameters (Å, º) top
Mn1—O1i2.1151 (18)C2—H2A1.00 (3)
Mn1—O52.1537 (16)C2—C31.389 (3)
Mn1—O2ii2.1806 (16)C3—C41.392 (3)
Mn1—O42.2228 (18)C3—C61.519 (3)
Mn1—N2iii2.2656 (18)C4—H4A1.00 (3)
Mn1—N12.2987 (19)C6—O31.251 (3)
O5—C121.242 (3)C6—O21.255 (3)
O4—H2W0.79 (4)O2—Mn1iv2.1806 (16)
O4—H1W0.92 (4)O1—Mn1i2.1151 (18)
N1—C11.336 (3)C10—N21.346 (3)
N1—C51.343 (3)C10—D40.95 (3)
C7—C81.388 (3)C9—N21.336 (3)
C7—C111.393 (3)C9—D30.93 (3)
C7—C121.521 (3)N2—Mn1v2.2656 (18)
C12—O11.232 (3)O6—H6A0.8499
C11—H11A0.98 (4)O6—C141.325 (5)
C11—C101.384 (3)C14—H14A0.8499
C5—C41.382 (3)C14—H14B0.8500
C5—D20.99 (3)C14—C131.397 (7)
C8—H8A0.95 (3)C13—H13A0.9599
C8—C91.387 (3)C13—H13C0.9600
C1—C21.390 (3)C13—H13B0.9602
C1—D10.94 (3)
O1i—Mn1—O599.32 (7)N1—C1—D1117 (2)
O1i—Mn1—O2ii90.33 (8)C2—C1—D1119 (2)
O5—Mn1—O2ii169.24 (7)H2A—C2—C3124.0 (19)
O1i—Mn1—O4177.07 (8)H2A—C2—C1117 (2)
O5—Mn1—O483.34 (7)C3—C2—C1119.1 (2)
O2ii—Mn1—O487.12 (8)C2—C3—C4117.3 (2)
O1i—Mn1—N2iii92.38 (8)C2—C3—C6120.46 (19)
O5—Mn1—N2iii88.71 (7)C4—C3—C6122.16 (19)
O2ii—Mn1—N2iii86.11 (7)H4A—C4—C5120.1 (17)
O4—Mn1—N2iii88.90 (7)H4A—C4—C3120.4 (17)
O1i—Mn1—N189.17 (8)C5—C4—C3119.5 (2)
O5—Mn1—N189.58 (7)O3—C6—O2126.75 (19)
O2ii—Mn1—N195.35 (7)O3—C6—C3117.77 (18)
O4—Mn1—N189.63 (7)O2—C6—C3115.47 (18)
N2iii—Mn1—N1177.87 (6)C6—O2—Mn1iv139.70 (15)
C12—O5—Mn1140.47 (14)C12—O1—Mn1i170.48 (18)
H2W—O4—H1W107 (3)N2—C10—C11123.2 (2)
H2W—O4—Mn1122 (3)N2—C10—D4118.4 (18)
H1W—O4—Mn1124 (2)C11—C10—D4118.4 (18)
C1—N1—C5116.52 (19)N2—C9—C8123.0 (2)
C1—N1—Mn1118.92 (15)N2—C9—D3114.5 (15)
C5—N1—Mn1124.44 (15)C8—C9—D3122.4 (15)
C8—C7—C11117.73 (19)C9—N2—C10117.39 (19)
C8—C7—C12121.61 (18)C9—N2—Mn1v122.39 (15)
C11—C7—C12120.64 (19)C10—N2—Mn1v119.92 (14)
O1—C12—O5127.2 (2)H6A—O6—C14110
O1—C12—C7116.5 (2)H14A—C14—H14B107.5
O5—C12—C7116.30 (18)H14A—C14—O6108
H11A—C11—C10124 (2)H14B—C14—O6109
H11A—C11—C7117 (2)H14A—C14—C13108
C10—C11—C7119.1 (2)H14B—C14—C13108
N1—C5—C4123.6 (2)O6—C14—C13116.1 (5)
N1—C5—D2115.9 (16)H13A—C13—H13C109.5
C4—C5—D2120.4 (16)H13A—C13—H13B109.5
H8A—C8—C9116.9 (16)H13C—C13—H13B109.5
H8A—C8—C7123.2 (16)H13A—C13—C14110
C9—C8—C7119.5 (2)H13C—C13—C14109
N1—C1—C2123.9 (2)H13B—C13—C14109
Symmetry codes: (i) x+2, y+1, z; (ii) x+1/2, y+3/2, z+1/2; (iii) x+1/2, y+1/2, z+1/2; (iv) x1/2, y+3/2, z1/2; (v) x1/2, y+1/2, z1/2.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
C11—H11A···O50.98 (4)2.44 (3)2.784 (3)100 (2)
O4—H2W···O60.79 (4)1.90 (4)2.680 (5)166 (4)
O6—H6A···O3ii0.851.902.729 (3)165
C11—H11A···O3vi0.98 (4)2.50 (4)3.404 (4)153 (3)
O4—H1W···O3vi0.92 (4)1.89 (4)2.793 (3)166 (3)
Symmetry codes: (ii) x+1/2, y+3/2, z+1/2; (vi) x+1, y+1, z.

Experimental details

Crystal data
Chemical formula[Mn(C6H2D2NO2)2(H2O)]·C2H6O
Mr367.24
Crystal system, space groupMonoclinic, P21/n
Temperature (K)293
a, b, c (Å)10.903 (2), 12.180 (2), 13.015 (3)
β (°) 110.02 (3)
V3)1623.9 (6)
Z4
Radiation typeMo Kα
µ (mm1)0.84
Crystal size (mm)0.20 × 0.20 × 0.20
Data collection
DiffractometerRigaku Mercury2 (2x2 bin mode)
diffractometer
Absorption correctionMulti-scan
(CrystalClear; Rigaku, 2005)
Tmin, Tmax0.795, 0.841
No. of measured, independent and
observed [I > 2σ(I)] reflections
16339, 3701, 3087
Rint0.042
(sin θ/λ)max1)0.649
Refinement
R[F2 > 2σ(F2)], wR(F2), S 0.040, 0.142, 1.06
No. of reflections3701
No. of parameters248
H-atom treatmentH atoms treated by a mixture of independent and constrained refinement
Δρmax, Δρmin (e Å3)0.52, 0.52

Computer programs: CrystalClear (Rigaku, 2005), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), SHELXTL (Sheldrick, 2008).

Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
C11—H11A···O50.98 (4)2.44 (3)2.784 (3)100 (2)
O4—H2W···O60.79 (4)1.90 (4)2.680 (5)166 (4)
O6—H6A···O3i0.851.902.729 (3)165
C11—H11A···O3ii0.98 (4)2.50 (4)3.404 (4)153 (3)
O4—H1W···O3ii0.92 (4)1.89 (4)2.793 (3)166 (3)
Symmetry codes: (i) x+1/2, y+3/2, z+1/2; (ii) x+1, y+1, z.
 

Acknowledgements

This work was supported by a Start-up Grant from Southeast University to Professor Ren-Gen Xiong.

References

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