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Acta Cryst. (2011). E67, m848    [ doi:10.1107/S1600536811020204 ]

Poly[tri-[mu]2-aqua-([mu]3-pyridine-2,4-dicarboxylato-[kappa]4N,O2:O2:O2')barium]

H. Pasdar, S. Siabi, B. Notash, H. Aghabozorg and N. Foroughifar

Abstract top

In the polymeric title compound, [Ba(C7H3NO4)(H2O)3]n, the BaII ion is ten-coordinated in an NO9 environment by one N atom and three O atoms from three pyridine-2,4-dicarboxylate (pydc) ligands and six water molecules. The [mu]3-pydc ligands and the bridging water molecules connect the Ba atoms into a layer parallel to (100). The crystal packing is stabilized by O-H...O and C-H...O hydrogen bonds.

Comment top

Pyridine dicarboxylic acid derivatives, depending on the composition and situation of carboxylic groups and the number of deprotonated carboxylic groups, can form wide variety of compounds from organic proton transfer compounds (Aghabozorg et al., 2011a,c,d) to discrete coordination compounds (Aghabozorg et al., 2008, 2011b; Noro et al., 2005; Pasdar et al., 2011a) and coordination polymers (Pasdar et al., 2011b).

The asymmetric unit of the title compound is shown in Fig. 1. Two carboxylate groups of the pyridine-2,4-dicarboxylate (pydc) ligand are deprotonated and BaII ion is ten-coordinated in an NO9 environment (Fig. 2). The crystal structure shows that the compound is a two-dimensional polymer (Fig. 3). O—H···O and C—H···O hydrogen bonds stabilize the crystal packing (Table 1).

Related literature top

For related compounds with pyridine dicarboxylic acid derivatives, see: Aghabozorg et al. (2008, 2011a,b,c,d); Noro et al. (2005); Pasdar et al. (2011a,b); Wang et al. (2007).

Experimental top

A mixture of Ba(NO3)2 (0.132 g), pyridine-2,4-dicarboxylic acid (0.085 g), 2,2'-bipyridine (0.156 g) in H2O (60 ml) was stirred at 40°C for 1 h. The solution was filtered, and the filtrate was stand at room temperature. After two weaks, colorless block-shaped crystals of the title compound were obtained.

Refinement top

H atoms of water molecules were found in a difference Fourier map and refined isotropically. H6B was refined with a distance restraint of O—H = 0.75 (3). C-bound H atoms were positioned geometrically and refined as riding atoms, with C—H = 0.93 Å and Uiso(H) = 1.2Ueq(C). The highest residual electron density was found at 0.80 Å from Ba1 atom and the deepest hole at 0.80 Å from Ba1 atom.

Computing details top

Data collection: X-AREA (Stoe & Cie, 2005); cell refinement: X-AREA (Stoe & Cie, 2005); data reduction: X-AREA (Stoe & Cie, 2005); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEP-3 (Farrugia, 1997); software used to prepare material for publication: WinGX (Farrugia, 1999).

Figures top
[Figure 1] Fig. 1. The asymmetric unit of the title compound, with displacement ellipsoids drawn at 50% probability level.
[Figure 2] Fig. 2. The coordination environment around BaII ion in the title compound. [Symmetry codes: (i) -x, -y+1, -z; (ii) x, -y+3/2, z-1/2; (iii) x, y, z-1; (iv) x, -y+3/2, z+1/2.]
[Figure 3] Fig. 3. A view of the two-dimensional structure of the title compound viewed down the a axis. H atoms have been omitted for clarity.
Poly[tri-µ2-aqua-(µ3-pyridine-2,4-dicarboxylato- κ4N,O2:O2:O2')barium] top
Crystal data top
[Ba(C7H3NO4)(H2O)3]F(000) = 680
Mr = 356.48Dx = 2.369 Mg m−3
Monoclinic, P21/cMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybcCell parameters from 2681 reflections
a = 11.079 (2) Åθ = 2.4–29.1°
b = 13.714 (3) ŵ = 4.00 mm−1
c = 6.5961 (13) ÅT = 298 K
β = 94.13 (3)°Block, colorless
V = 999.6 (3) Å30.39 × 0.38 × 0.33 mm
Z = 4
Data collection top
Stoe IPDS-2T
diffractometer		2681 independent reflections
Radiation source: fine-focus sealed tube2515 reflections with I > 2σ(I)
graphiteRint = 0.041
ω scansθmax = 29.1°, θmin = 2.4°
Absorption correction: numerical
(X-SHAPE and X-RED32; Stoe & Cie, 2005)		h = −15→13
Tmin = 0.410, Tmax = 0.460k = −18→18
7321 measured reflectionsl = −9→9
Refinement top
Refinement on F2Secondary atom site location: difference Fourier map
Least-squares matrix: fullHydrogen site location: inferred from neighbouring sites
R[F2 > 2σ(F2)] = 0.023H atoms treated by a mixture of independent and constrained refinement
wR(F2) = 0.057 w = 1/[σ2(Fo2) + (0.0343P)2 + 0.2871P]
where P = (Fo2 + 2Fc2)/3
S = 1.10(Δ/σ)max = 0.002
2681 reflectionsΔρmax = 2.20 e Å−3
170 parametersΔρmin = −0.60 e Å−3
1 restraintExtinction correction: SHELXL97 (Sheldrick, 2008), Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4
Primary atom site location: structure-invariant direct methodsExtinction coefficient: 0.0182 (7)
Crystal data top
[Ba(C7H3NO4)(H2O)3]V = 999.6 (3) Å3
Mr = 356.48Z = 4
Monoclinic, P21/cMo Kα radiation
a = 11.079 (2) ŵ = 4.00 mm−1
b = 13.714 (3) ÅT = 298 K
c = 6.5961 (13) Å0.39 × 0.38 × 0.33 mm
β = 94.13 (3)°
Data collection top
Stoe IPDS-2T
diffractometer		2681 independent reflections
Absorption correction: numerical
(X-SHAPE and X-RED32; Stoe & Cie, 2005)		2515 reflections with I > 2σ(I)
Tmin = 0.410, Tmax = 0.460Rint = 0.041
7321 measured reflectionsθmax = 29.1°
Refinement top
R[F2 > 2σ(F2)] = 0.023H atoms treated by a mixture of independent and constrained refinement
wR(F2) = 0.057Δρmax = 2.20 e Å−3
S = 1.10Δρmin = −0.60 e Å−3
2681 reflectionsAbsolute structure: ?
170 parametersFlack parameter: ?
1 restraintRogers parameter: ?
Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/Ueq
H6B−0.211 (3)0.577 (3)−0.019 (6)0.045 (11)*
O6−0.16582 (16)0.54989 (14)0.0517 (3)0.0245 (3)
O7−0.09028 (16)0.75423 (14)−0.1846 (3)0.0234 (3)
O50.02949 (17)0.45857 (14)0.2957 (3)0.0233 (3)
Ba10.074526 (10)0.638147 (8)0.073630 (17)0.01506 (7)
C20.4566 (2)0.63768 (15)0.5688 (4)0.0179 (4)
H20.46720.65080.70730.021*
N10.32012 (18)0.61649 (17)0.2709 (3)0.0223 (4)
C10.3411 (2)0.63417 (14)0.4713 (4)0.0166 (4)
C50.4167 (2)0.6023 (2)0.1646 (4)0.0290 (5)
H50.40400.59070.02580.035*
O10.13032 (15)0.66224 (14)0.4870 (3)0.0227 (3)
O20.24279 (19)0.65068 (17)0.7766 (3)0.0341 (5)
C60.2301 (2)0.64984 (16)0.5894 (4)0.0175 (4)
C70.6829 (2)0.62030 (18)0.5578 (4)0.0206 (4)
C30.5562 (2)0.62132 (17)0.4560 (4)0.0186 (4)
C40.5348 (2)0.6037 (2)0.2492 (4)0.0268 (5)
H40.59900.59300.16850.032*
O30.69566 (19)0.64657 (16)0.7382 (3)0.0340 (5)
O40.76764 (15)0.59321 (15)0.4537 (3)0.0278 (4)
H5A−0.019 (4)0.438 (3)0.361 (6)0.047 (11)*
H5B0.087 (4)0.448 (3)0.364 (6)0.048 (12)*
H7B−0.116 (4)0.792 (3)−0.116 (6)0.039 (10)*
H6A−0.209 (4)0.550 (3)0.164 (6)0.043 (10)*
H7A−0.155 (4)0.723 (3)−0.215 (7)0.060 (13)*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
O60.0208 (8)0.0317 (9)0.0211 (8)0.0043 (7)0.0026 (7)0.0017 (7)
O70.0204 (8)0.0251 (8)0.0246 (8)−0.0010 (7)0.0016 (7)−0.0014 (7)
O50.0192 (8)0.0316 (9)0.0191 (8)0.0005 (7)0.0014 (7)0.0040 (7)
Ba10.01347 (9)0.01844 (9)0.01335 (10)0.00009 (4)0.00141 (5)−0.00001 (4)
C20.0153 (10)0.0255 (11)0.0128 (10)−0.0002 (7)0.0004 (8)0.0007 (7)
N10.0169 (9)0.0350 (10)0.0148 (9)0.0023 (8)−0.0012 (7)−0.0034 (8)
C10.0128 (9)0.0217 (10)0.0156 (10)0.0007 (7)0.0026 (8)0.0003 (7)
C50.0177 (10)0.0535 (16)0.0157 (10)0.0048 (11)0.0003 (8)−0.0053 (11)
O10.0135 (7)0.0330 (8)0.0218 (8)0.0041 (6)0.0017 (6)0.0010 (7)
O20.0216 (9)0.0644 (14)0.0170 (9)−0.0006 (8)0.0052 (7)−0.0048 (8)
C60.0138 (9)0.0202 (9)0.0189 (10)−0.0006 (7)0.0042 (8)−0.0010 (8)
C70.0137 (9)0.0240 (9)0.0237 (11)−0.0005 (8)−0.0013 (8)0.0044 (9)
C30.0146 (9)0.0227 (9)0.0182 (10)0.0006 (8)0.0002 (8)0.0014 (8)
C40.0160 (10)0.0466 (15)0.0179 (11)0.0017 (10)0.0026 (8)−0.0032 (10)
O30.0237 (10)0.0518 (12)0.0252 (10)0.0041 (8)−0.0065 (8)−0.0064 (8)
O40.0156 (7)0.0406 (10)0.0272 (9)0.0030 (7)0.0023 (6)0.0028 (8)
Geometric parameters (Å, °) top
O6—H6B0.75 (3)C2—C11.391 (3)
O6—H6A0.91 (4)C2—C31.393 (3)
O7—H7B0.75 (4)C2—H20.9300
O7—H7A0.85 (5)N1—C51.336 (3)
O5—H5A0.77 (5)N1—C11.347 (3)
O5—H5B0.77 (5)C1—C61.518 (3)
Ba1—O12.7720 (19)C5—C41.385 (3)
Ba1—O2i2.805 (2)C5—H50.9300
Ba1—O1ii2.8728 (19)O1—C61.265 (3)
Ba1—O7iii2.9111 (18)O2—C61.233 (3)
Ba1—O72.8846 (19)C7—O31.242 (3)
Ba1—O6iv2.9115 (19)C7—O41.258 (3)
Ba1—O5iv2.935 (2)C7—C31.511 (3)
Ba1—O52.9263 (19)C3—C41.389 (3)
Ba1—N12.945 (2)C4—H40.9300
Ba1—O62.9190 (19)
Ba1iv—O6—Ba192.69 (5)O7—Ba1—N1147.33 (6)
Ba1iv—O6—H6B119 (3)O7iii—Ba1—N1113.88 (6)
Ba1—O6—H6B113 (3)O6iv—Ba1—N172.99 (6)
Ba1iv—O6—H6A117 (2)O6—Ba1—N1142.22 (6)
Ba1—O6—H6A119 (2)O5—Ba1—N183.07 (6)
H6B—O6—H6A98 (4)O5iv—Ba1—N1128.03 (6)
Ba1—O7—Ba1ii102.01 (6)O1—Ba1—C6ii77.88 (6)
Ba1—O7—H7B106 (3)O2i—Ba1—C6ii66.76 (6)
Ba1ii—O7—H7B105 (3)O1ii—Ba1—C6ii21.45 (5)
Ba1—O7—H7A110 (3)O7—Ba1—C6ii80.97 (6)
Ba1ii—O7—H7A133 (3)O7iii—Ba1—C6ii83.42 (5)
H7B—O7—H7A98 (4)O6iv—Ba1—C6ii125.75 (5)
Ba1—O5—Ba1iv92.06 (5)O6—Ba1—C6ii145.09 (5)
Ba1—O5—H5A138 (3)O5—Ba1—C6ii144.49 (6)
Ba1iv—O5—H5A93 (3)O5iv—Ba1—C6ii125.70 (5)
Ba1—O5—H5B107 (3)N1—Ba1—C6ii67.67 (6)
Ba1iv—O5—H5B131 (3)O1—Ba1—Ba1iv112.92 (4)
H5A—O5—H5B101 (4)O2i—Ba1—Ba1iv99.38 (5)
O1—Ba1—O2i124.57 (6)O1ii—Ba1—Ba1iv153.68 (4)
O1—Ba1—O1ii92.64 (5)O7—Ba1—Ba1iv98.00 (4)
O2i—Ba1—O1ii68.86 (6)O7iii—Ba1—Ba1iv109.33 (4)
O1—Ba1—O7127.00 (5)O6iv—Ba1—Ba1iv43.72 (4)
O2i—Ba1—O788.74 (6)O6—Ba1—Ba1iv43.58 (4)
O1ii—Ba1—O759.52 (5)O5—Ba1—Ba1iv44.06 (4)
O1—Ba1—O7iii60.32 (5)O5iv—Ba1—Ba1iv43.89 (4)
O2i—Ba1—O7iii145.70 (6)N1—Ba1—Ba1iv110.18 (5)
O1ii—Ba1—O7iii77.19 (5)C1—C2—C3119.0 (2)
O7—Ba1—O7iii69.42 (3)C1—C2—H2120.5
O1—Ba1—O6iv109.12 (5)C3—C2—H2120.5
O2i—Ba1—O6iv66.05 (6)C5—N1—C1116.9 (2)
O1ii—Ba1—O6iv134.70 (5)C5—N1—Ba1122.17 (16)
O7—Ba1—O6iv122.63 (6)C1—N1—Ba1120.28 (14)
O7iii—Ba1—O6iv148.11 (5)N1—C1—C2123.3 (2)
O1—Ba1—O6103.66 (6)N1—C1—C6116.1 (2)
O2i—Ba1—O6129.84 (6)C2—C1—C6120.7 (2)
O1ii—Ba1—O6126.43 (5)N1—C5—C4123.8 (2)
O7—Ba1—O670.32 (5)N1—C5—H5118.1
O7iii—Ba1—O668.38 (5)C4—C5—H5118.1
O6iv—Ba1—O687.31 (5)C6—O1—Ba1129.32 (14)
O1—Ba1—O569.02 (5)C6—O1—Ba1iii102.37 (14)
O2i—Ba1—O5123.15 (6)Ba1—O1—Ba1iii105.86 (6)
O1ii—Ba1—O5161.49 (5)C6—O2—Ba1v131.70 (16)
O7—Ba1—O5129.53 (5)O2—C6—O1124.5 (2)
O7iii—Ba1—O590.93 (5)O2—C6—C1118.5 (2)
O6iv—Ba1—O558.13 (5)O1—C6—C1117.0 (2)
O6—Ba1—O559.20 (6)O2—C6—Ba1iii92.52 (15)
O1—Ba1—O5iv156.43 (5)O1—C6—Ba1iii56.18 (12)
O2i—Ba1—O5iv71.89 (6)C1—C6—Ba1iii122.45 (13)
O1ii—Ba1—O5iv110.05 (5)O3—C7—O4124.8 (2)
O7—Ba1—O5iv64.24 (6)O3—C7—C3117.5 (2)
O7iii—Ba1—O5iv117.35 (5)O4—C7—C3117.8 (2)
O6iv—Ba1—O5iv59.19 (5)C4—C3—C2117.8 (2)
O6—Ba1—O5iv57.95 (5)C4—C3—C7121.5 (2)
O5—Ba1—O5iv87.94 (5)C2—C3—C7120.7 (2)
O1—Ba1—N156.21 (6)C5—C4—C3119.2 (2)
O2i—Ba1—N171.07 (6)C5—C4—H4120.4
O1ii—Ba1—N188.79 (6)C3—C4—H4120.4
Ba1ii—O7—Ba1—O1108.60 (7)C5—N1—C1—C2−0.2 (4)
Ba1ii—O7—Ba1—O2i−25.36 (7)Ba1—N1—C1—C2−171.43 (15)
Ba1ii—O7—Ba1—O1ii40.72 (5)C5—N1—C1—C6−179.8 (2)
Ba1ii—O7—Ba1—O7iii127.63 (8)Ba1—N1—C1—C69.0 (3)
Ba1ii—O7—Ba1—O6iv−85.61 (7)C3—C2—C1—N1−0.6 (3)
Ba1ii—O7—Ba1—O6−158.86 (7)C3—C2—C1—C6178.93 (19)
Ba1ii—O7—Ba1—O5−159.07 (5)C1—N1—C5—C40.7 (4)
Ba1ii—O7—Ba1—O5iv−95.80 (7)Ba1—N1—C5—C4171.8 (2)
Ba1ii—O7—Ba1—N125.18 (13)O2i—Ba1—O1—C6−24.9 (2)
Ba1ii—O7—Ba1—C6ii41.31 (6)O1ii—Ba1—O1—C6−90.95 (18)
Ba1ii—O7—Ba1—Ba1iv−124.66 (5)O7—Ba1—O1—C6−144.00 (18)
Ba1iv—O6—Ba1—O1−109.03 (5)O7iii—Ba1—O1—C6−164.6 (2)
Ba1iv—O6—Ba1—O2i55.35 (9)O6iv—Ba1—O1—C648.6 (2)
Ba1iv—O6—Ba1—O1ii147.19 (5)O6—Ba1—O1—C6140.52 (19)
Ba1iv—O6—Ba1—O7126.16 (6)O5—Ba1—O1—C691.6 (2)
Ba1iv—O6—Ba1—O7iii−158.90 (7)O5iv—Ba1—O1—C6104.5 (2)
Ba1iv—O6—Ba1—O6iv0.0N1—Ba1—O1—C6−4.18 (18)
Ba1iv—O6—Ba1—O5−54.04 (5)C6ii—Ba1—O1—C6−75.3 (2)
Ba1iv—O6—Ba1—O5iv54.86 (5)Ba1iv—Ba1—O1—C695.48 (19)
Ba1iv—O6—Ba1—N1−57.40 (10)O2i—Ba1—O1—Ba1iii96.43 (7)
Ba1iv—O6—Ba1—C6ii162.67 (7)O1ii—Ba1—O1—Ba1iii30.42 (8)
Ba1iv—O5—Ba1—O1174.89 (6)O7—Ba1—O1—Ba1iii−22.64 (8)
Ba1iv—O5—Ba1—O2i−66.73 (8)O7iii—Ba1—O1—Ba1iii−43.21 (5)
Ba1iv—O5—Ba1—O1ii166.78 (13)O6iv—Ba1—O1—Ba1iii170.00 (5)
Ba1iv—O5—Ba1—O753.60 (8)O6—Ba1—O1—Ba1iii−98.11 (6)
Ba1iv—O5—Ba1—O7iii117.34 (5)O5—Ba1—O1—Ba1iii−147.01 (7)
Ba1iv—O5—Ba1—O6iv−54.46 (5)O5iv—Ba1—O1—Ba1iii−134.14 (11)
Ba1iv—O5—Ba1—O653.36 (6)N1—Ba1—O1—Ba1iii117.19 (8)
Ba1iv—O5—Ba1—O5iv0.0C6ii—Ba1—O1—Ba1iii46.08 (6)
Ba1iv—O5—Ba1—N1−128.71 (6)Ba1iv—Ba1—O1—Ba1iii−143.15 (4)
Ba1iv—O5—Ba1—C6ii−162.71 (6)Ba1v—O2—C6—O114.9 (4)
O1—Ba1—N1—C5−174.1 (2)Ba1v—O2—C6—C1−165.84 (15)
O2i—Ba1—N1—C5−12.0 (2)Ba1v—O2—C6—Ba1iii64.7 (2)
O1ii—Ba1—N1—C5−80.1 (2)Ba1—O1—C6—O2−170.54 (18)
O7—Ba1—N1—C5−66.7 (3)Ba1iii—O1—C6—O266.7 (3)
O7iii—Ba1—N1—C5−155.5 (2)Ba1—O1—C6—C110.2 (3)
O6iv—Ba1—N1—C557.8 (2)Ba1iii—O1—C6—C1−112.54 (17)
O6—Ba1—N1—C5119.5 (2)Ba1—O1—C6—Ba1iii122.77 (18)
O5—Ba1—N1—C5116.6 (2)N1—C1—C6—O2168.6 (2)
O5iv—Ba1—N1—C534.7 (2)C2—C1—C6—O2−11.0 (3)
C6ii—Ba1—N1—C5−84.0 (2)N1—C1—C6—O1−12.2 (3)
Ba1iv—Ba1—N1—C581.3 (2)C2—C1—C6—O1168.2 (2)
O1—Ba1—N1—C1−3.35 (16)N1—C1—C6—Ba1iii−77.6 (2)
O2i—Ba1—N1—C1158.68 (19)C2—C1—C6—Ba1iii102.8 (2)
O1ii—Ba1—N1—C190.65 (18)C1—C2—C3—C41.0 (3)
O7—Ba1—N1—C1103.99 (18)C1—C2—C3—C7−177.42 (19)
O7iii—Ba1—N1—C115.24 (19)O3—C7—C3—C4172.7 (3)
O6iv—Ba1—N1—C1−131.43 (19)O4—C7—C3—C4−7.1 (4)
O6—Ba1—N1—C1−69.8 (2)O3—C7—C3—C2−8.9 (3)
O5—Ba1—N1—C1−72.70 (18)O4—C7—C3—C2171.3 (2)
O5iv—Ba1—N1—C1−154.60 (16)N1—C5—C4—C3−0.4 (5)
C6ii—Ba1—N1—C186.74 (18)C2—C3—C4—C5−0.5 (4)
Ba1iv—Ba1—N1—C1−108.02 (17)C7—C3—C4—C5177.9 (3)
Symmetry codes: (i) x, y, z−1; (ii) x, −y+3/2, z−1/2; (iii) x, −y+3/2, z+1/2; (iv) −x, −y+1, −z; (v) x, y, z+1.
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
C5—H5···O2i0.932.483.161 (3)130
O5—H5A···O1vi0.77 (5)2.14 (4)2.881 (3)162 (4)
O5—H5B···O4vii0.77 (5)2.02 (5)2.785 (3)174 (4)
O6—H6A···O4viii0.91 (4)2.03 (4)2.865 (3)151 (3)
O6—H6B···O3ix0.76 (4)2.08 (4)2.816 (3)167 (4)
O7—H7A···O3ix0.85 (5)1.96 (5)2.809 (3)175 (4)
O7—H7B···O4x0.75 (4)2.11 (4)2.810 (3)155 (4)
Symmetry codes: (i) x, y, z−1; (vi) −x, −y+1, −z+1; (vii) −x+1, −y+1, −z+1; (viii) x−1, y, z; (ix) x−1, y, z−1; (x) x−1, −y+3/2, z−1/2.
Table 1
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
C5—H5···O2i0.932.483.161 (3)130
O5—H5A···O1ii0.77 (5)2.14 (4)2.881 (3)162 (4)
O5—H5B···O4iii0.77 (5)2.02 (5)2.785 (3)174 (4)
O6—H6A···O4iv0.91 (4)2.03 (4)2.865 (3)151 (3)
O6—H6B···O3v0.76 (4)2.08 (4)2.816 (3)167 (4)
O7—H7A···O3v0.85 (5)1.96 (5)2.809 (3)175 (4)
O7—H7B···O4vi0.75 (4)2.11 (4)2.810 (3)155 (4)
Symmetry codes: (i) x, y, z−1; (ii) −x, −y+1, −z+1; (iii) −x+1, −y+1, −z+1; (iv) x−1, y, z; (v) x−1, y, z−1; (vi) x−1, −y+3/2, z−1/2.
Acknowledgements top

The authors gratefully acknowledge the Islamic Azad University, North Tehran Branch, for financial support.

references
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