supplementary materials


Acta Cryst. (2008). E64, i82-i83    [ doi:10.1107/S1600536808037884 ]

Na5(NH4)Mn3[B9P6O33(OH)3]·1.5H2O

Z.-S. Lin, Y.-X. Huang, Y. Prots, J.-T. Zhao and R. Kniep

Abstract top

The overall hexagonal framework of the title compound, pentasodium ammonium trimanganese(II) borophosphate sesquihydrate, consists of tube-like borophosphate anions, [infinity]1{[B3P2O11(OH)]4-}, made up of corner-sharing PO4 and BO4 tetrahedra and BO2(OH) triangles, forming ten-membered ring windows. The tubes are interconnected via distorted MnO6 octahedra, establishing a three-dimensional open-framework structure with two different types of ring-channels (12- and six-membered) that run along [001]. The 12-membered ring channels are occupied by NH4+ ions and water molecules. The ten-membered ring windows in the walls of the tubes are occupied by Na+ ions. The remaining Na+ ions and the water molecules, one of which is half-occupied, reside within the six-membered ring channels. The structural setup is consolidated by an O-H...O hydrogen bond between the OH group and an opposite O atom of the framework. Donor-acceptor distances ranging from 2.80 to 3.35 Å between the ammonium N atom, water O atoms and framework O atoms indicate further hydrogen-bonding interactions.

Comment top

In the past several years, borophosphates have attracted extensive attention due to their rich structural chemistry and potential applications as catalysts (Kniep et al., 1998; Ewald et al., 2007). Although a large variety of borophosphate anions has been reported, tube-like borophosphate anions are particularly rare (Liu et al., 2006; Yang et al., 2006a; Yang et al., 2006b). Up to now, only two manganese compounds containing borophosphate tubes, viz. Na2Mn[B3P2O11(OH)].0.67H2O (Yang et al., 2006a) and Na5(H3O)Mn3[B9P6O33(OH)3].2H2O (Yang et al., 2006b) are listed in the literature. Here, we report on an ammonium substituted sodium manganese borophosphate, Na5(NH4)Mn3[B9P6O33(OH)3].1.5H2O.

The crystal structure of the title compound comprises tube-like borophosphate anions, 1{[B3P2O11(OH)]4-}, which are built from 12-membered rings of alternating BO4 and PO4 tetrahedra, further interlinked by sharing common O-corners of neighbouring rings, and loop-branched by BO2(OH) triangles resulting in 10-membered ring windows on the walls of the tubes (Fig. 1). The manganese atoms are in a distorted octahedral coordination, surrounded by four oxygen atoms from phosphate tetrahedra (O1, O2, O5, O6) and two oxygen atoms from borate tetrahedra (O10, O11). The Mn-coordination octahedra interconnect the neighboring borophosphate tubes to form a three-dimensional framework with two different types of channels (Fig. 2), namely 6- and 12-membered ring channels. The 12-membered ring channels are occupied by NH4+ ions and water molecules; the 10-membered ring windows in the walls of the tubes are occupied by Na+ ions. The remaining Na+ ions and water molecules reside in the 6-membered ring channels. The structural setup is consolidated by an O—H···O hydrogen bond between the OH group and an opposite O atom of the framework. Donor-acceptor distances ranging from 2.8 to 3.35 Å between the ammonium N atom, water O atoms and framework O atoms indicate further hydrogen bonding interactions, but the corresponding H atoms were not located.

Related literature top

Reviews on the preparation, crystal chemistry and applications of borophosphates are given in Kniep et al. (1998) and Ewald et al. (2007). For isostructural compounds, see Yang, Li et al. (2006) for Na2Mn[B3P2O11(OH)].0.67H2O; Yang, Yu et al. (2006) for Na5(H3O)Mn3[B9P6O33(OH)3].2H2O; Liu et al. (2006) for Na6Cu3[B9P6O33(OH)3].2H2O.

Experimental top

Transparent, colourless single crystals of the title compound were synthesized hydrothermally from a mixture of H3BO3 (32.2 mmol), Mn(CH3COO)2.4H2O (3 mmol), (NH4)2HPO4 (6.4 mmol), NaF (5 mmol), and water (133.4 mmol). The educt mixture was transferred into a Teflon-lined stainless steel autoclave (internal volume 25 ml) and kept at 513 K for five days. The autoclave was cooled down to ambient temperature by removing out of the oven. The reaction products were washed with hot distilled water (333 K) until the boric acid was completely removed. Finally, the solids were dried in air at 333 K. Hexagonal prismatic crystals were selected for single-crystal diffraction. The NH4+ content was determined by elemental analysis and confirmed by IR spectroscopy.

Refinement top

The measured crystal was racemically twinned with an approximate twin fraction of 2:3. The hydrogen position bonded to O12 was found in a difference Fourier map and was refined freely. The hydrogen positions of the ammonium N atom and of the uncoordinated water atoms at O13 and O14 were not localized. The occupancy of O13 was refined to 0.50 (2). In the last refinement cycle this value was fixed to 0.50.

Computing details top

Data collection: CrystalClear (Rigaku/MSC, 2005); cell refinement: CrystalClear (Rigaku/MSC, 2005); data reduction: CrystalClear (Rigaku/MSC, 2005); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: DIAMOND (Brandenburg, 2004); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Figures top
[Figure 1] Fig. 1. Borophosphate tubes in the crystal structure of Na5(NH4)Mn3[B9P6O33(OH)3].1.5H2O interconnected by MnO6 coordination octahedra.
[Figure 2] Fig. 2. The overall framework of Na5(NH4)Mn3[B9P6O33(OH)3].1.5H2O viewed along [001], showing the resulting channel-system.
Pentasodium ammonium trimanganese(II) borophosphate sesquihydrate top
Crystal data top
Na5(NH4)Mn3[B9P6O33(OH)3]·1.5(H2O)Dx = 2.635 Mg m3
Mr = 2373.94Mo Kα radiation, λ = 0.71073 Å
Hexagonal, P63Cell parameters from 7346 reflections
Hall symbol: P 6cθ = 2.0–33.6°
a = 11.9331 (2) ŵ = 1.79 mm1
c = 12.1290 (4) ÅT = 295 K
V = 1495.76 (6) Å3Prism, colourless
Z = 10.08 × 0.04 × 0.04 mm
F(000) = 1164
Data collection top
Rigaku Mercury AFC7 CCD
diffractometer
2891 independent reflections
Radiation source: fine-focus sealed tube2784 reflections with I > 2σ(I)
graphiteRint = 0.030
ω–scansθmax = 30.0°, θmin = 2.0°
Absorption correction: multi-scan
(Blessing, 1995)
h = 1612
Tmin = 0.779, Tmax = 0.931k = 1616
12243 measured reflectionsl = 1616
Refinement top
Refinement on F2Secondary atom site location: difference Fourier map
Least-squares matrix: fullHydrogen site location: difference Fourier map
R[F2 > 2σ(F2)] = 0.040H atoms treated by a mixture of independent and constrained refinement
wR(F2) = 0.099 w = 1/[σ2(Fo2) + (0.0419P)2 + 2.5644P]
where P = (Fo2 + 2Fc2)/3
S = 1.12(Δ/σ)max < 0.001
2891 reflectionsΔρmax = 0.66 e Å3
191 parametersΔρmin = 0.81 e Å3
1 restraintAbsolute structure: Flack (1983), 1374 Friedel pairs
Primary atom site location: structure-invariant direct methodsFlack parameter: 0.43 (3)
Crystal data top
Na5(NH4)Mn3[B9P6O33(OH)3]·1.5(H2O)Z = 1
Mr = 2373.94Mo Kα radiation
Hexagonal, P63µ = 1.79 mm1
a = 11.9331 (2) ÅT = 295 K
c = 12.1290 (4) Å0.08 × 0.04 × 0.04 mm
V = 1495.76 (6) Å3
Data collection top
Rigaku Mercury AFC7 CCD
diffractometer
2891 independent reflections
Absorption correction: multi-scan
(Blessing, 1995)
2784 reflections with I > 2σ(I)
Tmin = 0.779, Tmax = 0.931Rint = 0.030
12243 measured reflectionsθmax = 30.0°
Refinement top
R[F2 > 2σ(F2)] = 0.040H atoms treated by a mixture of independent and constrained refinement
wR(F2) = 0.099Δρmax = 0.66 e Å3
S = 1.12Δρmin = 0.81 e Å3
2891 reflectionsAbsolute structure: Flack (1983), 1374 Friedel pairs
191 parametersFlack parameter: 0.43 (3)
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
xyzUiso*/UeqOcc. (<1)
Mn20.50444 (6)0.50073 (7)0.04544 (11)0.01330 (12)
P10.62139 (9)0.81051 (9)0.00965 (7)0.0111 (2)
P20.37924 (9)0.19394 (10)0.08857 (7)0.0120 (2)
B10.2928 (4)0.2502 (4)0.6961 (4)0.0118 (7)
B20.2992 (4)0.2504 (5)0.8974 (4)0.0151 (8)
B30.4938 (3)0.4004 (3)0.7921 (5)0.0159 (6)
O10.5750 (3)0.6785 (3)0.9591 (3)0.0169 (6)
O20.7011 (3)0.9084 (3)0.9177 (2)0.0162 (6)
O30.7199 (3)0.8385 (3)0.1043 (2)0.0150 (6)
O40.5126 (3)0.8296 (3)0.0515 (3)0.0177 (5)
O50.2925 (3)0.0953 (3)0.1789 (2)0.0152 (5)
O60.4181 (3)0.3245 (3)0.1360 (3)0.0179 (6)
O70.2859 (3)0.1639 (3)0.9886 (3)0.0182 (6)
O80.6857 (3)0.5090 (3)0.0545 (3)0.0175 (6)
O90.5731 (3)0.6364 (3)0.1961 (3)0.0147 (5)
O100.26579 (18)0.17981 (18)0.7974 (3)0.0132 (3)
O110.4355 (3)0.3637 (3)0.8933 (3)0.0157 (6)
O120.6273 (2)0.4785 (2)0.7907 (3)0.0235 (5)
H10.648 (7)0.498 (7)0.723 (6)0.07 (2)*
Na10.71486 (14)0.73129 (14)0.8014 (2)0.0273 (3)
Na20.33330.66670.9533 (3)0.0251 (6)
N0.00000.00000.0452 (12)0.0243 (10)
Na30.66670.33330.9477 (3)0.0248 (6)
O130.00000.00000.8007 (19)0.041 (2)*0.50
O140.33330.66670.7692 (9)0.090 (3)*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Mn20.0140 (2)0.01171 (19)0.0136 (2)0.00596 (15)0.00045 (17)0.00050 (14)
P10.0127 (4)0.0114 (4)0.0093 (4)0.0061 (3)0.0005 (3)0.0014 (3)
P20.0128 (4)0.0126 (4)0.0103 (4)0.0061 (4)0.0003 (4)0.0009 (3)
B10.0110 (17)0.0120 (17)0.0113 (18)0.0050 (14)0.0000 (15)0.0011 (15)
B20.018 (2)0.0153 (18)0.014 (2)0.0096 (16)0.0016 (16)0.0004 (16)
B30.0135 (13)0.0121 (12)0.0206 (17)0.0054 (10)0.0010 (19)0.0038 (19)
O10.0212 (13)0.0161 (12)0.0113 (14)0.0079 (11)0.0016 (11)0.0013 (10)
O20.0210 (14)0.0136 (12)0.0125 (12)0.0076 (11)0.0022 (11)0.0018 (10)
O30.0168 (13)0.0148 (12)0.0128 (14)0.0075 (10)0.0020 (11)0.0041 (10)
O40.0158 (12)0.0161 (12)0.0214 (14)0.0081 (11)0.0002 (11)0.0001 (11)
O50.0188 (13)0.0134 (12)0.0129 (12)0.0078 (10)0.0019 (11)0.0022 (10)
O60.0211 (13)0.0129 (12)0.0184 (15)0.0074 (11)0.0021 (12)0.0014 (11)
O70.0179 (14)0.0180 (13)0.0150 (15)0.0061 (11)0.0019 (12)0.0034 (11)
O80.0124 (11)0.0162 (12)0.0229 (14)0.0064 (10)0.0007 (11)0.0036 (11)
O90.0159 (12)0.0160 (12)0.0086 (12)0.0053 (10)0.0001 (11)0.0001 (11)
O100.0151 (8)0.0132 (8)0.0117 (8)0.0074 (7)0.0023 (13)0.0018 (13)
O110.0146 (12)0.0156 (12)0.0142 (14)0.0055 (10)0.0013 (11)0.0007 (11)
O120.0143 (10)0.0297 (12)0.0193 (13)0.0055 (9)0.0020 (14)0.0008 (15)
Na10.0289 (7)0.0381 (7)0.0212 (7)0.0215 (6)0.0001 (10)0.0012 (11)
Na20.0238 (8)0.0238 (8)0.0278 (15)0.0119 (4)0.0000.000
N0.0161 (12)0.0161 (12)0.041 (3)0.0081 (6)0.0000.000
Na30.0241 (8)0.0241 (8)0.0261 (14)0.0121 (4)0.0000.000
Geometric parameters (Å, °) top
Mn2—O82.118 (3)O5—B1ix1.475 (5)
Mn2—O1i2.126 (3)O5—Na1iv2.584 (3)
Mn2—O62.127 (3)O6—Na1iv2.434 (4)
Mn2—O4ii2.162 (3)O7—P2vii1.562 (3)
Mn2—O92.303 (3)O8—P2x1.505 (3)
Mn2—O11i2.326 (4)O8—Na3i2.376 (3)
Mn2—Na3i3.6069 (13)O9—B3iv1.354 (6)
Mn2—Na2i3.6582 (12)O9—B1iv1.493 (5)
P1—O1i1.513 (3)O10—Na1iii2.360 (2)
P1—O41.514 (3)O11—Mn2vii2.326 (4)
P1—O2i1.550 (3)O12—Na12.656 (3)
P1—O31.555 (3)O12—Na32.768 (4)
P1—Na2i3.0544 (12)O12—H10.86 (8)
P1—Na1i3.094 (3)Na1—O10x2.360 (2)
P2—O61.500 (3)Na1—O6vi2.434 (4)
P2—O8iii1.505 (3)Na1—O5vi2.584 (3)
P2—O51.562 (3)Na1—P1vii3.094 (3)
P2—O7i1.562 (3)Na1—P2vi3.118 (3)
P2—Na1iv3.118 (3)Na1—H12.66 (8)
P2—Na3i3.4270 (19)Na2—O142.232 (11)
B1—O101.430 (5)Na2—O4xi2.370 (3)
B1—O5v1.475 (5)Na2—O4vii2.370 (3)
B1—O3vi1.491 (5)Na2—O4xii2.370 (3)
B1—O9vi1.493 (5)Na2—O1viii2.817 (3)
B2—O101.416 (6)Na2—O1ii2.817 (3)
B2—O71.466 (6)Na2—P1xi3.0544 (12)
B2—O2ii1.494 (5)Na2—P1vii3.0544 (12)
B2—O111.509 (5)Na2—P1xii3.0544 (12)
B3—O9vi1.354 (6)Na2—Mn2xi3.6582 (12)
B3—O111.371 (6)Na2—Mn2xii3.6582 (12)
B3—O121.386 (4)Na3—O8xiii2.376 (3)
O1—P1vii1.513 (3)Na3—O8xiv2.376 (3)
O1—Mn2vii2.126 (3)Na3—O8vii2.376 (3)
O1—Na12.407 (4)Na3—O12iii2.768 (4)
O1—Na22.817 (3)Na3—O12x2.768 (4)
O2—B2viii1.494 (5)Na3—P2xiv3.4270 (19)
O2—P1vii1.550 (3)Na3—P2xiii3.4270 (19)
O2—Na12.614 (4)Na3—P2vii3.4270 (19)
O3—B1iv1.491 (5)Na3—Mn2xiii3.6069 (13)
O4—Mn2viii2.162 (3)Na3—Mn2xiv3.6069 (13)
O4—Na2i2.370 (3)Na3—Mn2vii3.6069 (13)
O8—Mn2—O1i95.34 (12)B3iv—O9—Mn2120.6 (2)
O8—Mn2—O689.88 (12)B1iv—O9—Mn2118.7 (2)
O1i—Mn2—O6174.49 (15)B2—O10—B1118.2 (2)
O8—Mn2—O4ii174.93 (18)B2—O10—Na1iii115.7 (3)
O1i—Mn2—O4ii88.22 (11)B1—O10—Na1iii119.3 (3)
O6—Mn2—O4ii86.46 (12)B3—O11—B2117.6 (3)
O8—Mn2—O986.00 (12)B3—O11—Mn2vii122.7 (2)
O1i—Mn2—O982.24 (11)B2—O11—Mn2vii116.6 (3)
O6—Mn2—O996.39 (14)B3—O12—Na1115.5 (2)
O4ii—Mn2—O990.91 (12)B3—O12—Na394.0 (3)
O8—Mn2—O11i93.89 (12)Na1—O12—Na3125.72 (15)
O1i—Mn2—O11i97.79 (14)B3—O12—H1106 (5)
O6—Mn2—O11i83.59 (12)Na1—O12—H181 (5)
O4ii—Mn2—O11i89.20 (12)Na3—O12—H1135 (5)
O9—Mn2—O11i179.89 (14)O10x—Na1—O1125.56 (16)
O1i—P1—O4113.43 (17)O10x—Na1—O6vi120.09 (16)
O1i—P1—O2i105.09 (17)O1—Na1—O6vi108.15 (9)
O4—P1—O2i112.12 (18)O10x—Na1—O5vi114.43 (12)
O1i—P1—O3111.51 (18)O1—Na1—O5vi111.70 (11)
O4—P1—O3109.4 (2)O6vi—Na1—O5vi57.80 (11)
O2i—P1—O3104.87 (18)O10x—Na1—O2117.98 (12)
O6—P2—O8iii114.34 (17)O1—Na1—O257.76 (11)
O6—P2—O5104.92 (18)O6vi—Na1—O2111.69 (11)
O8iii—P2—O5112.87 (17)O5vi—Na1—O267.76 (7)
O6—P2—O7i110.57 (19)O10x—Na1—O1280.64 (8)
O8iii—P2—O7i109.5 (2)O1—Na1—O1285.06 (12)
O5—P2—O7i104.11 (17)O6vi—Na1—O1279.46 (12)
O10—B1—O5v109.7 (3)O5vi—Na1—O12136.89 (14)
O10—B1—O3vi108.1 (3)O2—Na1—O12142.78 (14)
O5v—B1—O3vi108.4 (3)O14—Na2—O4xi120.17 (10)
O10—B1—O9vi110.9 (3)O14—Na2—O4vii120.17 (10)
O5v—B1—O9vi110.6 (3)O4xi—Na2—O4vii96.95 (13)
O3vi—B1—O9vi109.0 (3)O14—Na2—O4xii120.17 (10)
O10—B2—O7109.1 (4)O4xi—Na2—O4xii96.95 (13)
O10—B2—O2ii109.2 (3)O4vii—Na2—O4xii96.95 (13)
O7—B2—O2ii107.9 (3)O14—Na2—O1viii91.45 (9)
O10—B2—O11111.2 (3)O4xi—Na2—O1viii57.64 (10)
O7—B2—O11110.1 (4)O4vii—Na2—O1viii69.66 (9)
O2ii—B2—O11109.2 (4)O4xii—Na2—O1viii147.64 (16)
O9vi—B3—O11123.0 (3)O14—Na2—O1ii91.45 (9)
O9vi—B3—O12120.0 (5)O4xi—Na2—O1ii69.66 (10)
O11—B3—O12117.0 (5)O4vii—Na2—O1ii147.64 (16)
P1vii—O1—Mn2vii126.55 (19)O4xii—Na2—O1ii57.64 (10)
P1vii—O1—Na1101.82 (16)O1viii—Na2—O1ii119.937 (9)
Mn2vii—O1—Na1121.96 (15)O14—Na2—O191.45 (9)
P1vii—O1—Na283.98 (13)O4xi—Na2—O1147.64 (16)
Mn2vii—O1—Na294.45 (11)O4vii—Na2—O157.64 (10)
Na1—O1—Na2123.48 (15)O4xii—Na2—O169.66 (9)
B2viii—O2—P1vii135.3 (3)O1viii—Na2—O1119.937 (9)
B2viii—O2—Na1132.2 (3)O1ii—Na2—O1119.937 (8)
P1vii—O2—Na192.40 (14)O8xiii—Na3—O8xiv93.15 (13)
B1iv—O3—P1127.0 (3)O8xiii—Na3—O8vii93.15 (13)
P1—O4—Mn2viii127.87 (17)O8xiv—Na3—O8vii93.15 (13)
P1—O4—Na2i101.43 (16)O8xiii—Na3—O12iii120.52 (9)
Mn2viii—O4—Na2i107.56 (13)O8xiv—Na3—O12iii78.10 (11)
B1ix—O5—P2131.9 (3)O8vii—Na3—O12iii145.34 (9)
B1ix—O5—Na1iv133.0 (2)O8xiii—Na3—O12x78.10 (10)
P2—O5—Na1iv94.29 (14)O8xiv—Na3—O12x145.34 (9)
P2—O6—Mn2125.0 (2)O8vii—Na3—O12x120.52 (9)
P2—O6—Na1iv102.20 (17)O12iii—Na3—O12x77.89 (13)
Mn2—O6—Na1iv128.53 (15)O8xiii—Na3—O12145.34 (9)
B2—O7—P2vii127.6 (3)O8xiv—Na3—O12120.52 (9)
P2x—O8—Mn2128.59 (17)O8vii—Na3—O1278.10 (10)
P2x—O8—Na3i122.40 (16)O12iii—Na3—O1277.89 (13)
Mn2—O8—Na3i106.59 (13)O12x—Na3—O1277.89 (13)
B3iv—O9—B1iv119.0 (3)
Symmetry codes: (i) x, y, z−1; (ii) −x+y, −x+1, z; (iii) −y+1, xy, z; (iv) −x+1, −y+1, z−1/2; (v) xy, x, z+1/2; (vi) −x+1, −y+1, z+1/2; (vii) x, y, z+1; (viii) −y+1, xy+1, z; (ix) y, −x+y, z−1/2; (x) −x+y+1, −x+1, z; (xi) −y+1, xy+1, z+1; (xii) −x+y, −x+1, z+1; (xiii) −x+y+1, −x+1, z+1; (xiv) −y+1, xy, z+1.
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O12—H1···O4xv0.86 (7)2.11 (7)2.959 (3)167 (3)
O13—···.O10xvi..2.8035 (11).
O13—···.Nxvii..3.077 (16).
O14—···.O8v..3.284 (5).
O14—···.O6v..3.333 (3).
N—···.O13i..2.988 (16).
N—···.O3xviii..2.991 (3).
N—···.O7i..3.047 (3).
Symmetry codes: (xv) xy+1, x, z+1/2; (xvi) −x+y, −x, z; (xvii) −x, −y, z+1/2; (v) xy, x, z+1/2; (i) x, y, z−1; (xviii) x−1, y−1, z.
Table 1
Hydrogen-bond geometry (Å, °)
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D—H···AD—HH···AD···AD—H···A
O12—H1···O4i0.86 (7)2.11 (7)2.959 (3)167 (3)
O13—···.O10ii..2.8035 (11).
O13—···.Niii..3.077 (16).
O14—···.O8iv..3.284 (5).
O14—···.O6iv..3.333 (3).
N—···.O13v..2.988 (16).
N—···.O3vi..2.991 (3).
N—···.O7v..3.047 (3).
Symmetry codes: (i) xy+1, x, z+1/2; (ii) −x+y, −x, z; (iii) −x, −y, z+1/2; (iv) xy, x, z+1/2; (v) x, y, z−1; (vi) x−1, y−1, z.
Acknowledgements top

This project was supported by the doctoral joint project between the Chinese Academy of Sciences and the Max-Planck Society, the State '973' project (grant No. 2007CB936704) and the Major Basic Research Programs of Shanghai (grant No. 07DJ14001).

references
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