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Acta Cryst. (2009). E65, m1633    [ doi:10.1107/S1600536809048375 ]

Hexaaquazinc(II) bis(4-hydroxybenzenesulfonate) dihydrate

S. Gao and S. W. Ng

Abstract top

The asymmetric unit of the title hydrated molecular salt, [Zn(H2O)6](C6H5O4S)2·2H2O, contains two half-cations, two anions and two uncoordinated water molecules. Both cations are completed by crystallographic inversion symmetry, generating almost regular ZnO6 octahedra. In the crystal, the cations, anions and uncoordinated water molecules are linked by O-H...O hydrogen bonds, forming a three-dimensional network.

Related literature top

For the isostructural cobalt and nickel analogs of the title compound, see: Du et al. (2007) and Kosnic et al. (1992), respectively.

Experimental top

One millimolar quantities each of zinc dichloride and p-hydroxylbenzenesulfonic acid were dissolved in water; colourless prisms of (I) were isolated after a few days.

Refinement top

Carbon-bound H-atoms were placed in calculated positions (C—H = 0.93 Å) and were included in the refinement in the riding model approximation, with Uiso(H) = 1.2Ueq(C). The water H-atoms were located in a difference Fourier map, and were refined with a distance restraint of O–H = 0.85±0.01 Å; their Uiso values were refined.

Computing details top

Data collection: RAPID-AUTO (Rigaku, 1998); cell refinement: RAPID-AUTO (Rigaku, 1998); data reduction: CrystalClear (Rigaku/MSC, 2002); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: X-SEED (Barbour, 2001); software used to prepare material for publication: publCIF (Westrip, 2009).

Figures top
[Figure 1] Fig. 1. The molecular structure of (I) shown at the 50% probability level; hydrogen atoms are drawn as spheres of arbitrary radius. Unlabelled atoms bonded to Zn1 and Zn2 are generated by the symmetry operations (1–x, 1–y, 1–z) and (–x, 1–y, 1–z), respectively.
Hexaaquazinc(II) bis(4-hydroxybenzenesulfonate) dihydrate top
Crystal data top
[Zn(H2O)6](C6H5O4S)2·2H2OF(000) = 1152
Mr = 555.82Dx = 1.702 Mg m−3
Monoclinic, P21/cMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybcCell parameters from 12326 reflections
a = 11.7957 (5) Åθ = 3.2–27.5°
b = 7.2590 (4) ŵ = 1.40 mm−1
c = 25.3992 (11) ÅT = 293 K
β = 94.340 (1)°Prism, colorless
V = 2168.57 (18) Å30.19 × 0.19 × 0.15 mm
Z = 4
Data collection top
Rigaku R-AXIS RAPID IP
diffractometer		4940 independent reflections
Radiation source: fine-focus sealed tube3319 reflections with I > 2σ(I)
graphiteRint = 0.040
ω scanθmax = 27.5°, θmin = 3.2°
Absorption correction: multi-scan
(ABSCOR; Higashi, 1995)		h = −15→15
Tmin = 0.777, Tmax = 0.817k = −9→9
20669 measured reflectionsl = −29→32
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)] = 0.030Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.082H atoms treated by a mixture of independent and constrained refinement
S = 1.03 w = 1/[σ2(Fo2) + (0.0405P)2 + 0.0996P]
where P = (Fo2 + 2Fc2)/3
4940 reflections(Δ/σ)max = 0.001
355 parametersΔρmax = 0.33 e Å−3
18 restraintsΔρmin = −0.40 e Å−3
Crystal data top
[Zn(H2O)6](C6H5O4S)2·2H2OV = 2168.57 (18) Å3
Mr = 555.82Z = 4
Monoclinic, P21/cMo Kα radiation
a = 11.7957 (5) ŵ = 1.40 mm−1
b = 7.2590 (4) ÅT = 293 K
c = 25.3992 (11) Å0.19 × 0.19 × 0.15 mm
β = 94.340 (1)°
Data collection top
Rigaku R-AXIS RAPID IP
diffractometer		4940 independent reflections
Absorption correction: multi-scan
(ABSCOR; Higashi, 1995)		3319 reflections with I > 2σ(I)
Tmin = 0.777, Tmax = 0.817Rint = 0.040
20669 measured reflectionsθmax = 27.5°
Refinement top
R[F2 > 2σ(F2)] = 0.030H atoms treated by a mixture of independent and constrained refinement
wR(F2) = 0.082Δρmax = 0.33 e Å−3
S = 1.03Δρmin = −0.40 e Å−3
4940 reflectionsAbsolute structure: ?
355 parametersFlack parameter: ?
18 restraintsRogers 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.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/Ueq
Zn10.50000.50000.50000.02424 (10)
Zn20.00000.50000.50000.02415 (10)
S10.60479 (4)0.93447 (8)0.365706 (19)0.02769 (13)
S20.09496 (4)0.94376 (8)0.36609 (2)0.02778 (13)
O10.68413 (13)0.8168 (2)0.39683 (6)0.0436 (4)
O20.48651 (12)0.8933 (2)0.37348 (6)0.0367 (4)
O30.63007 (14)1.1293 (2)0.37462 (6)0.0410 (4)
O40.68090 (14)0.8205 (2)0.14198 (5)0.0405 (4)
O50.17678 (13)0.8315 (2)0.39749 (6)0.0393 (4)
O6−0.02227 (12)0.8971 (2)0.37348 (6)0.0385 (4)
O70.11697 (13)1.1408 (2)0.37451 (5)0.0359 (4)
O80.17676 (14)0.8359 (2)0.14229 (6)0.0412 (4)
O1W0.64682 (13)0.6615 (2)0.49333 (6)0.0300 (3)
O2W0.43907 (16)0.5631 (3)0.42376 (6)0.0425 (4)
O3W0.41407 (15)0.7258 (2)0.52682 (7)0.0412 (4)
O4W0.15569 (14)0.5586 (3)0.47077 (8)0.0463 (5)
O5W−0.08685 (14)0.5987 (2)0.43118 (6)0.0361 (4)
O6W−0.01038 (14)0.7610 (2)0.53290 (6)0.0349 (4)
O7W0.64137 (15)0.9889 (2)0.54413 (7)0.0371 (4)
O8W0.16394 (15)0.9973 (3)0.52453 (8)0.0410 (4)
C10.62536 (16)0.8899 (3)0.29904 (7)0.0243 (4)
C20.73038 (17)0.8248 (3)0.28587 (8)0.0290 (5)
H20.78710.79710.31210.035*
C30.74935 (17)0.8018 (3)0.23319 (8)0.0296 (5)
H30.81940.75910.22390.035*
C40.66492 (17)0.8420 (3)0.19444 (7)0.0271 (5)
C50.55995 (17)0.9073 (3)0.20759 (8)0.0323 (5)
H50.50310.93400.18130.039*
C60.54101 (17)0.9321 (3)0.26028 (8)0.0299 (5)
H60.47140.97710.26950.036*
C70.11767 (16)0.8998 (3)0.29945 (8)0.0254 (4)
C80.21941 (18)0.8220 (3)0.28634 (8)0.0315 (5)
H8A0.27340.78470.31280.038*
C90.23990 (18)0.8004 (3)0.23389 (8)0.0329 (5)
H90.30800.74870.22500.039*
C100.15988 (18)0.8550 (3)0.19468 (8)0.0283 (5)
C110.05795 (17)0.9337 (3)0.20744 (8)0.0305 (5)
H11A0.00400.97070.18100.037*
C120.03764 (17)0.9561 (3)0.26017 (8)0.0283 (5)
H12A−0.03001.00920.26910.034*
H40.7446 (14)0.769 (4)0.1384 (11)0.068 (9)*
H80.2355 (15)0.769 (3)0.1402 (11)0.060 (9)*
H110.659 (2)0.694 (3)0.4625 (5)0.048 (8)*
H120.642 (2)0.762 (2)0.5103 (9)0.056 (8)*
H210.459 (3)0.657 (3)0.4072 (11)0.084 (11)*
H220.3924 (17)0.494 (3)0.4062 (9)0.048 (8)*
H310.409 (2)0.761 (4)0.5581 (5)0.054 (8)*
H320.393 (2)0.813 (3)0.5066 (9)0.065 (10)*
H410.167 (2)0.639 (3)0.4480 (8)0.058 (8)*
H420.2171 (14)0.511 (3)0.4836 (10)0.065 (10)*
H51−0.063 (2)0.690 (3)0.4144 (10)0.073 (10)*
H52−0.122 (2)0.518 (3)0.4119 (9)0.047 (8)*
H610.0480 (18)0.831 (4)0.5340 (13)0.102 (13)*
H62−0.037 (2)0.768 (4)0.5627 (6)0.072 (10)*
H710.7069 (12)1.027 (4)0.5550 (10)0.053 (9)*
H720.603 (2)1.004 (4)0.5705 (8)0.084 (12)*
H810.205 (2)1.039 (4)0.5508 (8)0.059 (9)*
H820.121 (3)1.077 (4)0.5089 (13)0.112 (15)*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Zn10.02731 (18)0.0253 (2)0.02019 (17)−0.00203 (14)0.00222 (14)−0.00013 (14)
Zn20.02819 (18)0.0218 (2)0.02263 (18)−0.00079 (14)0.00322 (14)0.00139 (14)
S10.0340 (3)0.0312 (3)0.0185 (3)−0.0047 (2)0.0060 (2)0.0007 (2)
S20.0349 (3)0.0284 (3)0.0206 (3)−0.0069 (2)0.0056 (2)0.0014 (2)
O10.0478 (9)0.0573 (12)0.0259 (8)0.0082 (8)0.0034 (7)0.0106 (8)
O20.0347 (8)0.0444 (10)0.0323 (8)−0.0053 (7)0.0119 (7)0.0013 (7)
O30.0598 (10)0.0359 (10)0.0293 (8)−0.0147 (8)0.0151 (8)−0.0076 (7)
O40.0484 (10)0.0537 (12)0.0198 (7)0.0188 (8)0.0061 (7)−0.0021 (7)
O50.0461 (9)0.0444 (11)0.0272 (8)−0.0023 (8)0.0004 (7)0.0092 (7)
O60.0368 (8)0.0438 (11)0.0362 (9)−0.0103 (7)0.0122 (7)0.0016 (8)
O70.0511 (9)0.0290 (9)0.0292 (8)−0.0110 (7)0.0131 (7)−0.0067 (7)
O80.0520 (11)0.0486 (11)0.0235 (8)0.0167 (9)0.0068 (8)−0.0010 (7)
O1W0.0377 (8)0.0267 (9)0.0262 (8)−0.0057 (7)0.0065 (7)0.0017 (7)
O2W0.0637 (12)0.0380 (11)0.0237 (9)−0.0182 (9)−0.0102 (8)0.0058 (8)
O3W0.0605 (11)0.0352 (11)0.0288 (9)0.0136 (9)0.0087 (8)−0.0025 (8)
O4W0.0312 (9)0.0519 (12)0.0574 (12)0.0040 (9)0.0129 (9)0.0285 (10)
O5W0.0480 (10)0.0315 (10)0.0274 (9)−0.0078 (8)−0.0060 (8)0.0057 (8)
O6W0.0509 (10)0.0255 (9)0.0297 (9)−0.0042 (8)0.0112 (8)−0.0049 (7)
O7W0.0358 (9)0.0393 (10)0.0370 (9)−0.0047 (8)0.0078 (8)−0.0056 (8)
O8W0.0379 (10)0.0417 (11)0.0431 (11)−0.0049 (8)0.0017 (9)−0.0040 (9)
C10.0297 (11)0.0248 (12)0.0189 (9)−0.0033 (9)0.0048 (9)0.0011 (8)
C20.0312 (11)0.0305 (13)0.0247 (10)0.0054 (9)−0.0023 (9)−0.0004 (9)
C30.0303 (11)0.0314 (13)0.0274 (11)0.0086 (9)0.0053 (9)−0.0025 (9)
C40.0372 (12)0.0241 (12)0.0202 (10)0.0030 (9)0.0037 (9)−0.0026 (9)
C50.0308 (11)0.0413 (15)0.0241 (11)0.0051 (10)−0.0015 (9)−0.0007 (10)
C60.0260 (10)0.0362 (13)0.0278 (11)0.0016 (10)0.0045 (9)−0.0003 (9)
C70.0308 (11)0.0221 (12)0.0235 (10)−0.0046 (9)0.0024 (9)−0.0016 (9)
C80.0347 (12)0.0315 (13)0.0277 (11)0.0052 (10)−0.0019 (10)0.0004 (9)
C90.0343 (12)0.0332 (13)0.0316 (12)0.0089 (10)0.0050 (10)−0.0027 (10)
C100.0381 (12)0.0251 (12)0.0221 (10)0.0008 (9)0.0057 (9)−0.0004 (9)
C110.0308 (11)0.0332 (13)0.0267 (11)0.0052 (10)−0.0036 (9)0.0004 (9)
C120.0273 (11)0.0285 (12)0.0296 (12)0.0000 (9)0.0046 (9)−0.0021 (9)
Geometric parameters (Å, °) top
Zn1—O2W2.0654 (16)O4W—H410.840 (10)
Zn1—O2Wi2.0654 (16)O4W—H420.847 (10)
Zn1—O3W2.0692 (16)O5W—H510.846 (10)
Zn1—O3Wi2.0692 (16)O5W—H520.849 (10)
Zn1—O1Wi2.1087 (15)O6W—H610.854 (10)
Zn1—O1W2.1087 (15)O6W—H620.842 (10)
Zn2—O4Wii2.0763 (15)O7W—H710.847 (10)
Zn2—O4W2.0763 (15)O7W—H720.846 (10)
Zn2—O6W2.0780 (16)O8W—H810.849 (10)
Zn2—O6Wii2.0780 (16)O8W—H820.848 (10)
Zn2—O5W2.0849 (16)C1—C61.380 (3)
Zn2—O5Wii2.0849 (16)C1—C21.390 (3)
S1—O21.4549 (14)C2—C31.383 (3)
S1—O11.4555 (17)C2—H20.9300
S1—O31.4598 (17)C3—C41.377 (3)
S1—C11.7587 (19)C3—H30.9300
S2—O61.4495 (15)C4—C51.390 (3)
S2—O51.4542 (16)C5—C61.385 (3)
S2—O71.4666 (16)C5—H50.9300
S2—C71.763 (2)C6—H60.9300
O4—C41.369 (2)C7—C121.382 (3)
O4—H40.851 (10)C7—C81.389 (3)
O8—C101.367 (2)C8—C91.381 (3)
O8—H80.849 (10)C8—H8A0.9300
O1W—H110.843 (10)C9—C101.378 (3)
O1W—H120.849 (10)C9—H90.9300
O2W—H210.845 (10)C10—C111.391 (3)
O2W—H220.846 (10)C11—C121.388 (3)
O3W—H310.840 (10)C11—H11A0.9300
O3W—H320.841 (10)C12—H12A0.9300
O2W—Zn1—O2Wi180.0Zn1—O3W—H31128.5 (19)
O2W—Zn1—O3W89.34 (7)Zn1—O3W—H32121.7 (19)
O2Wi—Zn1—O3W90.66 (7)H31—O3W—H32108 (3)
O2W—Zn1—O3Wi90.66 (7)Zn2—O4W—H41125.1 (18)
O2Wi—Zn1—O3Wi89.34 (7)Zn2—O4W—H42122.1 (18)
O3W—Zn1—O3Wi180.0H41—O4W—H42112 (2)
O2W—Zn1—O1Wi88.23 (6)Zn2—O5W—H51122 (2)
O2Wi—Zn1—O1Wi91.77 (6)Zn2—O5W—H52115.8 (18)
O3W—Zn1—O1Wi89.32 (7)H51—O5W—H52114 (3)
O3Wi—Zn1—O1Wi90.68 (7)Zn2—O6W—H61119 (2)
O2W—Zn1—O1W91.77 (6)Zn2—O6W—H62117 (2)
O2Wi—Zn1—O1W88.23 (6)H61—O6W—H62107 (3)
O3W—Zn1—O1W90.68 (7)H71—O7W—H72104 (3)
O3Wi—Zn1—O1W89.32 (7)H81—O8W—H82114 (3)
O1Wi—Zn1—O1W180.0C6—C1—C2120.75 (18)
O4Wii—Zn2—O4W180.0C6—C1—S1120.10 (15)
O4Wii—Zn2—O6W87.85 (7)C2—C1—S1118.98 (16)
O4W—Zn2—O6W92.15 (7)C3—C2—C1119.08 (19)
O4Wii—Zn2—O6Wii92.15 (7)C3—C2—H2120.5
O4W—Zn2—O6Wii87.85 (7)C1—C2—H2120.5
O6W—Zn2—O6Wii180.0C4—C3—C2120.31 (18)
O4Wii—Zn2—O5W88.76 (7)C4—C3—H3119.8
O4W—Zn2—O5W91.24 (7)C2—C3—H3119.8
O6W—Zn2—O5W89.07 (7)O4—C4—C3121.75 (18)
O6Wii—Zn2—O5W90.93 (7)O4—C4—C5117.61 (19)
O4Wii—Zn2—O5Wii91.24 (7)C3—C4—C5120.64 (18)
O4W—Zn2—O5Wii88.76 (7)C6—C5—C4119.2 (2)
O6W—Zn2—O5Wii90.93 (7)C6—C5—H5120.4
O6Wii—Zn2—O5Wii89.07 (7)C4—C5—H5120.4
O5W—Zn2—O5Wii180.0C1—C6—C5120.01 (19)
O2—S1—O1112.93 (10)C1—C6—H6120.0
O2—S1—O3111.38 (10)C5—C6—H6120.0
O1—S1—O3111.66 (10)C12—C7—C8120.14 (19)
O2—S1—C1107.25 (9)C12—C7—S2119.64 (15)
O1—S1—C1106.56 (9)C8—C7—S2120.03 (16)
O3—S1—C1106.63 (9)C9—C8—C7119.7 (2)
O6—S2—O5113.46 (9)C9—C8—H8A120.2
O6—S2—O7111.65 (9)C7—C8—H8A120.2
O5—S2—O7111.33 (10)C10—C9—C8120.29 (19)
O6—S2—C7107.33 (9)C10—C9—H9119.9
O5—S2—C7106.41 (9)C8—C9—H9119.9
O7—S2—C7106.16 (9)O8—C10—C9122.13 (18)
C4—O4—H4109.9 (19)O8—C10—C11117.43 (19)
C10—O8—H8107.3 (19)C9—C10—C11120.44 (18)
Zn1—O1W—H11115.8 (17)C10—C11—C12119.2 (2)
Zn1—O1W—H12110.2 (17)C10—C11—H11A120.4
H11—O1W—H12105 (2)C12—C11—H11A120.4
Zn1—O2W—H21124 (2)C7—C12—C11120.23 (19)
Zn1—O2W—H22121.9 (19)C7—C12—H12A119.9
H21—O2W—H22114 (3)C11—C12—H12A119.9
O2—S1—C1—C637.7 (2)O6—S2—C7—C1244.6 (2)
O1—S1—C1—C6158.94 (18)O5—S2—C7—C12166.37 (17)
O3—S1—C1—C6−81.67 (19)O7—S2—C7—C12−74.94 (19)
O2—S1—C1—C2−146.98 (17)O6—S2—C7—C8−140.39 (17)
O1—S1—C1—C2−25.8 (2)O5—S2—C7—C8−18.6 (2)
O3—S1—C1—C293.60 (18)O7—S2—C7—C8100.09 (18)
C6—C1—C2—C3−0.3 (3)C12—C7—C8—C9−0.3 (3)
S1—C1—C2—C3−175.53 (16)S2—C7—C8—C9−175.31 (17)
C1—C2—C3—C4−0.4 (3)C7—C8—C9—C10−0.2 (3)
C2—C3—C4—O4−179.9 (2)C8—C9—C10—O8−179.9 (2)
C2—C3—C4—C50.5 (3)C8—C9—C10—C110.4 (3)
O4—C4—C5—C6−179.5 (2)O8—C10—C11—C12−179.8 (2)
C3—C4—C5—C60.1 (3)C9—C10—C11—C12−0.1 (3)
C2—C1—C6—C50.9 (3)C8—C7—C12—C110.6 (3)
S1—C1—C6—C5176.08 (17)S2—C7—C12—C11175.62 (17)
C4—C5—C6—C1−0.8 (3)C10—C11—C12—C7−0.4 (3)
Symmetry codes: (i) −x+1, −y+1, −z+1; (ii) −x, −y+1, −z+1.
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O1w—H11···O10.84 (1)1.93 (1)2.763 (2)169 (3)
O1w—H12···O7w0.85 (1)1.86 (1)2.707 (2)175 (2)
O2w—H21···O20.85 (1)1.95 (1)2.792 (2)172 (3)
O2w—H22···O4iii0.85 (1)1.92 (1)2.745 (2)166 (3)
O3w—H31···O3iv0.84 (1)1.97 (1)2.800 (2)168 (3)
O3w—H32···O7wiv0.84 (1)1.95 (1)2.790 (2)174 (3)
O4w—H41···O50.84 (1)1.91 (1)2.742 (2)174 (3)
O4w—H42···O1wi0.85 (1)2.08 (1)2.914 (2)167 (3)
O5w—H51···O60.85 (1)1.91 (1)2.755 (2)175 (3)
O5w—H52···O8v0.85 (1)1.98 (1)2.819 (2)168 (3)
O6w—H61···O8w0.85 (1)1.85 (1)2.698 (2)170 (3)
O6w—H62···O7vi0.84 (1)2.03 (1)2.8401 (19)163 (3)
O4—H4···O7iii0.85 (1)1.93 (1)2.777 (2)175 (3)
O8—H8···O3iii0.85 (1)1.95 (1)2.788 (2)172 (3)
Symmetry codes: (iii) −x+1, y−1/2, −z+1/2; (iv) −x+1, −y+2, −z+1; (i) −x+1, −y+1, −z+1; (v) −x, y−1/2, −z+1/2; (vi) −x, −y+2, −z+1.
Table 1
Selected geometric parameters (Å) top
Zn1—O2W2.0654 (16)Zn2—O4W2.0763 (15)
Zn1—O3W2.0692 (16)Zn2—O6W2.0780 (16)
Zn1—O1W2.1087 (15)Zn2—O5W2.0849 (16)
Table 2
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O1w—H11···O10.84 (1)1.93 (1)2.763 (2)169 (3)
O1w—H12···O7w0.85 (1)1.86 (1)2.707 (2)175 (2)
O2w—H21···O20.85 (1)1.95 (1)2.792 (2)172 (3)
O2w—H22···O4i0.85 (1)1.92 (1)2.745 (2)166 (3)
O3w—H31···O3ii0.84 (1)1.97 (1)2.800 (2)168 (3)
O3w—H32···O7wii0.84 (1)1.95 (1)2.790 (2)174 (3)
O4w—H41···O50.84 (1)1.91 (1)2.742 (2)174 (3)
O4w—H42···O1wiii0.85 (1)2.08 (1)2.914 (2)167 (3)
O5w—H51···O60.85 (1)1.91 (1)2.755 (2)175 (3)
O5w—H52···O8iv0.85 (1)1.98 (1)2.819 (2)168 (3)
O6w—H61···O8w0.85 (1)1.85 (1)2.698 (2)170 (3)
O6w—H62···O7v0.84 (1)2.03 (1)2.8401 (19)163 (3)
O4—H4···O7i0.85 (1)1.93 (1)2.777 (2)175 (3)
O8—H8···O3i0.85 (1)1.95 (1)2.788 (2)172 (3)
Symmetry codes: (i) −x+1, y−1/2, −z+1/2; (ii) −x+1, −y+2, −z+1; (iii) −x+1, −y+1, −z+1; (iv) −x, y−1/2, −z+1/2; (v) −x, −y+2, −z+1.
Acknowledgements top

We thank the Key Project of Natural Science Foundation of Heilongjiang Province (No. ZD200903), the Scientific Fund of Remarkable Teachers of Heilongjiang Province (No. 1054 G036), Heilongjiang University and the University of Malaya for supporting this study.

references
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