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Acta Cryst. (2012). E68, m1440-m1441    [ doi:10.1107/S1600536812043760 ]

catena-Poly[[[diaqua(1,10-phenanthroline-[kappa]2N,N')cobalt(II)]-[mu]-1H-benzimidazole-5,6-dicarboxylato-[kappa]2N3:O6] sesquihydrate}

D.-B. Xu, Y. Fang, D.-L. Jiang, Y. Zhu and M. Chen

Abstract top

In the title compound, {[Co(C9H4N2O4)(C12H8N2)(H2O)2]·1.5H2O}n, the CoII atom is hexacoordinated by one N atom and one O atom from two symmetry-related 1H-benzimidazole-5,6-dicarboxylate ligands, two N atoms from one 1,10-phenanthroline ligand (phen) and two water molecules. The dihedral angle between the 1H-benzimidazole-5,6-dicarboxylate and 1,10-phenanthroline ligands is 74.41 (4)°. The crystal packing is governed by intermolecular O-H...O and N-H...O hydrogen-bonding interactions. All water (coordinating and lattice) molecules take part in the hydrogen-bonding interactions. In addition, there are [pi]-[pi] stacking interactions between inversion-related phen ligands, the shortest centroid-centroid distance being 3.7536 (16) Å. One of the two lattice water molecules shows half-occupancy.

Comment top

The design and construction of supramolecular architectures have received considerable attention in recent years, in the structural investigation of 1H-benzimidazole-5,6-dicarboxylate complexes, it has been found that 1H-benzimidazole-5,6-dicarboxylate acid can function as a multidentate ligand (Lo et al., 2007; Gao et al., 2008; Yao et al., 2008), with versatile binding and coordination modes. 1,10-phenanthroline is also a good example for a bridging ligand that can link metal centers into extended networks, and a number of one-, two- and three- dimensional metal-1,10-phenanthroline frameworks have been generated (Chesnut et al., 1999). According to the procedure by Song et al. (Song et al., 2009), the reaction of 1H-benzimidazole-5,6-dicarboxylate acid with cobalt chloride in an alkaline aqueous solution yielded the CoII coordination polymer whose the crystal structure is reported here.

As illustrated in Fig. 1, the CoII atom exhibits a slightly distorted octahedral coordination sphere, defined by one N atom and one O atom [Co1i—N2 = 2.1304 (17) Å, Co1—O4 = 2.0582 (14) Å] from two different 1H-benzimidazole-5,6-dicarboxylate ligands, two N atoms [Co1—N3 = 2.1412 (18) Å, Co1—N4 = 2.1478 (18) Å] from one 1,10-phenanthroline ligand and two water molecules [Co1—OW1 = 2.1859 (15) Å, Co1—OW2 = 2.0689 (16) Å]. The metal atoms are linked by bidentate 1H-benzimidazole-5,6-dicarboxylate groups into one dimensional chain. Inter/intramolecular O—H···O and N—H···O hydrogen bonds between the carboxylate O atoms of 1H-benzimidazole-5,6-dicarboxylate and the coordinated water and solvent water molecules lead to a two-dimensional layer (Fig. 2). The layers are further self-assembled into a three-dimensional supramolecular network by intermolecular N-H···O hydrogen bonds between the imidazole units and carboxylate groups (Table 1). In the crystal structure, π-π stacking interactions between inversion-related phen ligands are also observed with a shortest centroid-centroid distance of 3.7536 (16) Å [between (N4/C16/C18/C19/C20/C21) and (N4/C16/C18/C19/C20/C21)viii with (viii) = 1-x, 1-y, 2-z)].

Related literature top

For general background to 1H-benzoimidazole-5,6-dicarboxylate complexes, see: Lo et al. (2007); Gao et al. (2008); Yao et al. (2008). For 1,10-phenanthroline as a bridging ligand, see: Chesnut et al. (1999). For a similar structure with NiII, see: Song et al. (2009).

Experimental top

According to the procedure by Song et al. (2009), a mixture of cobalt chloride (0.2 mmol), 1H-benzimidazole-5,6-dicarboxylate acid (0.2 mmol), 1,10-phenanthroline (0.2 mmol), NaOH (0.1 mmol) and H2O (15 mL) was placed in a 25 mL Teflon reactor, which was heated to 413 K for four days and then cooled to room temperature at a rate of 10 K h-1. The crystals obtained were washed with water and dryed in air.

Refinement top

Carbon and nitrogen bound H atoms were placed at calculated positions and were treated as riding on the parent C or N atoms with C—H = 0.93 Å, N—H = 0.86 Å, and Uiso(H) = 1.2Ueq (C,N). The water H-atoms were located in a difference Fourier map, and were refined with distance restraint of O—H = 0.85 (2) Å and Uiso(H) = 1.5Ueq (O).

Computing details top

Data collection: CrystalClear (Rigaku, 2008); cell refinement: CrystalClear (Rigaku, 2008); data reduction: CrystalClear (Rigaku, 2008); 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 displacement ellipsoids drawn at the 30% probability level [H-atoms are shown as spheres of arbitrary size; symmetry code: (i) = x+2, y-0.5, -z+1.5].
[Figure 2] Fig. 2. A view of the two-dimensional layer constructed by O-H···O and N-H···O hydrogen bonding interactions.
catena-Poly[[[diaqua(1,10-phenanthroline- κ2N,N')cobalt(II)]-µ-1H-benzimidazole-5,6- dicarboxylato-κ2N3:O6] sesquihydrate} top
Crystal data top
[Co(C9H4N2O4)(C12H8N2)(H2O)2]·1.5H2OF(000) = 1040
Mr = 506.33Dx = 1.615 Mg m−3
Monoclinic, P21/cMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybcCell parameters from 4850 reflections
a = 9.7250 (11) Åθ = 2.1–27.5°
b = 11.3956 (13) ŵ = 0.88 mm−1
c = 19.296 (2) ÅT = 173 K
β = 103.109 (2)°Prism, red
V = 2082.7 (4) Å30.30 × 0.24 × 0.20 mm
Z = 4
Data collection top
Rigaku Saturn724+
diffractometer		4797 independent reflections
Radiation source: fine-focus sealed tube3761 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.043
ω scansθmax = 27.5°, θmin = 2.1°
Absorption correction: multi-scan
(CrystalClear; Rigaku, 2008)		h = −11→12
Tmin = 0.776, Tmax = 0.838k = −14→14
17888 measured reflectionsl = −25→25
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)] = 0.036Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.090H-atom parameters constrained
S = 1.05 w = 1/[σ2(Fo2) + (0.0355P)2 + 1.1164P]
where P = (Fo2 + 2Fc2)/3
4796 reflections(Δ/σ)max = 0.002
331 parametersΔρmax = 0.42 e Å−3
12 restraintsΔρmin = −0.49 e Å−3
Crystal data top
[Co(C9H4N2O4)(C12H8N2)(H2O)2]·1.5H2OV = 2082.7 (4) Å3
Mr = 506.33Z = 4
Monoclinic, P21/cMo Kα radiation
a = 9.7250 (11) ŵ = 0.88 mm−1
b = 11.3956 (13) ÅT = 173 K
c = 19.296 (2) Å0.30 × 0.24 × 0.20 mm
β = 103.109 (2)°
Data collection top
Rigaku Saturn724+
diffractometer		3761 reflections with I > 2σ(I)
Absorption correction: multi-scan
(CrystalClear; Rigaku, 2008)		Rint = 0.043
Tmin = 0.776, Tmax = 0.838θmax = 27.5°
17888 measured reflectionsStandard reflections: ?
4797 independent reflections
Refinement top
R[F2 > 2σ(F2)] = 0.036H-atom parameters constrained
wR(F2) = 0.090Δρmax = 0.42 e Å−3
S = 1.05Δρmin = −0.49 e Å−3
4796 reflectionsAbsolute structure: ?
331 parametersFlack parameter: ?
12 restraintsRogers parameter: ?
Special details top

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

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/UeqOcc. (<1)
C11.3426 (2)0.48546 (18)0.68601 (11)0.0196 (4)
H11.43580.48780.68160.023*
C21.1479 (2)0.42156 (18)0.71347 (10)0.0162 (4)
C31.1260 (2)0.53227 (18)0.68112 (10)0.0158 (4)
C41.0404 (2)0.36045 (18)0.73494 (11)0.0184 (4)
H41.05620.28670.75590.022*
C50.9938 (2)0.58516 (18)0.66973 (10)0.0166 (4)
H50.97790.65810.64780.020*
C60.8856 (2)0.52623 (18)0.69192 (10)0.0155 (4)
C70.9086 (2)0.41394 (18)0.72378 (10)0.0162 (4)
C80.7474 (2)0.59071 (19)0.68525 (11)0.0202 (4)
C90.7931 (2)0.34525 (17)0.74648 (11)0.0173 (4)
C100.9568 (2)0.1841 (2)0.99354 (12)0.0285 (5)
H100.96090.11200.97150.034*
C111.0612 (3)0.2105 (3)1.05439 (13)0.0372 (6)
H111.13330.15731.07170.045*
C121.0563 (3)0.3156 (3)1.08825 (13)0.0393 (7)
H121.12540.33461.12850.047*
C130.9455 (3)0.3945 (2)1.06151 (12)0.0327 (6)
C140.8459 (2)0.3620 (2)0.99951 (11)0.0243 (5)
C150.9306 (3)0.5065 (3)1.09338 (13)0.0416 (7)
H150.99680.52901.13400.050*
C160.7329 (2)0.4407 (2)0.96903 (11)0.0242 (5)
C170.8237 (3)0.5794 (3)1.06611 (14)0.0420 (7)
H170.81600.65041.08870.050*
C180.7208 (3)0.5495 (2)1.00241 (13)0.0310 (6)
C190.6081 (3)0.6230 (2)0.96966 (15)0.0392 (6)
H190.59640.69580.98940.047*
C200.5165 (3)0.5876 (2)0.90918 (15)0.0380 (6)
H200.44160.63540.88750.046*
C210.5363 (3)0.4778 (2)0.87982 (13)0.0321 (6)
H210.47310.45450.83840.039*
N11.28761 (18)0.39420 (15)0.71539 (9)0.0195 (4)
H1A1.33110.33100.73210.023*
N21.25154 (17)0.57083 (15)0.66427 (9)0.0172 (4)
N30.85195 (19)0.25700 (17)0.96580 (9)0.0220 (4)
N40.64096 (19)0.40575 (16)0.90847 (9)0.0237 (4)
O10.71519 (16)0.66394 (16)0.63497 (9)0.0359 (4)
O20.67695 (16)0.57174 (14)0.73083 (8)0.0279 (4)
O30.68891 (16)0.31022 (14)0.69976 (8)0.0250 (4)
O40.81500 (15)0.32296 (12)0.81218 (7)0.0191 (3)
OW10.51234 (15)0.24220 (13)0.77827 (8)0.0184 (3)
H1C0.56150.24960.74720.028*
H1D0.45890.18240.77230.028*
OW20.55560 (16)0.15584 (16)0.92303 (8)0.0280 (4)
H2C0.46770.16570.90390.042*
H2D0.57340.15560.96730.042*
OW30.5829 (3)0.1182 (2)1.06327 (10)0.0635 (7)
H3C0.64030.13831.10270.095*
H3D0.52100.07011.06880.095*
OW41.2843 (5)0.0672 (3)1.0004 (2)0.0451 (10)0.50
H4C1.31670.00040.99240.068*0.50
H4D1.26090.11160.96370.068*0.50
Co10.69156 (3)0.23795 (2)0.870080 (14)0.01613 (9)
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
C10.0132 (10)0.0213 (11)0.0250 (11)0.0005 (8)0.0060 (8)−0.0015 (9)
C20.0125 (10)0.0178 (11)0.0182 (10)0.0033 (8)0.0030 (8)−0.0001 (8)
C30.0138 (10)0.0170 (10)0.0175 (10)−0.0015 (8)0.0052 (8)−0.0020 (8)
C40.0180 (10)0.0161 (10)0.0210 (10)0.0003 (8)0.0046 (8)0.0028 (8)
C50.0161 (10)0.0153 (10)0.0185 (10)0.0020 (8)0.0039 (8)0.0007 (8)
C60.0119 (10)0.0168 (10)0.0174 (10)0.0010 (8)0.0024 (8)−0.0010 (8)
C70.0142 (10)0.0184 (10)0.0171 (10)−0.0038 (8)0.0057 (8)−0.0022 (8)
C80.0133 (10)0.0188 (11)0.0284 (11)−0.0023 (9)0.0044 (9)−0.0007 (9)
C90.0152 (10)0.0137 (10)0.0234 (10)0.0012 (8)0.0056 (8)−0.0010 (8)
C100.0252 (12)0.0360 (14)0.0239 (11)0.0017 (10)0.0049 (10)0.0073 (10)
C110.0267 (13)0.0548 (18)0.0270 (13)0.0008 (12)0.0000 (10)0.0131 (12)
C120.0298 (14)0.0624 (19)0.0221 (12)−0.0170 (13)−0.0018 (10)0.0036 (12)
C130.0300 (13)0.0483 (16)0.0196 (11)−0.0168 (12)0.0056 (10)−0.0046 (11)
C140.0245 (12)0.0306 (13)0.0188 (11)−0.0090 (10)0.0070 (9)−0.0028 (9)
C150.0444 (17)0.0558 (18)0.0243 (12)−0.0224 (15)0.0071 (12)−0.0168 (12)
C160.0269 (12)0.0262 (12)0.0219 (11)−0.0056 (10)0.0107 (9)−0.0048 (9)
C170.0540 (18)0.0423 (17)0.0350 (14)−0.0205 (14)0.0213 (13)−0.0217 (12)
C180.0363 (14)0.0304 (13)0.0315 (13)−0.0106 (11)0.0185 (11)−0.0100 (10)
C190.0492 (17)0.0260 (13)0.0501 (16)−0.0004 (12)0.0273 (14)−0.0116 (12)
C200.0418 (15)0.0285 (14)0.0464 (16)0.0112 (12)0.0157 (13)−0.0016 (12)
C210.0322 (14)0.0299 (14)0.0337 (13)0.0068 (11)0.0062 (11)−0.0038 (11)
N10.0151 (9)0.0175 (9)0.0266 (9)0.0037 (7)0.0058 (7)0.0038 (7)
N20.0122 (8)0.0169 (9)0.0230 (9)−0.0003 (7)0.0052 (7)−0.0012 (7)
N30.0179 (9)0.0298 (11)0.0179 (9)−0.0026 (8)0.0030 (7)0.0025 (8)
N40.0235 (10)0.0254 (10)0.0234 (9)0.0014 (8)0.0080 (8)−0.0030 (8)
O10.0180 (8)0.0454 (11)0.0462 (11)0.0108 (8)0.0115 (8)0.0256 (9)
O20.0198 (8)0.0337 (9)0.0343 (9)0.0079 (7)0.0149 (7)0.0098 (7)
O30.0210 (8)0.0329 (9)0.0204 (8)−0.0104 (7)0.0030 (6)0.0012 (7)
O40.0172 (7)0.0199 (8)0.0198 (7)−0.0028 (6)0.0036 (6)0.0014 (6)
OW10.0139 (7)0.0203 (8)0.0216 (7)−0.0007 (6)0.0055 (6)0.0004 (6)
OW20.0193 (8)0.0442 (11)0.0215 (8)0.0020 (8)0.0066 (7)0.0014 (8)
OW30.101 (2)0.0582 (16)0.0250 (10)0.0048 (13)0.0016 (11)−0.0050 (10)
OW40.057 (3)0.031 (2)0.053 (2)0.0131 (19)0.027 (2)0.0075 (18)
Co10.01335 (15)0.01823 (16)0.01708 (14)0.00037 (11)0.00402 (10)−0.00106 (11)
Geometric parameters (Å, º) top
C1—N21.318 (3)C15—C171.342 (4)
C1—N11.352 (3)C15—H150.9300
C1—H10.9300C16—N41.360 (3)
C2—N11.387 (2)C16—C181.414 (3)
C2—C41.395 (3)C17—C181.439 (4)
C2—C31.402 (3)C17—H170.9300
C3—C51.392 (3)C18—C191.411 (4)
C3—N21.403 (2)C19—C201.359 (4)
C4—C71.391 (3)C19—H190.9300
C4—H40.9300C20—C211.405 (3)
C5—C61.395 (3)C20—H200.9300
C5—H50.9300C21—N41.327 (3)
C6—C71.415 (3)C21—H210.9300
C6—C81.512 (3)N1—H1A0.8600
C7—C91.513 (3)N2—Co1i2.1304 (17)
C8—O21.250 (3)N3—Co12.1412 (18)
C8—O11.264 (3)N4—Co12.1478 (18)
C9—O31.259 (2)O4—Co12.0582 (14)
C9—O41.263 (2)OW1—Co12.1859 (15)
C10—N31.330 (3)OW1—H1C0.85
C10—C111.400 (3)OW1—H1D0.85
C10—H100.9300OW2—Co12.0689 (16)
C11—C121.370 (4)OW2—H2C0.86
C11—H110.9300OW2—H2D0.83
C12—C131.409 (4)OW3—H3C0.87
C12—H120.9300OW3—H3D0.84
C13—C141.408 (3)OW4—H4C0.85
C13—C151.438 (4)OW4—H4D0.86
C14—N31.369 (3)Co1—N2ii2.1304 (17)
C14—C161.437 (3)
N2—C1—N1113.57 (18)C18—C17—H17119.5
N2—C1—H1123.2C19—C18—C16116.9 (2)
N1—C1—H1123.2C19—C18—C17124.1 (2)
N1—C2—C4132.52 (19)C16—C18—C17118.9 (2)
N1—C2—C3105.23 (17)C20—C19—C18120.0 (2)
C4—C2—C3122.24 (18)C20—C19—H19120.0
C5—C3—C2120.06 (18)C18—C19—H19120.0
C5—C3—N2130.47 (19)C19—C20—C21119.2 (2)
C2—C3—N2109.46 (17)C19—C20—H20120.4
C7—C4—C2117.49 (19)C21—C20—H20120.4
C7—C4—H4121.3N4—C21—C20123.0 (2)
C2—C4—H4121.3N4—C21—H21118.5
C3—C5—C6118.51 (19)C20—C21—H21118.5
C3—C5—H5120.7C1—N1—C2107.13 (17)
C6—C5—H5120.7C1—N1—H1A126.4
C5—C6—C7120.88 (18)C2—N1—H1A126.4
C5—C6—C8117.26 (18)C1—N2—C3104.61 (17)
C7—C6—C8121.72 (17)C1—N2—Co1i123.74 (14)
C4—C7—C6120.81 (18)C3—N2—Co1i130.77 (13)
C4—C7—C9116.55 (18)C10—N3—C14117.8 (2)
C6—C7—C9122.64 (18)C10—N3—Co1128.73 (16)
O2—C8—O1125.1 (2)C14—N3—Co1113.40 (14)
O2—C8—C6118.34 (19)C21—N4—C16118.0 (2)
O1—C8—C6116.54 (18)C21—N4—Co1128.41 (16)
O3—C9—O4125.49 (19)C16—N4—Co1113.55 (15)
O3—C9—C7119.11 (18)C9—O4—Co1130.78 (13)
O4—C9—C7115.30 (18)Co1—OW1—H1C96
N3—C10—C11123.2 (2)Co1—OW1—H1D116.4
N3—C10—H10118.4H1C—OW1—H1D114.1
C11—C10—H10118.4Co1—OW2—H2C114
C12—C11—C10119.4 (2)Co1—OW2—H2D120.0
C12—C11—H11120.3H2C—OW2—H2D113.4
C10—C11—H11120.3H3C—OW3—H3D113.6
C11—C12—C13119.3 (2)H4C—OW4—H4D114.7
C11—C12—H12120.3O4—Co1—OW2176.10 (6)
C13—C12—H12120.3O4—Co1—N2ii91.56 (6)
C14—C13—C12117.7 (2)OW2—Co1—N2ii89.40 (7)
C14—C13—C15118.7 (2)O4—Co1—N391.15 (6)
C12—C13—C15123.6 (2)OW2—Co1—N392.41 (7)
N3—C14—C13122.6 (2)N2ii—Co1—N399.75 (7)
N3—C14—C16117.51 (19)O4—Co1—N488.60 (6)
C13—C14—C16119.9 (2)OW2—Co1—N490.59 (7)
C17—C15—C13121.7 (2)N2ii—Co1—N4177.70 (7)
C17—C15—H15119.1N3—Co1—N477.95 (7)
C13—C15—H15119.1O4—Co1—OW190.32 (6)
N4—C16—C18122.8 (2)OW2—Co1—OW185.92 (6)
N4—C16—C14117.5 (2)N2ii—Co1—OW189.04 (6)
C18—C16—C14119.7 (2)N3—Co1—OW1171.05 (6)
C15—C17—C18121.0 (2)N4—Co1—OW193.26 (6)
C15—C17—H17119.5
Symmetry codes: (i) −x+2, y+1/2, −z+3/2; (ii) −x+2, y−1/2, −z+3/2.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
OW1—H1C···O30.851.832.650 (2)160
OW2—H2D···OW30.831.882.693 (2)164
N1—H1A···OW1iii0.862.052.837 (2)151
OW1—H1D···O2iv0.851.822.654 (2)168
OW2—H2C···O1iv0.861.772.619 (2)172 (3)
OW3—H3C···O3v0.871.922.726 (3)155 (3)
OW3—H3D···OW4vi0.842.382.940 (5)125
OW4—H4C···OW3vii0.852.102.891 (4)154 (5)
OW4—H4C···OW2vii0.852.543.166 (4)131 (5)
OW4—H4D···O1ii0.862.062.837 (4)151
Symmetry codes: (ii) −x+2, y−1/2, −z+3/2; (iii) x+1, y, z; (iv) −x+1, y−1/2, −z+3/2; (v) x, −y+1/2, z+1/2; (vi) x−1, y, z; (vii) −x+2, −y, −z+2.
Selected bond lengths (Å) top
N2—Co1i2.1304 (17)OW1—Co12.1859 (15)
N3—Co12.1412 (18)OW2—Co12.0689 (16)
N4—Co12.1478 (18)Co1—N2ii2.1304 (17)
O4—Co12.0582 (14)
Symmetry codes: (i) −x+2, y+1/2, −z+3/2; (ii) −x+2, y−1/2, −z+3/2.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
OW1—H1C···O30.851.832.650 (2)160
OW2—H2D···OW30.831.882.693 (2)164.3
N1—H1A···OW1iii0.862.052.837 (2)151.2
OW1—H1D···O2iv0.851.822.654 (2)168.4
OW2—H2C···O1iv0.861.772.619 (2)172 (3)
OW3—H3C···O3v0.871.922.726 (3)155 (3)
OW3—H3D···OW4vi0.842.382.940 (5)124.8
OW4—H4C···OW3vii0.852.102.891 (4)154 (5)
OW4—H4C···OW2vii0.852.543.166 (4)131 (5)
OW4—H4D···O1ii0.862.062.837 (4)150.5
Symmetry codes: (ii) −x+2, y−1/2, −z+3/2; (iii) x+1, y, z; (iv) −x+1, y−1/2, −z+3/2; (v) x, −y+1/2, z+1/2; (vi) x−1, y, z; (vii) −x+2, −y, −z+2.
Acknowledgements top

The authors thank Jiangsu University for supporting this work.

references
References top

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