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Acta Cryst. (2008). E64, o876-o877    [ doi:10.1107/S1600536808010465 ]

(E)-2-[4-(Dimethylamino)styryl]-1-methylquinolinium iodide sesquihydrate

S. Chantrapromma, T. Kobkeatthawin, K. Chanawanno, C. Karalai and H.-K. Fun

Abstract top

In the title compound, C20H21N2+.I-·1.5H2O, the cation exists in the E configuration and is not planar. The dihedral angle between the quinolinium and dimethylaminophenyl rings is 9.26 (6)°. The O atom of one of the solvent water molecules lies on a twofold rotation axis. In the crystal structure, the cations form one-dimensional zigzag chains along the [001] direction. The cations are linked to water molecules and iodide ions through weak C-H...O and C-H...I interactions, respectively. Water molecules and iodide ions form O-H...O and O-H...I hydrogen bonds, which stabilize the crystal structure. A C-H...[pi] interaction is also present.

Comment top

Organic molecules with large π systems have been extensively used in attempts to obtain non-linear optical (NLO) materials (Chia et al., 1995; Dittrich et al., 2003; Marder et al., 1994; Nogi et al., 2000; Otero et al., 2002; Pan et al., 1996; Sato et al., 1999). We have previously synthesized and crystallized several ionic organic salts of quinolinium derivatives which have a conjugate π system to study their non-linear optical properties (Chantrapromma et al., 2006; 2007a; 2007b; 2007c; 2007d; Jindawong et al., 2005). Previous investigations by Marder et al., 1994, Pan et al., 1996 and Umezawa et al., 2000 reported that 1-methyl-4-(2-(4-(dimethylamino)phenyl)ethenyl)pyridinium p-toluenesulfonate (DAST) is a promising second-order NLO material. Based on this information and our previous investigation (Chantrapromma et al., 2007c), we have designed and synthesized the title compound (I) with the replacement of the 3-hydroxy-4-methoxyphenyl ring in the cation of 2-[(E)-(3-hydroxy-4-methoxyphenyl)ethenyl]-1-methylquinolinium iodide monohydrate which showed second-order NLO properties (Chantrapromma et al., 2007c) by the 4-dimethylaminophenyl ring and its crystal structure was reported here. However since second-order NLO effects are created only when chromophores are arranged in a non-centrosymmetric manner, the title compound, which crystallized in the centrosymmetric space group C2/c, does not exhibit any second-order NLO properties.

The asymmetric unit of the title compound consists of one C20H21N2+ cation, one I- anion and 1.5 H2O molecules. The remaining cell contents are generated by symmetry with the O2W atom (symmetry code: -x, y, 1/2 - z) lying on a two-fold rotation axis. The cation exists in the E configuration with respect to the C10═C11 double bond [1.357 (2) Å] and is not planar as indicated by a dihedral angle of 9.26 (6)° between the quinolinium and the dimethylaminophenyl rings. This value is relatively wider than the corresponding angle (3.41 (7)°) reported for the closely related structure of the 4-methoxybenzenesulfonate salt of the same cation (Kobkeatthawin et al., 2008). This may be due to packing effects involving the different counterions. The orientation of the ethenyl unit with respect to the quinolinium and the dimethylaminophenyl rings can be indicated by the torsion angles C8–C9–C10–C11 = 8.5 (2)° and C10–C11–C12–C17 = -1.2 (2)°. The bond lengths and angles are in normal ranges (Allen et al., 1987) and are comparable to those in closely related structures (Chantrapromma et al., 2006; 2007a; 2007b; 2007c; Kobkeatthawin et al., 2008).

In the crystal packing (Fig. 2), the cations form one-dimensional zigzag chains along the [0 0 1] direction. Water molecules contribute to an O2W—H1W2···O1W hydrogen bond. The cations are linked to water molecules and iodide ions through weak C—H···O and C—H···I interactions respectively (Table 1). Water molecules and iodide ions are interconnected by O—H···I hydrogen bonds (O1W—H1W1···I1 and O1W—H2W1···I1 symmetry codes: -x, y, 1/2 - z and x, 1 - y, -1/2 + z, respectively). The crystal is further stabilized by O—H···O and O—H···I hydrogen bonds together with weak C—H···O and C—H···I interactions. A C2—H2A···π interaction to the dimethylaminophenyl ring [C12–C17] was also observed: C2—H2A = 0.93; H2A···Cgi = 3.0219; C2—Cg1i = 3.7648 (17) Å; C2—H2A···Cg1i = 138°. [Cg1i is the centroid of the C12–C17 ring (symmetry code: (i): x, 1 - y, -1/2 + z)].

Related literature top

For bond lengths and angles, see: Allen et al. (1987). For background to non-linear optical (NLO) materials research, see: Chia et al. (1995); Marder et al. (1994); Otero et al. (2002); Pan et al. (1996). For related structures, see for example: Chantrapromma et al. (2006, 2007a,b,c,d); Dittrich et al. (2003); Jindawong et al. (2005); Kobkeatthawin et al. (2008); Nogi et al. (2000); Sato et al. (1999); Umezawa et al. (2000). Cg1 is the

centroid of the C12–C17 ring.

Experimental top

The title compound was synthesized by mixing a 1:1:1 molar ratio solution of 1,2-dimethylquinolinium iodide (2.00 g, 7.01 mmol), dimethylaminobenzaldehyde (1.05 g, 7.01 mmol) and piperidine (0.70 g, 7.01 mmol) in hot methanol (50 ml). The resulting solution was refluxed for 6 h under a nitrogen atmosphere. The resulting solid was filtered off, washed with methanol and recrystallized from methanol to give green crystals. Single crystals of the title compound suitable for x-ray structure determination were recrystalized from methanol/ethanol solvent (1:1 v/v) by slow evaporation of the solvent at room temperature after a few weeks. (Mp. 491–493 K).

Refinement top

Water hydrogen atoms were located in a difference map and refined isotropically. H atoms attached to C were placed in calculated positions with d(C—H) = 0.93 Å, Uiso=1.2Ueq(C) for aromatic and CH, 0.96 Å, Uiso = 1.5Ueq(C) for CH3 atoms. A rotating group model was used for the methyl groups. The highest residual electron density peak is located at 0.57 Å from I1 and the deepest hole is located at 0.46 Å from I1.

Computing details top

Data collection: APEX2 (Bruker, 2005); cell refinement: APEX2 (Bruker, 2005); data reduction: SAINT (Bruker, 2005); program(s) used to solve structure: SHELXTL (Sheldrick, 2008); program(s) used to refine structure: SHELXTL (Sheldrick, 2008); molecular graphics: SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXTL (Sheldrick, 2008) and PLATON (Spek, 2003).

Figures top
[Figure 1] Fig. 1. The asymmetric unit of (I) showing 50% probability displacement ellipsoids and the atom-numbering scheme. O2W (symmetry code: -x, y, 1/2 - z) lies on a two-fold rotation axis.
[Figure 2] Fig. 2. The crystal packing of (I) viewed along the a axis, showing the one-dimensional zigzag chains of the cations running along the c direction. The O—H···O and O—H···I hydrogen bonds and weak C—H···O interactions are drawn as dashed lines.
(E)-2-[4-(Dimethylamino)styryl]-1-methylquinolinium iodide sesquihydrate top
Crystal data top
C20H21N2+·I–·1.5H2OF000 = 1784
Mr = 443.31Dx = 1.571 Mg m−3
Monoclinic, C2/cMelting point = 491–493 K
Hall symbol: -C 2ycMo Kα radiation
λ = 0.71073 Å
a = 20.8997 (4) ÅCell parameters from 8240 reflections
b = 10.5941 (2) Åθ = 2.1–35.0º
c = 18.4020 (4) ŵ = 1.72 mm−1
β = 113.047 (1)ºT = 100.0 (1) K
V = 3749.24 (13) Å3Block, green
Z = 80.52 × 0.35 × 0.12 mm
Data collection top
Bruker SMART APEX2 CCD area-detector
diffractometer		8240 independent reflections
Radiation source: fine-focus sealed tube7476 reflections with I > 2σ(I)
Monochromator: graphiteRint = 0.032
Detector resolution: 8.33 pixels mm-1θmax = 35.0º
T = 100.0(1) Kθmin = 2.1º
ω scansh = −33→33
Absorption correction: multi-scan
(SADABS; Bruker, 2005)		k = −17→15
Tmin = 0.469, Tmax = 0.818l = −29→28
50083 measured reflections
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)] = 0.029H atoms treated by a mixture of
independent and constrained refinement
wR(F2) = 0.068  w = 1/[σ2(Fo2) + (0.0246P)2 + 7.1959P]
where P = (Fo2 + 2Fc2)/3
S = 1.07(Δ/σ)max = 0.001
8240 reflectionsΔρmax = 1.58 e Å−3
237 parametersΔρmin = −0.80 e Å−3
Primary atom site location: structure-invariant direct methodsExtinction correction: none
Crystal data top
C20H21N2+·I–·1.5H2OV = 3749.24 (13) Å3
Mr = 443.31Z = 8
Monoclinic, C2/cMo Kα
a = 20.8997 (4) ŵ = 1.72 mm−1
b = 10.5941 (2) ÅT = 100.0 (1) K
c = 18.4020 (4) Å0.52 × 0.35 × 0.12 mm
β = 113.047 (1)º
Data collection top
Bruker SMART APEX2 CCD area-detector
diffractometer		8240 independent reflections
Absorption correction: multi-scan
(SADABS; Bruker, 2005)		7476 reflections with I > 2σ(I)
Tmin = 0.469, Tmax = 0.818Rint = 0.032
50083 measured reflections
Refinement top
R[F2 > 2σ(F2)] = 0.029237 parameters
wR(F2) = 0.068H atoms treated by a mixture of
independent and constrained refinement
S = 1.07Δρmax = 1.58 e Å−3
8240 reflectionsΔρmin = −0.80 e Å−3
Special details top

Experimental. The low-temparture data was collected with the Oxford Cryosystem Cobra low-temperature attachment.

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
I10.121463 (5)0.401312 (10)0.489877 (6)0.02205 (3)
N10.34849 (6)0.45765 (12)−0.04848 (7)0.0163 (2)
N20.16088 (7)0.23257 (14)0.29101 (8)0.0219 (2)
C10.39458 (7)0.47357 (14)−0.08616 (8)0.0173 (2)
C20.38630 (8)0.57284 (16)−0.13994 (9)0.0222 (3)
H2A0.34880.6279−0.15250.027*
C30.43416 (9)0.58795 (17)−0.17383 (10)0.0249 (3)
H3A0.42830.6533−0.20960.030*
C40.49123 (8)0.50731 (18)−0.15566 (9)0.0244 (3)
H4A0.52340.5202−0.17840.029*
C50.49965 (8)0.40906 (16)−0.10418 (9)0.0219 (3)
H5A0.53710.3542−0.09290.026*
C60.45170 (8)0.39079 (14)−0.06823 (8)0.0183 (2)
C70.45974 (8)0.29187 (16)−0.01383 (9)0.0211 (3)
H7A0.49610.2347−0.00290.025*
C80.41475 (8)0.27979 (15)0.02247 (9)0.0201 (3)
H8A0.42060.21420.05810.024*
C90.35853 (7)0.36612 (14)0.00684 (8)0.0163 (2)
C100.31320 (7)0.35646 (14)0.04900 (8)0.0173 (2)
H10A0.27350.40650.03260.021*
C110.32564 (7)0.27790 (14)0.11134 (8)0.0170 (2)
H11A0.36520.22750.12650.020*
C120.28250 (7)0.26643 (13)0.15584 (8)0.0157 (2)
C130.29994 (7)0.18026 (14)0.21874 (9)0.0185 (2)
H13A0.33960.13090.23090.022*
C140.26020 (7)0.16647 (14)0.26310 (9)0.0193 (2)
H14A0.27300.10750.30380.023*
C150.20009 (7)0.24148 (14)0.24709 (8)0.0164 (2)
C160.18213 (7)0.32815 (14)0.18357 (8)0.0177 (2)
H16A0.14280.37830.17140.021*
C170.22208 (7)0.33918 (14)0.13970 (8)0.0176 (2)
H17A0.20880.39640.09810.021*
C180.17497 (9)0.13464 (18)0.35064 (10)0.0262 (3)
H18A0.22310.13760.38570.039*
H18B0.16460.05350.32540.039*
H18C0.14650.14810.38010.039*
C190.09759 (8)0.30611 (17)0.27093 (10)0.0236 (3)
H19A0.10780.39390.26800.035*
H19B0.07920.29430.31070.035*
H19C0.06400.27880.22080.035*
C200.28968 (8)0.54554 (16)−0.07022 (10)0.0223 (3)
H20A0.25750.5167−0.04830.034*
H20B0.26670.5494−0.12670.034*
H20C0.30640.6280−0.04990.034*
O1W0.03830 (8)0.54957 (15)0.13362 (8)0.0297 (3)
O2W0.00000.6809 (2)0.25000.0385 (5)
H1W20.0114 (15)0.633 (3)0.2215 (16)0.044 (8)*
H1W10.0030 (16)0.508 (3)0.1033 (17)0.049 (8)*
H2W10.0580 (16)0.564 (3)0.1055 (18)0.049 (8)*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
I10.01875 (5)0.02151 (5)0.02561 (5)−0.00453 (3)0.00840 (4)−0.00498 (4)
N10.0154 (5)0.0157 (5)0.0180 (5)0.0016 (4)0.0066 (4)0.0002 (4)
N20.0203 (5)0.0256 (6)0.0235 (6)0.0038 (5)0.0126 (5)0.0083 (5)
C10.0164 (5)0.0198 (6)0.0165 (5)−0.0016 (5)0.0073 (5)−0.0020 (5)
C20.0190 (6)0.0242 (7)0.0236 (6)0.0016 (5)0.0086 (5)0.0041 (5)
C30.0233 (7)0.0289 (8)0.0229 (7)−0.0006 (6)0.0096 (6)0.0055 (6)
C40.0214 (6)0.0351 (8)0.0199 (6)−0.0028 (6)0.0117 (5)−0.0010 (6)
C50.0180 (6)0.0282 (7)0.0205 (6)0.0022 (5)0.0088 (5)−0.0032 (5)
C60.0177 (6)0.0205 (6)0.0167 (5)0.0008 (5)0.0067 (5)−0.0009 (5)
C70.0198 (6)0.0222 (7)0.0219 (6)0.0056 (5)0.0088 (5)0.0013 (5)
C80.0183 (6)0.0222 (7)0.0214 (6)0.0051 (5)0.0095 (5)0.0036 (5)
C90.0168 (5)0.0153 (5)0.0173 (5)0.0002 (4)0.0072 (5)−0.0001 (4)
C100.0182 (6)0.0172 (6)0.0185 (6)−0.0002 (5)0.0093 (5)−0.0006 (5)
C110.0149 (5)0.0184 (6)0.0179 (5)0.0003 (4)0.0068 (4)−0.0001 (5)
C120.0149 (5)0.0160 (5)0.0162 (5)−0.0004 (4)0.0060 (4)0.0004 (4)
C130.0162 (5)0.0194 (6)0.0199 (6)0.0031 (5)0.0071 (5)0.0036 (5)
C140.0183 (6)0.0193 (6)0.0208 (6)0.0028 (5)0.0082 (5)0.0057 (5)
C150.0152 (5)0.0173 (6)0.0164 (5)−0.0011 (4)0.0059 (4)0.0012 (4)
C160.0161 (5)0.0182 (6)0.0187 (6)0.0021 (4)0.0067 (5)0.0036 (5)
C170.0179 (6)0.0179 (6)0.0178 (6)0.0023 (5)0.0079 (5)0.0041 (5)
C180.0246 (7)0.0310 (8)0.0267 (7)0.0036 (6)0.0143 (6)0.0111 (6)
C190.0213 (6)0.0291 (8)0.0236 (6)0.0040 (6)0.0125 (5)0.0025 (6)
C200.0209 (6)0.0212 (7)0.0284 (7)0.0064 (5)0.0133 (6)0.0048 (5)
O1W0.0271 (6)0.0340 (7)0.0266 (6)0.0012 (5)0.0092 (5)0.0014 (5)
O2W0.0530 (13)0.0278 (10)0.0469 (12)0.0000.0329 (11)0.000
Geometric parameters (Å, °) top
N1—C91.3614 (19)C11—C121.4411 (19)
N1—C11.4001 (18)C11—H11A0.9300
N1—C201.4672 (19)C12—C131.406 (2)
N2—C151.3611 (18)C12—C171.408 (2)
N2—C191.453 (2)C13—C141.381 (2)
N2—C181.454 (2)C13—H13A0.9300
C1—C21.408 (2)C14—C151.417 (2)
C1—C61.412 (2)C14—H14A0.9300
C2—C31.380 (2)C15—C161.417 (2)
C2—H2A0.9300C16—C171.3750 (19)
C3—C41.397 (2)C16—H16A0.9300
C3—H3A0.9300C17—H17A0.9300
C4—C51.372 (2)C18—H18A0.9600
C4—H4A0.9300C18—H18B0.9600
C5—C61.414 (2)C18—H18C0.9600
C5—H5A0.9300C19—H19A0.9600
C6—C71.413 (2)C19—H19B0.9600
C7—C81.356 (2)C19—H19C0.9600
C7—H7A0.9300C20—H20A0.9600
C8—C91.427 (2)C20—H20B0.9600
C8—H8A0.9300C20—H20C0.9600
C9—C101.4442 (19)O1W—H1W10.85 (3)
C10—C111.357 (2)O1W—H2W10.79 (3)
C10—H10A0.9300O2W—H1W20.83 (3)
C9—N1—C1121.43 (12)C12—C11—H11A117.4
C9—N1—C20121.57 (12)C13—C12—C17116.83 (12)
C1—N1—C20116.99 (12)C13—C12—C11120.29 (13)
C15—N2—C19120.78 (13)C17—C12—C11122.88 (13)
C15—N2—C18120.43 (13)C14—C13—C12122.22 (13)
C19—N2—C18118.03 (12)C14—C13—H13A118.9
N1—C1—C2121.28 (13)C12—C13—H13A118.9
N1—C1—C6119.48 (13)C13—C14—C15120.47 (13)
C2—C1—C6119.22 (13)C13—C14—H14A119.8
C3—C2—C1119.55 (15)C15—C14—H14A119.8
C3—C2—H2A120.2N2—C15—C14121.92 (13)
C1—C2—H2A120.2N2—C15—C16120.54 (13)
C2—C3—C4121.53 (15)C14—C15—C16117.54 (12)
C2—C3—H3A119.2C17—C16—C15120.92 (13)
C4—C3—H3A119.2C17—C16—H16A119.5
C5—C4—C3119.73 (14)C15—C16—H16A119.5
C5—C4—H4A120.1C16—C17—C12122.01 (13)
C3—C4—H4A120.1C16—C17—H17A119.0
C4—C5—C6120.29 (14)C12—C17—H17A119.0
C4—C5—H5A119.9N2—C18—H18A109.5
C6—C5—H5A119.9N2—C18—H18B109.5
C1—C6—C7118.76 (13)H18A—C18—H18B109.5
C1—C6—C5119.66 (14)N2—C18—H18C109.5
C7—C6—C5121.58 (14)H18A—C18—H18C109.5
C8—C7—C6120.39 (14)H18B—C18—H18C109.5
C8—C7—H7A119.8N2—C19—H19A109.5
C6—C7—H7A119.8N2—C19—H19B109.5
C7—C8—C9121.03 (14)H19A—C19—H19B109.5
C7—C8—H8A119.5N2—C19—H19C109.5
C9—C8—H8A119.5H19A—C19—H19C109.5
N1—C9—C8118.76 (13)H19B—C19—H19C109.5
N1—C9—C10120.71 (13)N1—C20—H20A109.5
C8—C9—C10120.53 (13)N1—C20—H20B109.5
C11—C10—C9123.26 (13)H20A—C20—H20B109.5
C11—C10—H10A118.4N1—C20—H20C109.5
C9—C10—H10A118.4H20A—C20—H20C109.5
C10—C11—C12125.20 (13)H20B—C20—H20C109.5
C10—C11—H11A117.4H1W1—O1W—H2W1102 (3)
C9—N1—C1—C2−176.27 (14)C7—C8—C9—N13.4 (2)
C20—N1—C1—C22.6 (2)C7—C8—C9—C10−176.63 (14)
C9—N1—C1—C61.9 (2)N1—C9—C10—C11−171.48 (14)
C20—N1—C1—C6−179.27 (14)C8—C9—C10—C118.5 (2)
N1—C1—C2—C3177.80 (15)C9—C10—C11—C12179.05 (14)
C6—C1—C2—C3−0.3 (2)C10—C11—C12—C13179.02 (14)
C1—C2—C3—C4−0.4 (3)C10—C11—C12—C17−1.2 (2)
C2—C3—C4—C51.3 (3)C17—C12—C13—C140.0 (2)
C3—C4—C5—C6−1.4 (2)C11—C12—C13—C14179.79 (14)
N1—C1—C6—C71.5 (2)C12—C13—C14—C15−1.0 (2)
C2—C1—C6—C7179.65 (14)C19—N2—C15—C14−176.95 (15)
N1—C1—C6—C5−177.92 (13)C18—N2—C15—C14−7.2 (2)
C2—C1—C6—C50.2 (2)C19—N2—C15—C163.7 (2)
C4—C5—C6—C10.6 (2)C18—N2—C15—C16173.45 (15)
C4—C5—C6—C7−178.79 (15)C13—C14—C15—N2−178.22 (15)
C1—C6—C7—C8−2.3 (2)C13—C14—C15—C161.2 (2)
C5—C6—C7—C8177.06 (15)N2—C15—C16—C17178.95 (15)
C6—C7—C8—C9−0.1 (2)C14—C15—C16—C17−0.5 (2)
C1—N1—C9—C8−4.2 (2)C15—C16—C17—C12−0.5 (2)
C20—N1—C9—C8176.94 (14)C13—C12—C17—C160.7 (2)
C1—N1—C9—C10175.76 (13)C11—C12—C17—C16−179.05 (14)
C20—N1—C9—C10−3.1 (2)
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O1W—H1W1···I1i0.85 (3)2.74 (3)3.5832 (16)172 (3)
O2W—H1W2···O1W0.83 (3)2.10 (3)2.9164 (19)167 (3)
O1W—H2W1···I1ii0.79 (3)2.94 (3)3.7267 (16)174 (3)
C3—H3A···O2Wiii0.932.603.371 (2)141
C7—H7A···I1iv0.933.043.9290 (18)161
C17—H17A···I1ii0.933.013.8784 (14)157
C2—H2A···Cg1ii0.933.023.7648 (17)138
Symmetry codes: (i) −x, y, −z+1/2; (ii) x, −y+1, z−1/2; (iii) −x+1/2, −y+3/2, −z; (iv) x+1/2, −y+1/2, z−1/2.
Table 1
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O1W—H1W1···I1i0.85 (3)2.74 (3)3.5832 (16)172 (3)
O2W—H1W2···O1W0.83 (3)2.10 (3)2.9164 (19)167 (3)
O1W—H2W1···I1ii0.79 (3)2.94 (3)3.7267 (16)174 (3)
C3—H3A···O2Wiii0.932.603.371 (2)141
C7—H7A···I1iv0.933.043.9290 (18)161
C17—H17A···I1ii0.933.013.8784 (14)157
C2—H2A···Cg1ii0.933.023.7648 (17)138
Symmetry codes: (i) −x, y, −z+1/2; (ii) x, −y+1, z−1/2; (iii) −x+1/2, −y+3/2, −z; (iv) x+1/2, −y+1/2, z−1/2.
Acknowledgements top

The authors thank the Prince of Songkla University for financial support. The authors also thank the Malaysian Government and Universiti Sains Malaysia for Research University Golden Goose grant No. 1001/PFIZIK/811012.

references
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