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

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ISSN: 2056-9890

Triphen­yl(tetra­hydro­furan)­aluminium(III)

aDepartment of Chemistry, National Chung Hsing University, Taichung 402, Taiwan
*Correspondence e-mail: hmgau@dragon.nchu.edu.tw

(Received 24 September 2008; accepted 6 October 2008; online 11 October 2008)

In the title compound, [Al(C6H5)3(C4H8O)], the Al atom has a distorted tetra­hedral geometry. The C—Al—C angles range from 113.25 (7) to 116.27 (8)°, much larger than the O—Al—C angles, which range from 103.39 (7) to 103.90 (6)°. The tetra­hydro­furan ring adopts an envelope conformation. The crystal packing is stabilized by C—H⋯π inter­actions.

Related literature

For general background, see: Chen et al. (2007[Chen, C.-A., Wu, K.-H. & Gau, H.-M. (2007). Angew. Chem. Int. Ed. 46, 5373-5376.]); Ku et al. (2007[Ku, S.-L., Hui, X.-P., Chen, C.-A., Kuo, Y.-Y. & Gau, H.-M. (2007). Chem. Commun. pp. 3847-3849.]); Wu & Gau (2006[Wu, K.-H. & Gau, H.-M. (2006). J. Am. Chem. Soc. 128, 14808-14809.]). For related structures, see: Barber et al. (1982[Barber, M., Liptak, D. & Oliver, J. P. (1982). Organometallics, 1, 1307-1311.]); De Mel & Oliver (1989[De Mel, V. S. J. & Oliver, J. P. (1989). Organometallics, 8, 827-830.]); Jerius et al. (1986[Jerius, J. J., Hahn, J. M., Rahman, A. F. M. M., Mols, O., Ilsley, W. H. & Oliver, J. P. (1986). Organometallics, 5, 1812-1814.]); Malone & McDonald (1967[Malone, J. F. & McDonald, W. S. (1967). Chem. Commun. pp. 444-445.]).

[Scheme 1]

Experimental

Crystal data
  • [Al(C6H5)3(C4H8O)]

  • Mr = 330.38

  • Monoclinic, P 21 /c

  • a = 9.649 (2) Å

  • b = 12.966 (3) Å

  • c = 16.038 (4) Å

  • β = 104.210 (4)°

  • V = 1945.2 (8) Å3

  • Z = 4

  • Mo Kα radiation

  • μ = 0.11 mm−1

  • T = 293 (2) K

  • 0.58 × 0.42 × 0.21 mm

Data collection
  • Bruker SMART 1000 CCD diffractometer

  • Absorption correction: multi-scan (SADABS; Sheldrick, 1996[Sheldrick, G. M. (1996). SADABS. University of Göttingen, Germany.]) Tmin = 0.799, Tmax = 1.000 (expected range = 0.781–0.977)

  • 10682 measured reflections

  • 3804 independent reflections

  • 2971 reflections with I > 2σ(I)

  • Rint = 0.023

Refinement
  • R[F2 > 2σ(F2)] = 0.048

  • wR(F2) = 0.162

  • S = 1.33

  • 3804 reflections

  • 217 parameters

  • H-atom parameters constrained

  • Δρmax = 0.27 e Å−3

  • Δρmin = −0.23 e Å−3

Table 1
Selected bond lengths (Å)

Al1—O1 1.8972 (13)
Al1—C1 1.9783 (18)
Al1—C13 1.9800 (18)
Al1—C7 1.9809 (19)

Table 2
Hydrogen-bond geometry (Å, °)

Cg1 is the centroid of the C1–C6 ring.

D—H⋯A D—H H⋯A DA D—H⋯A
C16—H16⋯Cg1i 0.93 2.78 3.654 (4) 156
C19—H19ACg1ii 0.97 2.81 3.600 (4) 139
Symmetry codes: (i) x+1, y, z; (ii) [-x+1, y+{\script{1\over 2}}, -z+{\script{1\over 2}}].

Data collection: SMART (Siemens, 1996[Siemens (1996). SMART and SAINT. Siemens Analytical X-ray Instruments Inc., Madison, Wisconsin, USA.]); cell refinement: SAINT (Siemens, 1996[Siemens (1996). SMART and SAINT. Siemens Analytical X-ray Instruments 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

Triphenylaluminium was first reported 40 years ago and the solid-state study revealed a dimeric Al2Ph6 structure bridging through two phenyl groups (Malone & McDonald, 1967). For synthesis of monomeric triarylaluminium complexes, two synthetic strategies were used. The first route employed a reaction of dimesitylmercury with Al/HgCl2, furnishing three-coordinate trimesitylaluminium (Jerius et al., 1986) which possesses a trigonal planar structure. The second synthetic route used a strategy of providing an additional neutral ligand, such as tetrahydrofuran (THF) or diethyl ether (OEt2), giving four-coordinate monomeric AlAr3(L) complexes (L = THF or OEt2) (Barber et al., 1982; De Mel & Oliver, 1989). In addition to structural studies, organoaluminium reagents had been demonstrated as excellent nucleophiles in organic synthesis, owing to their higher reactivity and the Lewis acidity of the aluminium center. Recently, we reported applications of AlAr3(THF) in asymmetric aryl additions to aldehydes (Wu & Gau, 2006) and to ketones (Chen et al., 2007) and in coupling reactions (Ku et al., 2007). Due to their diversified applications in catalysis, we report herein the synthesis and structure of a four-coordinate triphenylaluminium compound, [Al(C6H5)3(OC4H8)].

The molecule of the title compound contains unsubstituted phenyl ligands and has a distorted tetrahedral geometry around the aluminium metal center (Fig. 1). The Al—O(THF) bond length of 1.8972 (13) Å is shorter by 0.08 Å than the Al—C bond distances of 1.9783 (18), 1.9800 (18) and 1.9809 (19) Å. This complex has similar Al—C bond distances with the four-coordinate (o-tol)3Al(OEt2) complex (Barber et al., 1982). The C—Al—O bond angles in the title complex [103.39 (7), 103.90 (6) and 103.75 (7)°] are much smaller than the C—Al—C bond angles [113.25 (7), 114.23 (7) and 116.27 (8)°]. In contrast, the bulky mesityl ligands in trimesityl(tetrahydrofuran)aluminium complex (De Mel & Oliver, 1989) repel each other, giving longer Al—C bond lengths of 2.011 (7), 2.020 (7) and 2.021 (6) Å. Similarly, the Al—O(THF) bond distance of 1.969 (5) Å in the above complex is longer by 0.07 Å than the Al—O(THF) bond length in the title compound.

Related literature top

For general background, see: Chen et al. (2007); Ku et al. (2007); Wu & Gau (2006). For related structures, see: Barber et al. (1982); De Mel & Oliver (1989); Jerius et al. (1986); Malone & McDonald (1967).

Experimental top

A solution of phenylmagnesium bromide (90.0 mmol) in THF (50 ml) was slowly added to a solution of AlCl3 (4.00 g, 30.0 mmol) in THF (20 ml) at 273 K. The mixture was stirred at room temperature for 12 h and the solvent was removed under reduced pressure to afford a residue which was extracted with toluene (2 × 40 ml). The extracts were combined and concentrated to about 50 ml. Colourless crystals of the title compound (8.92 g, 90.0% yield) were obtained by cooling the concentrated solution at 273 K. The above synthetic procedures were conducted strictly under nitrogen atmosphere. 1H NMR (CDCl3, 400 MHz): δ 7.80–7.76 (m, 6H), 7.34–7.30 (m, 9H), 4.16 (m, 4H), 2.01 (m, 4H) p.p.m. 13C{1H} NMR (CDCl3, 100 MHz): δ 146.74, 137.99, 127.50, 127.06, 75.59, 24.97 p.p.m. Analysis calculated for C22H23OAl: C 79.97, H 7.02%;found: C 79.44, H 6.75%.

Refinement top

All H atoms were fixed geometrically [C—H = 0.93 Å or 0.97 Å] and treated as riding, with Uiso(H) = 1.2Ueq(C). The C atoms of the tetrahydrofuran ring display large displacement parameters, but no suitable refinement model for disorder was found.

Computing details top

Data collection: SMART (Siemens, 1996); cell refinement: SAINT (Siemens, 1996); data reduction: SAINT (Siemens, 1996); 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
[Figure 1] Fig. 1. The molecular structure of the title compound with atom-numbering and displacement ellipsoids drawn at the 20% probability level.
Triphenyl(tetrahydrofuran)aluminium(III) top
Crystal data top
[Al(C6H5)3(C4H8O)]F(000) = 704
Mr = 330.38Dx = 1.128 Mg m3
Monoclinic, P21/cMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybcCell parameters from 3804 reflections
a = 9.649 (2) Åθ = 2.1–26.0°
b = 12.966 (3) ŵ = 0.11 mm1
c = 16.038 (4) ÅT = 293 K
β = 104.210 (4)°Block, colourless
V = 1945.1 (8) Å30.58 × 0.42 × 0.21 mm
Z = 4
Data collection top
Bruker SMART 1000 CCD
diffractometer
3804 independent reflections
Radiation source: fine-focus sealed tube2971 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.023
ϕ and ω scansθmax = 26.0°, θmin = 2.1°
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)
h = 811
Tmin = 0.799, Tmax = 1.000k = 1515
10682 measured reflectionsl = 1917
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)] = 0.048Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.162H-atom parameters constrained
S = 1.33 w = 1/[σ2(Fo2) + (0.0861P)2]
where P = (Fo2 + 2Fc2)/3
3804 reflections(Δ/σ)max = 0.001
217 parametersΔρmax = 0.27 e Å3
0 restraintsΔρmin = 0.23 e Å3
Crystal data top
[Al(C6H5)3(C4H8O)]V = 1945.1 (8) Å3
Mr = 330.38Z = 4
Monoclinic, P21/cMo Kα radiation
a = 9.649 (2) ŵ = 0.11 mm1
b = 12.966 (3) ÅT = 293 K
c = 16.038 (4) Å0.58 × 0.42 × 0.21 mm
β = 104.210 (4)°
Data collection top
Bruker SMART 1000 CCD
diffractometer
3804 independent reflections
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)
2971 reflections with I > 2σ(I)
Tmin = 0.799, Tmax = 1.000Rint = 0.023
10682 measured reflections
Refinement top
R[F2 > 2σ(F2)] = 0.0480 restraints
wR(F2) = 0.162H-atom parameters constrained
S = 1.33Δρmax = 0.27 e Å3
3804 reflectionsΔρmin = 0.23 e Å3
217 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
Al10.39089 (5)0.69171 (4)0.18576 (3)0.0508 (2)
O10.38837 (12)0.75437 (10)0.29228 (7)0.0623 (4)
C10.27597 (17)0.56515 (13)0.18473 (11)0.0540 (4)
C20.2690 (2)0.51194 (15)0.25899 (13)0.0668 (5)
H20.31580.53940.31200.080*
C30.1956 (3)0.42013 (17)0.25735 (17)0.0818 (6)
H30.19360.38720.30860.098*
C40.1265 (2)0.37804 (17)0.1810 (2)0.0880 (7)
H40.07710.31620.17970.106*
C50.1296 (2)0.42674 (18)0.10597 (17)0.0836 (7)
H50.08200.39820.05360.100*
C60.2037 (2)0.51857 (15)0.10780 (13)0.0672 (5)
H60.20530.55030.05600.081*
C70.30029 (18)0.79599 (14)0.09920 (11)0.0573 (4)
C80.1554 (2)0.78994 (17)0.05675 (12)0.0699 (5)
H80.10160.73560.07000.084*
C90.0891 (2)0.8605 (2)0.00356 (15)0.0864 (7)
H90.00750.85350.03020.104*
C100.1641 (3)0.9404 (2)0.02443 (16)0.0907 (7)
H100.11950.98790.06580.109*
C110.3064 (3)0.95119 (19)0.01558 (16)0.0891 (7)
H110.35811.00650.00190.107*
C120.3729 (2)0.87933 (16)0.07661 (13)0.0728 (5)
H120.46940.88750.10320.087*
C130.59598 (17)0.66352 (14)0.19566 (12)0.0585 (4)
C140.6632 (2)0.68758 (18)0.13121 (16)0.0793 (6)
H140.61300.72370.08310.095*
C150.8054 (3)0.6587 (3)0.1368 (3)0.1220 (12)
H150.84780.67520.09240.146*
C160.8810 (3)0.6072 (3)0.2061 (4)0.1415 (18)
H160.97590.58980.21000.170*
C170.8176 (3)0.5803 (2)0.2712 (3)0.1196 (13)
H170.86860.54300.31830.144*
C180.6776 (2)0.60906 (17)0.26599 (15)0.0801 (6)
H180.63640.59160.31080.096*
C190.5018 (3)0.8175 (3)0.3436 (2)0.1160 (11)
H19A0.54750.85720.30660.139*
H19B0.57340.77420.38050.139*
C200.4384 (3)0.8856 (2)0.39505 (16)0.0958 (8)
H20A0.43830.95600.37450.115*
H20B0.49210.88340.45470.115*
C210.2920 (3)0.8503 (2)0.38737 (17)0.1015 (8)
H21A0.28600.81150.43810.122*
H21B0.22750.90870.38150.122*
C220.2540 (3)0.7853 (2)0.31139 (17)0.0937 (8)
H22A0.19520.82320.26340.112*
H22B0.20130.72530.32230.112*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Al10.0376 (3)0.0578 (3)0.0577 (3)0.00033 (19)0.0129 (2)0.0031 (2)
O10.0481 (7)0.0751 (8)0.0667 (8)0.0054 (6)0.0197 (5)0.0148 (6)
C10.0381 (8)0.0582 (10)0.0664 (10)0.0042 (7)0.0140 (7)0.0002 (7)
C20.0603 (11)0.0681 (11)0.0727 (12)0.0086 (9)0.0174 (9)0.0077 (9)
C30.0814 (15)0.0639 (12)0.1086 (17)0.0102 (11)0.0395 (13)0.0202 (12)
C40.0677 (13)0.0540 (11)0.152 (2)0.0030 (10)0.0462 (15)0.0004 (14)
C50.0645 (13)0.0739 (14)0.1074 (17)0.0091 (10)0.0115 (12)0.0236 (12)
C60.0617 (11)0.0696 (12)0.0696 (11)0.0052 (9)0.0147 (8)0.0044 (9)
C70.0494 (10)0.0637 (10)0.0604 (10)0.0024 (8)0.0166 (7)0.0036 (7)
C80.0520 (11)0.0782 (13)0.0780 (13)0.0087 (9)0.0132 (9)0.0022 (10)
C90.0616 (12)0.1058 (18)0.0869 (14)0.0202 (13)0.0090 (10)0.0061 (13)
C100.0918 (17)0.0995 (17)0.0817 (14)0.0332 (14)0.0230 (12)0.0228 (12)
C110.0990 (19)0.0791 (14)0.0950 (16)0.0011 (13)0.0351 (14)0.0210 (12)
C120.0638 (12)0.0744 (12)0.0788 (12)0.0017 (10)0.0149 (9)0.0058 (10)
C130.0416 (8)0.0606 (10)0.0736 (11)0.0031 (8)0.0149 (7)0.0177 (8)
C140.0639 (12)0.0857 (14)0.0985 (15)0.0117 (10)0.0397 (11)0.0284 (12)
C150.0816 (19)0.121 (2)0.192 (3)0.0240 (18)0.088 (2)0.069 (2)
C160.0459 (14)0.126 (3)0.248 (5)0.0064 (16)0.027 (2)0.093 (3)
C170.0632 (16)0.0994 (19)0.167 (3)0.0286 (14)0.0274 (18)0.054 (2)
C180.0604 (12)0.0780 (14)0.0911 (14)0.0149 (10)0.0020 (10)0.0159 (11)
C190.0805 (16)0.150 (3)0.124 (2)0.0434 (17)0.0365 (15)0.0780 (19)
C200.115 (2)0.0881 (16)0.0816 (15)0.0032 (15)0.0194 (13)0.0192 (12)
C210.125 (2)0.0909 (17)0.1067 (19)0.0074 (16)0.0622 (16)0.0156 (14)
C220.0641 (13)0.122 (2)0.1058 (17)0.0033 (13)0.0421 (12)0.0335 (15)
Geometric parameters (Å, º) top
Al1—O11.8972 (13)C11—C121.389 (3)
Al1—C11.9783 (18)C11—H110.93
Al1—C131.9800 (18)C12—H120.93
Al1—C71.9809 (19)C13—C141.384 (3)
O1—C191.450 (2)C13—C181.398 (3)
O1—C221.460 (2)C14—C151.404 (4)
C1—C21.392 (3)C14—H140.93
C1—C61.397 (3)C15—C161.346 (5)
C2—C31.382 (3)C15—H150.93
C2—H20.93C16—C171.378 (5)
C3—C41.357 (3)C16—H160.93
C3—H30.93C17—C181.384 (4)
C4—C51.365 (3)C17—H170.93
C4—H40.93C18—H180.93
C5—C61.385 (3)C19—C201.443 (3)
C5—H50.93C19—H19A0.97
C6—H60.93C19—H19B0.97
C7—C121.384 (3)C20—C211.461 (4)
C7—C81.399 (2)C20—H20A0.97
C8—C91.371 (3)C20—H20B0.97
C8—H80.93C21—C221.452 (3)
C9—C101.351 (4)C21—H21A0.97
C9—H90.93C21—H21B0.97
C10—C111.372 (4)C22—H22A0.97
C10—H100.93C22—H22B0.97
O1—Al1—C1103.39 (7)C11—C12—H12118.9
O1—Al1—C13103.90 (6)C14—C13—C18116.2 (2)
C1—Al1—C13113.25 (7)C14—C13—Al1122.74 (16)
O1—Al1—C7103.75 (7)C18—C13—Al1120.78 (16)
C1—Al1—C7114.23 (7)C15—C14—C13121.4 (3)
C13—Al1—C7116.27 (8)C15—C14—H14119.3
C19—O1—C22108.11 (17)C13—C14—H14119.3
C19—O1—Al1125.33 (13)C14—C15—C16120.6 (3)
C22—O1—Al1121.01 (12)C14—C15—H15119.7
C2—C1—C6115.02 (17)C16—C15—H15119.7
C2—C1—Al1123.27 (13)C15—C16—C17119.9 (3)
C6—C1—Al1121.56 (14)C15—C16—H16120.0
C3—C2—C1122.9 (2)C17—C16—H16120.0
C3—C2—H2118.6C18—C17—C16119.6 (3)
C1—C2—H2118.6C18—C17—H17120.2
C4—C3—C2120.0 (2)C16—C17—H17120.2
C4—C3—H3120.0C17—C18—C13122.2 (3)
C2—C3—H3120.0C17—C18—H18118.9
C3—C4—C5119.8 (2)C13—C18—H18118.9
C3—C4—H4120.1O1—C19—C20107.6 (2)
C5—C4—H4120.1O1—C19—H19A110.2
C6—C5—C4120.1 (2)C20—C19—H19A110.2
C6—C5—H5119.9O1—C19—H19B110.2
C4—C5—H5119.9C20—C19—H19B110.2
C5—C6—C1122.2 (2)H19A—C19—H19B108.5
C5—C6—H6118.9C19—C20—C21107.3 (2)
C1—C6—H6118.9C19—C20—H20A110.3
C12—C7—C8115.23 (18)C21—C20—H20A110.3
C12—C7—Al1123.50 (14)C19—C20—H20B110.3
C8—C7—Al1121.27 (15)C21—C20—H20B110.3
C9—C8—C7122.9 (2)H20A—C20—H20B108.5
C9—C8—H8118.6C22—C21—C20107.0 (2)
C7—C8—H8118.6C22—C21—H21A110.3
C10—C9—C8120.1 (2)C20—C21—H21A110.3
C10—C9—H9120.0C22—C21—H21B110.3
C8—C9—H9120.0C20—C21—H21B110.3
C9—C10—C11119.8 (2)H21A—C21—H21B108.6
C9—C10—H10120.1C21—C22—O1106.4 (2)
C11—C10—H10120.1C21—C22—H22A110.5
C12—C11—C10119.8 (2)O1—C22—H22A110.5
C12—C11—H11120.1C21—C22—H22B110.5
C10—C11—H11120.1O1—C22—H22B110.5
C7—C12—C11122.2 (2)H22A—C22—H22B108.6
C7—C12—H12118.9
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
C16—H16···Cg1i0.932.783.654 (4)156
C19—H19A···Cg1ii0.972.813.600 (4)139
Symmetry codes: (i) x+1, y, z; (ii) x+1, y+1/2, z+1/2.

Experimental details

Crystal data
Chemical formula[Al(C6H5)3(C4H8O)]
Mr330.38
Crystal system, space groupMonoclinic, P21/c
Temperature (K)293
a, b, c (Å)9.649 (2), 12.966 (3), 16.038 (4)
β (°) 104.210 (4)
V3)1945.1 (8)
Z4
Radiation typeMo Kα
µ (mm1)0.11
Crystal size (mm)0.58 × 0.42 × 0.21
Data collection
DiffractometerBruker SMART 1000 CCD
diffractometer
Absorption correctionMulti-scan
(SADABS; Sheldrick, 1996)
Tmin, Tmax0.799, 1.000
No. of measured, independent and
observed [I > 2σ(I)] reflections
10682, 3804, 2971
Rint0.023
(sin θ/λ)max1)0.618
Refinement
R[F2 > 2σ(F2)], wR(F2), S 0.048, 0.162, 1.33
No. of reflections3804
No. of parameters217
H-atom treatmentH-atom parameters constrained
Δρmax, Δρmin (e Å3)0.27, 0.23

Computer programs: SMART (Siemens, 1996), SAINT (Siemens, 1996), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), SHELXTL (Sheldrick, 2008).

Selected bond lengths (Å) top
Al1—O11.8972 (13)Al1—C131.9800 (18)
Al1—C11.9783 (18)Al1—C71.9809 (19)
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
C16—H16···Cg1i0.932.783.654 (4)156
C19—H19A···Cg1ii0.972.813.600 (4)139
Symmetry codes: (i) x+1, y, z; (ii) x+1, y+1/2, z+1/2.
 

Acknowledgements

This work was supported by a grant from the National Science Council of Taiwan (grant No. NSC 96-2113-M-005-007-MY3)

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