supplementary materials


is5268 scheme

Acta Cryst. (2013). E69, o839    [ doi:10.1107/S1600536813011690 ]

1-Methyl-3-(2-oxo-2H-chromen-3-yl)-1H-imidazol-3-ium picrate

N. V. Tuyen, L. T. Anh, A. A. Festa, L. G. Voskressensky and V. N. Khrustalev

Abstract top

The title salt, C13H11N2O2+·C6H2N3O7-, is the unexpected product of a domino reaction of 3-cyanomethyl-1-methylimidazolium chloride with salicylic aldehyde in the presence of picric acid. In the cation, the 1H-imidazole ring is twisted by 63.2 (1)° from the 2H-chromen plane. In the crystal, cations and anions are alternately stacked along the a axis through [pi]-[pi] stacking interactions between the almost parallel aromatic rings [centroid-centroid distances = 3.458 (2) and 3.678 (2) Å]. The stacks are further linked by C-H...O hydrogen bonds into a two-tier layer parallel to (001).

Comment top

Coumarin derivatives are known to possess a range of different biological activities (Kontogiorgis et al., 2012). The title compound, C13H11N2O2+.C6H2N3O7- (I), is the unexpected product of Knoevenagel condensation of 3-(cyanomethyl)-1-methylimidazolium chloride with salicylic aldehyde followed by the hydrolysis of imino-group and the formation of ammonium salt with picric acid (Fig. 1; Voskressensky et al., 2012a,b).

The cation and anion of I form a tight ionic pair by the C9—H9···O3 hydrogen bond (Table 1) as well as the ππ stacking interactions between the almost parallel aromatic moieties [the dihedral angle between the mean planes of the 2H-chromen (cation) and benzene (anion) fragments is 3.55 (7)°; the shortest C8···C17 distance is 3.280 (5) Å; Fig. 2]. The 1H-imidazole ring is twisted at 63.2 (1)° from the 2H-chromen plane. In the crystal, the tight ionic pairs form stacks along the a axis by the ππ stacking interactions (Fig. 3). The stacks are further bound by the C—H···O hydrogen bonds into two-tier layers parallel to (001) (Fig. 4).

Related literature top

For recent review on coumarin-based drug patents, see: Kontogiorgis et al. (2012). For analogous domino reactions, see: Voskressensky et al. (2012a,b). For related compounds, see: Yu et al. (2006); Morris et al. (2011).

Experimental top

A solid Na2CO3 (67.0 mg, 0.63 mmol) was added to a stirred solution of 3-(cyanomethyl)-1-methylimidazolium chloride (500 mg, 3.2 mmol) and salicylic aldehyde (350 mg, 2.9 mmol) in a mixture of methanol (4 ml) and water (1 ml) at reflux. The reaction mixture was heated at reflux for 1 h. Then picric acid (870 mg, 3.8 mmol) was added to the solution. The formed precipitate was filtered-off and washed with acetone (3x) to give 630 mg of yellow crystals of I. The yield is 48%. M.p. = 459 K (decomp.). 1H NMR (DMSO-d6, 400 MHz): δ = 4.04 (3H, s, Me), 7.54 (1H, t, J = 7.5 Hz, H6'), 7.63 (1H, d, J = 8.3 Hz, H5'), 7.79–7.85 (1H, m, H7'), 7.87–7.92 (1H, m, H8'), 7.98–8.01 (1H, m, H5), 8.16–8.19 (1H, m, H4), 8.61 (2H, s, picric acid CH), 8.70 (1H, s, H4'), 9.71 (1H, bs, H2); 13C NMR (DMSO-d6, 100 MHz): δ = 36.2, 116.4, 117.6, 121.6, 122.5, 123.7, 124.2, 125.1 (2 C), 125.5, 129.4, 133.5, 137.3, 137.5, 141.8, 152.4 (2 C), 156.1, 160.8. Anal. Calcd for C13H11N2O2.C6H2N3O7: C 50.12, H 2.88, N 15.38; found: C 50.34, H 3.01, N 15.53.

Refinement top

H atoms were placed in calculated positions with C—H = 0.95 Å (CH) and 0.98 Å (CH3) and refined in the riding model with fixed isotropic displacement parameters [Uiso(H) = 1.5Ueq(C) for the CH3 group and 1.2Ueq(C) for the CH groups].

Computing details top

Data collection: APEX2 (Bruker, 2005); cell refinement: SAINT (Bruker, 2001); data reduction: SAINT (Bruker, 2001); 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 domino reaction of 3-(cyanomethyl)-1-methylimidazolium chloride with salicylic aldehyde.
[Figure 2] Fig. 2. The molecular structure of the title compound. Displacement ellipsoids are shown at the 50% probability level. H atoms are presented as small spheres of arbitrary radius. Dashed line indicates the (N)C(N+)—H···O- hydrogen bond between cation and anion.
[Figure 3] Fig. 3. A portion of crystal packing of the title compound demonstrating the stacks along the a axis. Dashed lines indicate the intermolecular C—H···O hydrogen bonds.
[Figure 4] Fig. 4. The two-tier layers of the title compound parallel to (001). Dashed lines indicate the intermolecular C—H···O hydrogen bonds.
1-Methyl-3-(2-oxo-2H-chromen-3-yl)-1H-imidazol-3-ium picrate top
Crystal data top
C13H11N2O2+·C6H2N3O7F(000) = 468
Mr = 455.34Dx = 1.634 Mg m3
Monoclinic, P21Mo Kα radiation, λ = 0.71073 Å
Hall symbol: P 2ybCell parameters from 3018 reflections
a = 6.8142 (12) Åθ = 2.5–30.2°
b = 8.1942 (14) ŵ = 0.13 mm1
c = 16.832 (3) ÅT = 100 K
β = 100.081 (4)°Plate, yellow
V = 925.3 (3) Å30.30 × 0.21 × 0.03 mm
Z = 2
Data collection top
Bruker APEXII CCD
diffractometer
4415 independent reflections
Radiation source: fine-focus sealed tube3734 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.040
φ and ω scansθmax = 28.0°, θmin = 2.5°
Absorption correction: multi-scan
(SADABS; Bruker, 2003)
h = 99
Tmin = 0.961, Tmax = 0.996k = 1010
10390 measured reflectionsl = 2222
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)] = 0.063Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.165H-atom parameters constrained
S = 1.00 w = 1/[σ2(Fo2) + (0.0754P)2 + 1.86P]
where P = (Fo2 + 2Fc2)/3
4415 reflections(Δ/σ)max < 0.001
299 parametersΔρmax = 0.46 e Å3
1 restraintΔρmin = 0.36 e Å3
Crystal data top
C13H11N2O2+·C6H2N3O7V = 925.3 (3) Å3
Mr = 455.34Z = 2
Monoclinic, P21Mo Kα radiation
a = 6.8142 (12) ŵ = 0.13 mm1
b = 8.1942 (14) ÅT = 100 K
c = 16.832 (3) Å0.30 × 0.21 × 0.03 mm
β = 100.081 (4)°
Data collection top
Bruker APEXII CCD
diffractometer
4415 independent reflections
Absorption correction: multi-scan
(SADABS; Bruker, 2003)
3734 reflections with I > 2σ(I)
Tmin = 0.961, Tmax = 0.996Rint = 0.040
10390 measured reflectionsθmax = 28.0°
Refinement top
R[F2 > 2σ(F2)] = 0.063H-atom parameters constrained
wR(F2) = 0.165Δρmax = 0.46 e Å3
S = 1.00Δρmin = 0.36 e Å3
4415 reflectionsAbsolute structure: ?
299 parametersFlack parameter: ?
1 restraintRogers parameter: ?
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
O10.8777 (4)0.6678 (3)0.68430 (15)0.0183 (5)
O20.8003 (4)0.5232 (4)0.57153 (15)0.0233 (6)
N10.9223 (4)0.2275 (4)0.64703 (18)0.0166 (6)
N20.8021 (5)0.0278 (4)0.57071 (17)0.0166 (6)
C20.8643 (5)0.5224 (5)0.6430 (2)0.0168 (7)
C30.9351 (5)0.3784 (5)0.6911 (2)0.0161 (7)
C41.0105 (5)0.3861 (5)0.7702 (2)0.0171 (7)
H41.05610.28990.79920.021*
C4A1.0217 (5)0.5408 (5)0.8103 (2)0.0165 (7)
C51.0926 (5)0.5595 (5)0.8937 (2)0.0178 (7)
H51.13650.46660.92570.021*
C61.0989 (5)0.7124 (5)0.9295 (2)0.0205 (8)
H61.14610.72400.98570.025*
C71.0352 (6)0.8501 (5)0.8821 (2)0.0223 (8)
H71.04200.95520.90630.027*
C80.9623 (6)0.8333 (5)0.8001 (2)0.0208 (8)
H80.91690.92570.76810.025*
C8A0.9570 (5)0.6794 (5)0.7661 (2)0.0172 (7)
C90.7529 (5)0.1610 (5)0.6077 (2)0.0163 (7)
H90.62180.20180.60640.020*
C101.0046 (6)0.0076 (5)0.5862 (2)0.0198 (7)
H101.07720.07860.56710.024*
C111.0826 (5)0.1338 (5)0.6341 (2)0.0200 (7)
H111.21950.15360.65460.024*
C120.6615 (6)0.0810 (5)0.5195 (2)0.0218 (8)
H12A0.52470.05170.52450.033*
H12B0.68780.19420.53690.033*
H12C0.67840.06940.46320.033*
O30.4485 (4)0.3812 (3)0.67659 (15)0.0185 (5)
O40.7273 (4)0.1650 (4)0.85805 (18)0.0293 (7)
O50.4224 (5)0.1338 (4)0.79424 (19)0.0310 (7)
O60.6970 (5)0.6752 (4)1.02204 (16)0.0283 (7)
O70.5824 (4)0.8942 (3)0.95546 (17)0.0245 (6)
O80.4486 (4)0.8736 (3)0.66647 (16)0.0239 (6)
O90.2690 (4)0.6684 (4)0.61602 (16)0.0257 (6)
N30.5653 (5)0.2192 (4)0.82445 (19)0.0197 (6)
N40.6210 (5)0.7472 (4)0.95962 (18)0.0183 (6)
N50.3876 (5)0.7341 (4)0.67024 (19)0.0185 (6)
C130.4738 (5)0.4649 (4)0.7400 (2)0.0134 (7)
C140.5419 (5)0.3960 (5)0.8190 (2)0.0173 (7)
C150.5962 (5)0.4837 (4)0.8896 (2)0.0155 (7)
H150.64810.43100.93910.019*
C160.5717 (5)0.6527 (5)0.8853 (2)0.0159 (7)
C170.5037 (5)0.7321 (5)0.8138 (2)0.0161 (7)
H170.49080.84760.81250.019*
C180.4542 (5)0.6419 (5)0.7437 (2)0.0170 (7)
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
O10.0207 (13)0.0180 (13)0.0157 (12)0.0025 (11)0.0020 (9)0.0013 (10)
O20.0339 (16)0.0201 (14)0.0150 (12)0.0014 (12)0.0016 (11)0.0006 (11)
N10.0142 (14)0.0172 (15)0.0181 (14)0.0001 (12)0.0019 (11)0.0001 (12)
N20.0237 (16)0.0131 (14)0.0130 (13)0.0004 (12)0.0036 (11)0.0013 (11)
C20.0155 (17)0.0147 (16)0.0212 (17)0.0007 (14)0.0062 (13)0.0011 (14)
C30.0155 (16)0.0133 (16)0.0195 (17)0.0007 (13)0.0035 (13)0.0003 (14)
C40.0178 (16)0.0163 (17)0.0175 (16)0.0021 (14)0.0042 (13)0.0010 (14)
C4A0.0121 (16)0.0192 (18)0.0181 (16)0.0026 (14)0.0024 (13)0.0024 (14)
C50.0141 (17)0.0206 (18)0.0175 (17)0.0014 (14)0.0002 (13)0.0014 (14)
C60.0167 (17)0.022 (2)0.0212 (18)0.0030 (14)0.0002 (13)0.0036 (15)
C70.0167 (18)0.022 (2)0.028 (2)0.0007 (15)0.0040 (15)0.0101 (16)
C80.0208 (19)0.0158 (18)0.0256 (19)0.0011 (14)0.0038 (15)0.0010 (14)
C8A0.0176 (17)0.0215 (18)0.0117 (15)0.0002 (14)0.0010 (12)0.0012 (14)
C90.0165 (16)0.0163 (17)0.0157 (16)0.0008 (14)0.0018 (12)0.0001 (13)
C100.0222 (18)0.0217 (19)0.0169 (16)0.0009 (15)0.0072 (13)0.0022 (14)
C110.0163 (17)0.0250 (19)0.0192 (18)0.0025 (15)0.0047 (13)0.0031 (14)
C120.029 (2)0.0186 (19)0.0155 (17)0.0051 (15)0.0025 (14)0.0030 (14)
O30.0197 (13)0.0201 (13)0.0150 (12)0.0018 (11)0.0011 (9)0.0049 (11)
O40.0266 (15)0.0243 (15)0.0361 (16)0.0091 (12)0.0032 (12)0.0054 (13)
O50.0455 (18)0.0140 (14)0.0279 (15)0.0041 (13)0.0089 (13)0.0022 (11)
O60.0429 (17)0.0229 (15)0.0172 (13)0.0014 (13)0.0000 (12)0.0038 (11)
O70.0348 (15)0.0170 (14)0.0207 (13)0.0002 (12)0.0022 (11)0.0069 (11)
O80.0346 (16)0.0162 (13)0.0212 (13)0.0008 (12)0.0061 (11)0.0041 (11)
O90.0310 (15)0.0231 (14)0.0195 (13)0.0036 (12)0.0057 (11)0.0036 (11)
N30.0268 (17)0.0144 (15)0.0179 (15)0.0034 (13)0.0041 (12)0.0003 (12)
N40.0204 (15)0.0182 (16)0.0172 (15)0.0036 (12)0.0060 (12)0.0042 (12)
N50.0173 (15)0.0213 (16)0.0170 (15)0.0028 (13)0.0036 (12)0.0008 (12)
C130.0079 (16)0.0186 (18)0.0135 (15)0.0004 (12)0.0015 (12)0.0022 (12)
C140.0168 (17)0.0131 (17)0.0212 (18)0.0002 (14)0.0011 (13)0.0017 (14)
C150.0147 (17)0.0144 (17)0.0184 (17)0.0013 (13)0.0051 (13)0.0006 (13)
C160.0157 (16)0.0160 (17)0.0159 (16)0.0011 (14)0.0023 (12)0.0057 (14)
C170.0129 (16)0.0159 (17)0.0205 (17)0.0002 (14)0.0062 (13)0.0020 (14)
C180.0148 (17)0.0170 (18)0.0182 (17)0.0003 (14)0.0002 (13)0.0003 (14)
Geometric parameters (Å, º) top
O1—C21.374 (5)C10—C111.361 (6)
O1—C8A1.391 (4)C10—H100.9500
O2—C21.206 (5)C11—H110.9500
N1—C91.342 (5)C12—H12A0.9800
N1—C111.383 (5)C12—H12B0.9800
N1—C31.437 (5)C12—H12C0.9800
N2—C91.328 (5)O3—C131.255 (4)
N2—C101.369 (5)O4—N31.232 (4)
N2—C121.472 (5)O5—N31.235 (4)
C2—C31.464 (5)O6—N41.237 (4)
C3—C41.342 (5)O7—N41.232 (4)
C4—C4A1.433 (5)O8—N51.222 (4)
C4—H40.9500O9—N51.232 (4)
C4A—C8A1.387 (5)N3—C141.458 (5)
C4A—C51.410 (5)N4—C161.460 (4)
C5—C61.387 (6)N5—C181.452 (5)
C5—H50.9500C13—C141.445 (5)
C6—C71.405 (6)C13—C181.458 (5)
C6—H60.9500C14—C151.383 (5)
C7—C81.389 (6)C15—C161.395 (5)
C7—H70.9500C15—H150.9500
C8—C8A1.383 (5)C16—C171.376 (5)
C8—H80.9500C17—C181.383 (5)
C9—H90.9500C17—H170.9500
C2—O1—C8A122.7 (3)C11—C10—H10126.4
C9—N1—C11109.4 (3)N2—C10—H10126.4
C9—N1—C3125.0 (3)C10—C11—N1106.1 (3)
C11—N1—C3125.5 (3)C10—C11—H11126.9
C9—N2—C10109.8 (3)N1—C11—H11126.9
C9—N2—C12125.5 (3)N2—C12—H12A109.5
C10—N2—C12124.7 (3)N2—C12—H12B109.5
O2—C2—O1118.7 (3)H12A—C12—H12B109.5
O2—C2—C3125.7 (4)N2—C12—H12C109.5
O1—C2—C3115.6 (3)H12A—C12—H12C109.5
C4—C3—N1122.0 (3)H12B—C12—H12C109.5
C4—C3—C2122.9 (4)O4—N3—O5124.3 (3)
N1—C3—C2115.1 (3)O4—N3—C14117.8 (3)
C3—C4—C4A119.3 (4)O5—N3—C14117.9 (3)
C3—C4—H4120.4O7—N4—O6124.6 (3)
C4A—C4—H4120.4O7—N4—C16117.1 (3)
C8A—C4A—C5117.8 (3)O6—N4—C16118.3 (3)
C8A—C4A—C4119.1 (3)O8—N5—O9123.7 (3)
C5—C4A—C4123.1 (4)O8—N5—C18118.2 (3)
C6—C5—C4A120.6 (4)O9—N5—C18118.0 (3)
C6—C5—H5119.7O3—C13—C14122.9 (3)
C4A—C5—H5119.7O3—C13—C18125.4 (3)
C5—C6—C7119.7 (3)C14—C13—C18111.5 (3)
C5—C6—H6120.1C15—C14—C13125.6 (3)
C7—C6—H6120.1C15—C14—N3116.9 (3)
C8—C7—C6120.3 (4)C13—C14—N3117.4 (3)
C8—C7—H7119.8C14—C15—C16117.4 (3)
C6—C7—H7119.8C14—C15—H15121.3
C8A—C8—C7118.7 (4)C16—C15—H15121.3
C8A—C8—H8120.6C17—C16—C15122.3 (3)
C7—C8—H8120.6C17—C16—N4119.4 (3)
C8—C8A—C4A122.8 (3)C15—C16—N4118.3 (3)
C8—C8A—O1116.8 (3)C16—C17—C18119.2 (3)
C4A—C8A—O1120.5 (3)C16—C17—H17120.4
N2—C9—N1107.4 (3)C18—C17—H17120.4
N2—C9—H9126.3C17—C18—N5116.2 (3)
N1—C9—H9126.3C17—C18—C13123.9 (3)
C11—C10—N2107.3 (3)N5—C18—C13119.9 (3)
C8A—O1—C2—O2177.4 (3)C12—N2—C10—C11178.7 (3)
C8A—O1—C2—C31.3 (5)N2—C10—C11—N10.6 (4)
C9—N1—C3—C4120.5 (4)C9—N1—C11—C100.5 (4)
C11—N1—C3—C464.0 (5)C3—N1—C11—C10176.6 (3)
C9—N1—C3—C260.9 (5)O3—C13—C14—C15171.5 (4)
C11—N1—C3—C2114.6 (4)C18—C13—C14—C154.2 (5)
O2—C2—C3—C4177.9 (4)O3—C13—C14—N35.2 (5)
O1—C2—C3—C40.7 (5)C18—C13—C14—N3179.1 (3)
O2—C2—C3—N10.7 (5)O4—N3—C14—C1550.5 (5)
O1—C2—C3—N1179.4 (3)O5—N3—C14—C15130.3 (4)
N1—C3—C4—C4A179.1 (3)O4—N3—C14—C13126.5 (4)
C2—C3—C4—C4A0.6 (5)O5—N3—C14—C1352.7 (5)
C3—C4—C4A—C8A1.0 (5)C13—C14—C15—C164.2 (5)
C3—C4—C4A—C5177.9 (3)N3—C14—C15—C16179.1 (3)
C8A—C4A—C5—C60.7 (5)C14—C15—C16—C172.4 (5)
C4—C4A—C5—C6179.6 (3)C14—C15—C16—N4177.7 (3)
C4A—C5—C6—C70.4 (5)O7—N4—C16—C176.1 (5)
C5—C6—C7—C81.3 (6)O6—N4—C16—C17174.2 (3)
C6—C7—C8—C8A1.1 (6)O7—N4—C16—C15174.1 (3)
C7—C8—C8A—C4A0.1 (6)O6—N4—C16—C155.6 (5)
C7—C8—C8A—O1178.3 (3)C15—C16—C17—C181.2 (5)
C5—C4A—C8A—C81.0 (5)N4—C16—C17—C18179.0 (3)
C4—C4A—C8A—C8179.9 (4)C16—C17—C18—N5178.7 (3)
C5—C4A—C8A—O1177.3 (3)C16—C17—C18—C131.5 (5)
C4—C4A—C8A—O11.6 (5)O8—N5—C18—C1727.9 (5)
C2—O1—C8A—C8179.8 (3)O9—N5—C18—C17150.4 (3)
C2—O1—C8A—C4A1.8 (5)O8—N5—C18—C13149.4 (3)
C10—N2—C9—N10.1 (4)O9—N5—C18—C1332.3 (5)
C12—N2—C9—N1179.0 (3)O3—C13—C18—C17172.8 (3)
C11—N1—C9—N20.3 (4)C14—C13—C18—C172.8 (5)
C3—N1—C9—N2176.4 (3)O3—C13—C18—N54.3 (5)
C9—N2—C10—C110.4 (4)C14—C13—C18—N5179.9 (3)
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
C5—H5···O7i0.952.583.349 (4)138
C9—H9···O30.952.333.122 (5)140
C10—H10···O9ii0.952.513.303 (5)141
C11—H11···O3iii0.952.423.196 (5)139
C11—H11···O5iii0.952.513.231 (5)132
C12—H12A···O2iv0.982.583.360 (5)137
C12—H12B···O2v0.982.483.448 (5)171
C12—H12C···O3iv0.982.393.269 (4)148
C12—H12C···O9iv0.982.423.160 (5)132
C17—H17···O5vi0.952.403.345 (5)172
Symmetry codes: (i) x+2, y1/2, z+2; (ii) x+1, y1, z; (iii) x+1, y, z; (iv) x+1, y1/2, z+1; (v) x, y1, z; (vi) x, y+1, z.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
C5—H5···O7i0.952.583.349 (4)138
C9—H9···O30.952.333.122 (5)140
C10—H10···O9ii0.952.513.303 (5)141
C11—H11···O3iii0.952.423.196 (5)139
C11—H11···O5iii0.952.513.231 (5)132
C12—H12A···O2iv0.982.583.360 (5)137
C12—H12B···O2v0.982.483.448 (5)171
C12—H12C···O3iv0.982.393.269 (4)148
C12—H12C···O9iv0.982.423.160 (5)132
C17—H17···O5vi0.952.403.345 (5)172
Symmetry codes: (i) x+2, y1/2, z+2; (ii) x+1, y1, z; (iii) x+1, y, z; (iv) x+1, y1/2, z+1; (v) x, y1, z; (vi) x, y+1, z.
Acknowledgements top

The authors are grateful to the Russian Foundation for Basic Research (project No. 12–03-93000-Viet-a) and the Vietnam Academy of Science and Technology (grant VAST.HTQT.NGA. 06/2012–2013) for the financial support of this work.

references
References top

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