metal-organic compounds\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

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catena-Poly[zinc-μ3-{3,3′-[(1,7-dioxa-4,10-di­aza­cyclo­do­decane-4,10-di­yl)bis­­(methyl­ene)]dibenzoato}]

aCenter for Functional Nanoscale Materials, Department of Chemistry, Clark Atlanta University, 223 James P. Brawley Drive, Atlanta, GA 30314, USA, and bDepartment of Chemistry, Emory University, Atlanta, GA 30322, USA
*Correspondence e-mail: cingram@cau.edu

(Received 16 September 2012; accepted 18 October 2012; online 27 October 2012)

The ZnII ion in the title compound, [Zn(C24H28N2O6)]n, is located on a twofold rotation axis and is at the midpoint of a crown-4 moiety of 3,3′-[(1,7-dioxa-4,10-diaza­cyclo­dodecane-4,10-di­yl)bis­(methyl­ene)]dibenzoate anion. It is octahedrally coordinated by two N atoms and two O atoms of the crown moiety from one ligand and two carboxyl­ate O atoms from two bridging intra-chain ligands. Metallomacrocyclic rings are identified in the structure. The metallomacrocycle contains two ZnII ions and 14 atoms from the bridging ligands. Repetition of these units gives rise to an infinite zigzag chain along [101]. C—H⋯O hydrogen bonds occur.

Related literature

For coordination polymers including metal-organic framework structures, see: Bai et al. (2012[Bai, S.-Q., Yong, A. M., Hu, J. J., Young, D. J., Zhang, X., Zong, Y., Xu, J., Zuo, J.-L. & Hor, T. S. A. (2012). CrystEngComm, 14, 961-971.]); Janiak (2003[Janiak, C. (2003). Dalton Trans. pp. 2781-2804.]); Kitagawa et al. (2004[Kitagawa, S., Kitaura, R. & Noro, S. (2004). Angew. Chem. Int. Ed. 43, 2334-2375.]); Li et al. (2012[Li, J.-R., Sculley, J. & Zhou, H.-C. (2012). Chem. Rev. 112, 869-932.]); Liao et al. (2012[Liao, L., Ingram, C. W., Vandeveer, D., Hardcastle, K., Solntsev, K. M., Sabo, D., Zhang, Z. J. & Weber, R. T. (2012). Inorg. Chim. Acta, 391, 1-9.]); Liu et al. (2012[Liu, Y.-Y., Li, J., Ma, J.-F., Ma, J.-C. & Yang, J. (2012). CrystEngComm, 14, 169-177.]); O'Keeffe et al. (2000[O'Keeffe, M., Eddaoudi, M., Li, H., Reineke, T. & Yaghi, O. M. (2000). J. Solid State Chem. 152, 3-20.]); Suh et al. (2012[Suh, M. P., Park, H. J., Prasad, T. K. & Lim, D.-W. (2012). Chem. Rev. 112, 782-835.]); Yoon et al. (2012[Yoon, M., Srirambalaji, R. & Kim, K. (2012). Chem. Rev. 112, 1196-1231.]).

[Scheme 1]

Experimental

Crystal data
  • [Zn(C24H28N2O6)]

  • Mr = 505.87

  • Monoclinic, C 2/c

  • a = 20.7264 (15) Å

  • b = 8.9791 (7) Å

  • c = 13.9745 (19) Å

  • β = 127.200 (4)°

  • V = 2071.5 (4) Å3

  • Z = 4

  • Cu Kα radiation

  • μ = 2.05 mm−1

  • T = 173 K

  • 0.48 × 0.14 × 0.11 mm

Data collection
  • Bruker D8 diffractometer with an APEXII detector

  • Absorption correction: multi-scan (SADABS; Bruker, 2008[Bruker (2008). SADABS. Bruker AXS Inc., Madison, Wisconsin, USA.]) Tmin = 0.414, Tmax = 0.685

  • 4413 measured reflections

  • 1684 independent reflections

  • 1538 reflections with I > 2σ(I)

  • Rint = 0.029

Refinement
  • R[F2 > 2σ(F2)] = 0.033

  • wR(F2) = 0.086

  • S = 1.06

  • 1684 reflections

  • 150 parameters

  • H-atom parameters constrained

  • Δρmax = 0.47 e Å−3

  • Δρmin = −0.21 e Å−3

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A D—H H⋯A DA D—H⋯A
C5—H5A⋯O1i 0.99 2.57 3.224 (3) 124
C12—H12⋯O2ii 0.95 2.43 3.256 (3) 145
Symmetry codes: (i) [-x+{\script{1\over 2}}, -y+{\script{1\over 2}}, -z+1]; (ii) -x, -y+1, -z.

Data collection: APEX2 (Bruker, 2011[Bruker (2011). APEX2. Bruker AXS Inc., Madison, Wisconsin, USA.]); cell refinement: SAINT (Bruker, 2009[Bruker (2009). 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: OLEX2 (Dolomanov et al., 2009[Dolomanov, O. V., Bourhis, L. J., Gildea, R. J., Howard, J. A. K. & Puschmann, H. (2009). J. Appl. Cryst. 42, 339-341.]); software used to prepare material for publication: OLEX2.

Supporting information


Comment top

The title compound 1 is the first of a series of coordination polymers that were synthesized from the ligand LH2, 3,3'-((1,7-dioxa-4,10-diazacyclododecane-4,10-diyl)bis(methylene)) dibenzoic acid. In these new compounds the metal atoms are positioned in the center of the organic linker.

The asymmetric unit of 1 contains a ZnII ion and a deprotonated ligand L with formula C24H28N2O6Zn. The ZnII ion is 6-coordinate in a distorted octahedral geometry being bound to two nitrogen atoms and two oxygen atoms of the crown (1,7-diaza-12-crown-4) and two carboxylic oxygen atoms, one from each of two additional intra-chain ligands (Fig. 1). The Zn1—O1, Zn1—O3 and Zn1—N1 bond lengths are 1.978 (2) Å, 2.287 (2) Å and 2.242 (2) Å, respectively. The O1—Zn1—O1 angle is 107.54 (8)°. The shortest distance between two neighboring ZnII ions in a chain is 9.019 (1) Å.

The ZnII ion of the Zn(crown-4)2+ unit is located on a 2-fold rotation axis. The symmetry independent atoms consist of one half of the ligand with the rotation axis generating the second half of the ligand at the Zn atom. Bond circuits consisting of sixteen-membered metallomacrocycle rings can be identified in the structure. Each ring contains two ZnII ions and fourteen non-H atoms of the ligand. Each ZnII ion is a node for three ligands and two connected rings (Fig. 1). The pair of benzene moieties within a metallomacrocycle ring are co-planar within one standard deviation (0.0 (2)°) and the dihedral angle between this plane and the plane of the next two nearest phenyl rings along the 1-D chain is 68.79 (5)°. Repetition of these units creates a 1-D polymer network with an infinite number of these rings.

Related literature top

For coordination polymers including metal-organic framework structures, see: Bai et al. (2012); Janiak (2003); Kitagawa et al. (2004); Li et al. (2012); Liao et al. (2012); Liu et al. (2012); O'Keeffe et al. (2000); Suh et al. (2012); Yoon et al. (2012).

Experimental top

The title compound was synthesized in an autoclave by mixing the ligand, 3,3'-((1,7-dioxa-4,10-diazacyclododecane-4,10-diyl)bis(methylene))dibenzoic acid, LH2 (2x10-6 mol), Zn(NO3)2.6H2O (6x10-6 mol, 1.79 mg), H2O (0.60 ml) and pyridine (2x10-3 ml) in a vial of 2 ml capacity. The mixture was heated at 85 °C for 7 d and then cooled to ambient temperature. The white crystals were collected and washed with H2O by filtration. Elem. anal. calcd. C24H28N2O6Zn %: C, 56.98; H, 5.58; N, 5.54; Found: C, 56.99; H, 5.79; N, 5.58.

Refinement top

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.

Computing details top

Data collection: APEX2 (Bruker, 2011); cell refinement: SAINT (Bruker, 2009); data reduction: SAINT (Bruker, 2009); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: OLEX2 (Dolomanov et al., 2009); software used to prepare material for publication: OLEX2 (Dolomanov et al., 2009).

Figures top
[Figure 1] Fig. 1. A portion of the one-dimensional chain of 1 showing the 16-membered metallomacrocycle rings.
catena-poly[zinc-µ3-{3,3'-[(1,7-dioxa-4,10-diazacyclododecane-4,10- diyl)bis(methylene)]dibenzoato}] top
Crystal data top
[Zn(C24H28N2O6)]F(000) = 1056
Mr = 505.87Dx = 1.622 Mg m3
Monoclinic, C2/cCu Kα radiation, λ = 1.54184 Å
a = 20.7264 (15) ÅCell parameters from 3719 reflections
b = 8.9791 (7) Åθ = 4.0–67.7°
c = 13.9745 (19) ŵ = 2.05 mm1
β = 127.200 (4)°T = 173 K
V = 2071.5 (4) Å3Column, colourless
Z = 40.48 × 0.14 × 0.11 mm
Data collection top
Bruker D8
diffractometer with an APEXII detector
1684 independent reflections
Radiation source: sealed tube1538 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.029
Detector resolution: 512 pixels mm-1θmax = 65.1°, θmin = 5.4°
ϕ and ω scans with a narrow frame widthh = 2418
Absorption correction: multi-scan
(SADABS; Bruker, 2008)
k = 1010
Tmin = 0.414, Tmax = 0.685l = 1415
4413 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)] = 0.033Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.086H-atom parameters constrained
S = 1.06 w = 1/[σ2(Fo2) + (0.0475P)2 + 2.6887P]
where P = (Fo2 + 2Fc2)/3
1684 reflections(Δ/σ)max < 0.001
150 parametersΔρmax = 0.47 e Å3
0 restraintsΔρmin = 0.21 e Å3
Crystal data top
[Zn(C24H28N2O6)]V = 2071.5 (4) Å3
Mr = 505.87Z = 4
Monoclinic, C2/cCu Kα radiation
a = 20.7264 (15) ŵ = 2.05 mm1
b = 8.9791 (7) ÅT = 173 K
c = 13.9745 (19) Å0.48 × 0.14 × 0.11 mm
β = 127.200 (4)°
Data collection top
Bruker D8
diffractometer with an APEXII detector
1684 independent reflections
Absorption correction: multi-scan
(SADABS; Bruker, 2008)
1538 reflections with I > 2σ(I)
Tmin = 0.414, Tmax = 0.685Rint = 0.029
4413 measured reflections
Refinement top
R[F2 > 2σ(F2)] = 0.0330 restraints
wR(F2) = 0.086H-atom parameters constrained
S = 1.06Δρmax = 0.47 e Å3
1684 reflectionsΔρmin = 0.21 e Å3
150 parameters
Special details top

Experimental. Absorption correction: SADABS (Bruker-AXS, 2008) was used for absorption correction. R(int) was 0.0732 before and 0.0388 after correction. The ratio of minimum to maximum transmission is 0.6049. The λ/2 correction factor is not present.

Geometry. Bond distances, angles etc. have been calculated using the rounded fractional coordinates. All su's are estimated from the variances of the (full) variance-covariance matrix. The cell esds are taken into account in the estimation of distances, angles and torsion angles

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
Zn10.000000.05167 (4)0.250000.0179 (1)
O10.07654 (9)0.18185 (17)0.24895 (14)0.0224 (4)
O20.00673 (9)0.3592 (2)0.11945 (15)0.0325 (5)
O30.55621 (9)0.63747 (19)0.71322 (13)0.0254 (4)
N10.39893 (11)0.5343 (2)0.56752 (16)0.0193 (5)
C10.06066 (13)0.3070 (3)0.19486 (19)0.0213 (6)
C20.13504 (13)0.3887 (2)0.22567 (19)0.0196 (6)
C30.21035 (13)0.3696 (2)0.33678 (19)0.0201 (6)
C40.28068 (13)0.4328 (2)0.36223 (19)0.0200 (6)
C50.36142 (13)0.4041 (2)0.48377 (19)0.0206 (6)
C60.33847 (13)0.5969 (3)0.58104 (19)0.0232 (6)
C70.37416 (14)0.7072 (3)0.6834 (2)0.0258 (7)
C80.42804 (14)0.6450 (3)0.52237 (19)0.0247 (6)
C90.49985 (14)0.7382 (3)0.6195 (2)0.0255 (7)
C100.27460 (13)0.5139 (3)0.2717 (2)0.0229 (6)
C110.19960 (14)0.5358 (3)0.1619 (2)0.0235 (7)
C120.12995 (13)0.4758 (2)0.13890 (19)0.0204 (6)
H30.213900.311900.396800.0240*
H5A0.353700.323300.524200.0250*
H5B0.400200.367500.469900.0250*
H6A0.313300.514200.594500.0280*
H6B0.295200.647000.505200.0280*
H7A0.389800.800300.664200.0310*
H7B0.334200.731800.698100.0310*
H8A0.382700.712700.465700.0300*
H8B0.443400.591500.476900.0300*
H9A0.525100.789400.586900.0310*
H9B0.482200.814300.650500.0310*
H100.321900.554100.285300.0280*
H110.196000.593000.101600.0280*
H120.078800.494000.064100.0250*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Zn10.0119 (2)0.0195 (2)0.0178 (2)0.00000.0066 (2)0.0000
O10.0161 (7)0.0227 (8)0.0279 (8)0.0020 (6)0.0131 (7)0.0024 (6)
O20.0159 (8)0.0391 (10)0.0287 (9)0.0006 (7)0.0063 (7)0.0110 (7)
O30.0178 (8)0.0244 (8)0.0205 (7)0.0034 (6)0.0045 (7)0.0021 (6)
N10.0142 (9)0.0212 (10)0.0184 (9)0.0003 (7)0.0077 (8)0.0012 (7)
C10.0181 (11)0.0250 (12)0.0170 (10)0.0017 (9)0.0086 (9)0.0012 (9)
C20.0181 (11)0.0191 (11)0.0197 (10)0.0019 (8)0.0104 (9)0.0012 (8)
C30.0198 (11)0.0198 (11)0.0183 (10)0.0003 (9)0.0103 (9)0.0006 (8)
C40.0174 (11)0.0195 (11)0.0197 (10)0.0005 (8)0.0094 (9)0.0016 (8)
C50.0154 (10)0.0218 (11)0.0192 (10)0.0000 (8)0.0077 (9)0.0007 (8)
C60.0129 (10)0.0291 (12)0.0198 (10)0.0051 (9)0.0058 (9)0.0020 (9)
C70.0181 (11)0.0279 (12)0.0226 (11)0.0082 (9)0.0077 (9)0.0021 (9)
C80.0207 (11)0.0264 (12)0.0190 (10)0.0014 (9)0.0078 (10)0.0043 (9)
C90.0217 (11)0.0237 (12)0.0234 (11)0.0022 (9)0.0096 (10)0.0059 (9)
C100.0198 (11)0.0249 (11)0.0233 (11)0.0033 (9)0.0126 (10)0.0020 (9)
C110.0244 (12)0.0239 (12)0.0208 (11)0.0004 (9)0.0130 (10)0.0022 (9)
C120.0170 (11)0.0197 (11)0.0180 (10)0.0021 (8)0.0071 (9)0.0008 (8)
Geometric parameters (Å, º) top
Zn1—O11.978 (2)C6—C71.515 (3)
Zn1—O1i1.978 (2)C8—C91.522 (4)
Zn1—O3ii2.2869 (19)C10—C111.388 (4)
Zn1—N1ii2.2422 (19)C11—C121.384 (4)
Zn1—O3iii2.2869 (19)C3—H30.9500
Zn1—N1iii2.2422 (19)C5—H5A0.9900
O1—C11.281 (3)C5—H5B0.9900
O2—C11.225 (3)C6—H6A0.9900
O3—C91.431 (3)C6—H6B0.9900
O3—C7iv1.429 (3)C7—H7A0.9900
N1—C51.497 (3)C7—H7B0.9900
N1—C61.486 (4)C8—H8A0.9900
N1—C81.487 (4)C8—H8B0.9900
C1—C21.516 (4)C9—H9A0.9900
C2—C31.395 (3)C9—H9B0.9900
C2—C121.392 (3)C10—H100.9500
C3—C41.395 (4)C11—H110.9500
C4—C51.521 (3)C12—H120.9500
C4—C101.398 (3)
O1—Zn1—O1i107.54 (8)N1—C8—C9114.72 (19)
O1—Zn1—O3ii163.85 (7)O3—C9—C8106.6 (2)
O1—Zn1—N1ii90.57 (8)C4—C10—C11120.2 (3)
O1—Zn1—O3iii85.26 (7)C10—C11—C12121.0 (2)
O1—Zn1—N1iii113.40 (8)C2—C12—C11119.8 (2)
O1i—Zn1—O3ii85.26 (7)C2—C3—H3119.00
O1i—Zn1—N1ii113.40 (8)C4—C3—H3119.00
O1i—Zn1—O3iii163.85 (7)N1—C5—H5A108.00
O1i—Zn1—N1iii90.57 (8)N1—C5—H5B108.00
O3ii—Zn1—N1ii75.01 (7)C4—C5—H5A108.00
O3ii—Zn1—O3iii84.09 (7)C4—C5—H5B108.00
O3ii—Zn1—N1iii75.37 (7)H5A—C5—H5B107.00
O3iii—Zn1—N1ii75.37 (7)N1—C6—H6A109.00
N1ii—Zn1—N1iii139.73 (7)N1—C6—H6B109.00
O3iii—Zn1—N1iii75.01 (7)C7—C6—H6A109.00
Zn1—O1—C1126.99 (19)C7—C6—H6B109.00
C7iv—O3—C9114.52 (19)H6A—C6—H6B108.00
Zn1ii—O3—C9115.48 (17)C6—C7—H7A110.00
Zn1ii—O3—C7iv116.02 (14)C6—C7—H7B110.00
C5—N1—C6108.4 (2)H7A—C7—H7B109.00
C5—N1—C8110.03 (19)O3iv—C7—H7A110.00
Zn1ii—N1—C5107.87 (12)O3iv—C7—H7B110.00
C6—N1—C8113.0 (2)N1—C8—H8A109.00
Zn1ii—N1—C6105.39 (13)N1—C8—H8B109.00
Zn1ii—N1—C8111.90 (16)C9—C8—H8A109.00
O1—C1—O2126.6 (3)C9—C8—H8B109.00
O1—C1—C2113.7 (2)H8A—C8—H8B108.00
O2—C1—C2119.6 (2)O3—C9—H9A110.00
C1—C2—C3121.2 (2)O3—C9—H9B110.00
C1—C2—C12119.7 (2)C8—C9—H9A110.00
C3—C2—C12118.9 (3)C8—C9—H9B110.00
C2—C3—C4121.8 (2)H9A—C9—H9B109.00
C3—C4—C5119.4 (2)C4—C10—H10120.00
C3—C4—C10118.2 (2)C11—C10—H10120.00
C5—C4—C10122.4 (3)C10—C11—H11119.00
N1—C5—C4116.19 (17)C12—C11—H11120.00
N1—C6—C7113.6 (2)C2—C12—H12120.00
O3iv—C7—C6106.6 (2)C11—C12—H12120.00
O1i—Zn1—O1—C136.0 (2)O1—C1—C2—C328.4 (3)
N1ii—Zn1—O1—C1150.78 (19)O1—C1—C2—C12146.7 (2)
O3iii—Zn1—O1—C1133.96 (19)O2—C1—C2—C3155.2 (2)
N1iii—Zn1—O1—C162.5 (2)O2—C1—C2—C1229.7 (4)
Zn1—O1—C1—O211.7 (4)C1—C2—C3—C4173.8 (2)
Zn1—O1—C1—C2172.18 (15)C12—C2—C3—C41.3 (3)
C7iv—O3—C9—C8176.1 (2)C1—C2—C12—C11172.2 (2)
Zn1ii—O3—C9—C837.4 (3)C3—C2—C12—C113.0 (3)
C9—O3—C7iv—C6iv160.4 (2)C2—C3—C4—C5178.2 (2)
C6—N1—C5—C453.7 (3)C2—C3—C4—C101.7 (3)
C8—N1—C5—C470.4 (3)C3—C4—C5—N1110.1 (2)
Zn1ii—N1—C5—C4167.3 (2)C10—C4—C5—N173.5 (3)
C5—N1—C6—C7167.45 (19)C3—C4—C10—C113.0 (4)
C8—N1—C6—C770.3 (2)C5—C4—C10—C11179.4 (2)
Zn1ii—N1—C6—C752.2 (2)N1—C6—C7—O3iv49.9 (3)
C5—N1—C8—C9151.4 (2)N1—C8—C9—O345.1 (3)
C6—N1—C8—C987.3 (3)C4—C10—C11—C121.3 (4)
Zn1ii—N1—C8—C931.5 (3)C10—C11—C12—C21.7 (4)
Symmetry codes: (i) x, y, z+1/2; (ii) x+1/2, y+1/2, z+1; (iii) x1/2, y+1/2, z1/2; (iv) x+1, y, z+3/2.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
C5—H5A···O1ii0.992.573.224 (3)124
C12—H12···O2v0.952.433.256 (3)145
Symmetry codes: (ii) x+1/2, y+1/2, z+1; (v) x, y+1, z.

Experimental details

Crystal data
Chemical formula[Zn(C24H28N2O6)]
Mr505.87
Crystal system, space groupMonoclinic, C2/c
Temperature (K)173
a, b, c (Å)20.7264 (15), 8.9791 (7), 13.9745 (19)
β (°) 127.200 (4)
V3)2071.5 (4)
Z4
Radiation typeCu Kα
µ (mm1)2.05
Crystal size (mm)0.48 × 0.14 × 0.11
Data collection
DiffractometerBruker D8
diffractometer with an APEXII detector
Absorption correctionMulti-scan
(SADABS; Bruker, 2008)
Tmin, Tmax0.414, 0.685
No. of measured, independent and
observed [I > 2σ(I)] reflections
4413, 1684, 1538
Rint0.029
(sin θ/λ)max1)0.588
Refinement
R[F2 > 2σ(F2)], wR(F2), S 0.033, 0.086, 1.06
No. of reflections1684
No. of parameters150
H-atom treatmentH-atom parameters constrained
Δρmax, Δρmin (e Å3)0.47, 0.21

Computer programs: APEX2 (Bruker, 2011), SAINT (Bruker, 2009), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), OLEX2 (Dolomanov et al., 2009).

Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
C5—H5A···O1i0.99002.57003.224 (3)124
C12—H12···O2ii0.95002.43003.256 (3)145
Symmetry codes: (i) x+1/2, y+1/2, z+1; (ii) x, y+1, z.
 

Acknowledgements

Financial support for this work by USA NSF/CREST/CFNM Award No. HRD-1137751 is gratefully acknowledged.

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