metal-organic compounds\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

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ISSN: 2056-9890

catena-Poly[[[tri­aqua­copper(II)]-μ2-pyrazine-2,3-di­carboxyl­ato] monohydrate]

aDepartment of Chemistry, Sichuan University of Science and Engineering, Zigong, 643000, People's Republic of China, and bDepartment of Materials and Chemical Engineering, Sichuan University of Science and Engineering, Zigong 643000,People's Republic of China
*Correspondence e-mail: wuweipingzg@126.com

(Received 16 November 2007; accepted 24 November 2007; online 6 December 2007)

The Cu atom in the title complex, {[Cu(C6H2N2O4)(H2O)3]·H2O}n or {[Cu(L)(H2O)3]·H2O}n (L is pyrazine-2,3-dicarbox­yl­ate), displays octa­hedral coordination formed by the ligand L and three coordinated water mol­ecules. The ligand L is tridentate, with one N atom of the pyrazine ring and one O atom of one carboxyl­ate group forming a chelate ring, whereas one O atom from the second carboxyl­ate group is coordinated to another Cu atom. The ligand L links mol­ecules to form an infinite chain parallel to the [101] direction. The chains are further linked through O—H⋯O and O—H⋯N hydrogen bonds involving the water mol­ecules to build up a three-dimensional network.

Related literature

For related literature, see: Gokel et al. (2004[Gokel, G. W., Leevy, W. M. & Weber, W. E. (2004). Chem. Rev. 104, 2723-2750.]); Shan et al. (2001[Shan, B. Z., Zhao, O., Goswami, N., Eichhorn, D. M. & Rillema, D. P. (2001). Coord. Chem. Rev. 211, 117-144.]); Starosta & Leciejewicz (2005[Starosta, W. & Leciejewicz, J. (2005). J. Coord. Chem. 58, 963-968.]); Takusagawa & Shimada (1973[Takusagawa, T. & Shimada, A. (1973). Chem. Lett. pp. 1121-1126.]); Tombul et al. (2007[Tombul, M., Güven, K. & Büyükgüngör, O. (2007). Acta Cryst. E63, m1783-m1784.]); Ptasiewicz-Bak & Leciejewicz (1997a[Ptasiewicz-Bak, H. & Leciejewicz, J. (1997a). Pol. J. Chem. 71, 493-500.],b[Ptasiewicz-Bak, H. & Leciejewicz, J. (1997b). Pol. J. Chem. 71, 1603-1610.]).

[Scheme 1]

Experimental

Crystal data
  • [Cu(C6H2N2O4)(H2O)3]·H2O

  • Mr = 301.70

  • Monoclinic, C c

  • a = 8.4254 (4) Å

  • b = 18.0692 (8) Å

  • c = 7.4187 (3) Å

  • β = 114.4120 (10)°

  • V = 1028.45 (8) Å3

  • Z = 4

  • Mo Kα radiation

  • μ = 2.16 mm−1

  • T = 298 (2) K

  • 0.28 × 0.25 × 0.17 mm

Data collection
  • Bruker APEXII area-detector diffractometer

  • Absorption correction: multi-scan (SADABS; Sheldrick, 2004[Sheldrick, G. M. (2004). SADABS. University of Göttingen, Germany.]) Tmin = 0.519, Tmax = 0.660

  • 3079 measured reflections

  • 1683 independent reflections

  • 1654 reflections with I > 2σ(I)

  • Rint = 0.067

Refinement
  • R[F2 > 2σ(F2)] = 0.036

  • wR(F2) = 0.089

  • S = 1.11

  • 1683 reflections

  • 156 parameters

  • 2 restraints

  • H-atom parameters constrained

  • Δρmax = 0.67 e Å−3

  • Δρmin = −1.08 e Å−3

  • Absolute structure: Flack (1983[Flack, H. D. (1983). Acta Cryst. A39, 876-881.]), 758 Friedel pairs

  • Flack parameter: −0.04 (2)

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A D—H H⋯A DA D—H⋯A
O5—H5A⋯O7i 0.84 2.30 2.953 (4) 134
O5—H5B⋯O8 0.86 1.79 2.630 (6) 168
O6—H6A⋯O4ii 0.85 1.98 2.837 (4) 176
O6—H6A⋯O3ii 0.85 2.57 3.170 (5) 128
O6—H6B⋯O4iii 0.84 1.87 2.691 (4) 167
O7—H7A⋯N2iii 0.83 1.98 2.817 (5) 177
O7—H7B⋯O1iv 0.83 1.90 2.719 (6) 166
O8—H8A⋯O4iv 0.85 2.02 2.864 (6) 176
O8—H8B⋯O1v 0.85 2.11 2.898 (6) 155
Symmetry codes: (i) [x, -y, z+{\script{1\over 2}}]; (ii) [x+{\script{1\over 2}}, y-{\script{1\over 2}}, z]; (iii) [x+{\script{1\over 2}}, -y+{\script{1\over 2}}, z-{\script{1\over 2}}]; (iv) [x-{\script{1\over 2}}, -y+{\script{1\over 2}}, z-{\script{1\over 2}}]; (v) [x-{\script{1\over 2}}, y-{\script{1\over 2}}, z].

Data collection: APEX2 (Bruker, 2004[Bruker (2004). APEX2 (Version 1.22) and SAINT (Version 6.36A). Bruker AXS Inc., Madison, Wisconsin, USA.]); cell refinement: SAINT (Bruker, 2004[Bruker (2004). APEX2 (Version 1.22) and SAINT (Version 6.36A). Bruker AXS Inc., Madison, Wisconsin, USA.]); data reduction: SAINT; program(s) used to solve structure: SHELXS97 (Sheldrick, 1997[Sheldrick, G. M. (1997). SHELXS97 and SHELXL97. University of Göttingen, Germany.]); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997[Sheldrick, G. M. (1997). SHELXS97 and SHELXL97. University of Göttingen, Germany.]); molecular graphics: ORTEPIII (Burnett & Johnson, 1996[Burnett, M. N. & Johnson, C. K. (1996). ORTEPIII. Report ORNL-6895. Oak Ridge National Laboratory, Tennessee, USA.]) and PLATON (Spek, 2003[Spek, A. L. (2003). J. Appl. Cryst. 36, 7-13.]); software used to prepare material for publication: SHELXL97.

Supporting information


Comment top

Transition metal complexes with bipyridine derivatives are suitable models for the study of excited state dynamics. In addition, they are of interest for the development of light-energy conversion devices and optical sensors (Gokel et al., 2004; Shan et al., 2001). Since the single-crystal X-ray analysis of pyrazine-2,3 dicarboxylic acid was first determined (Takusagawa & Shimada, 1973), a variety of metal-organic compounds of pyrazine-2,3-dicarboxylic acid have been characterized crystallographically, due to growing interest in supramolecular chemistry (Tombul et al., 2007). These include the calcium (Ptasiewicz-Bak & Leciejewicz, 1997a; Starosta & Leciejewicz, 2005) and magnesium (Ptasiewicz-Bak & Leciejewicz, 1997b) complexes. In this paper, we report the synthesis and crystal structure of the title complex,(I).

The CuII ion displays octahedral coordination formed by the one L ligand and three coordinated water molecules. The ligand L is tridentate with the N atom of the pyridine ring and one O atom of one carboxylate forming a chelate ring whereas one O atom from the second carboxylate is coordinated to another Cu atom (Fig. 1). Then the ligand L links molecules to form an infinite chain parallel to the [1 O 1] direction. The chains are further linked through O—H···O and O—H···N involving the water molecules to build up a three dimensionnal network (Table 1).

Related literature top

For related literature, see: Gokel et al. (2004); Shan et al. (2001); Starosta & Leciejewicz (2005); Takusagawa & Shimada (1973); Tombul et al. (2007); Ptasiewicz-Bak & Leciejewicz (1997a,b).

Experimental top

L (0.031 g, 0.018 mmol), CuSO4 (0.018 g, 0.016 mmol) and NaOH(0.048 mmol,0.12 mmol), were added in a mixed solvent of ethanol, the mixture was heated for three hours under reflux. during the process stirring and influx were required. The resultant was then filtered to give a pure solution which was infiltrated by diethyl ether freely in a closed vessel, two weeks later some single crystals of the size suitable for X-Ray diffraction analysis.

Refinement top

All H atoms attached to C atoms were fixed geometrically and treated as riding with C—H = 0.93 Å and Uiso(H) = 1.2Ueq(C). H atoms of water molecule were located in difference Fourier maps and included in the subsequent refinement using restraints (O—H= 0.85 (1)Å and H···H= 1.39 (2) Å) with Uiso(H) = 1.5Ueq(O). In the last stage of refinement, they were treated as riding on their parent O atoms.

Structure description top

Transition metal complexes with bipyridine derivatives are suitable models for the study of excited state dynamics. In addition, they are of interest for the development of light-energy conversion devices and optical sensors (Gokel et al., 2004; Shan et al., 2001). Since the single-crystal X-ray analysis of pyrazine-2,3 dicarboxylic acid was first determined (Takusagawa & Shimada, 1973), a variety of metal-organic compounds of pyrazine-2,3-dicarboxylic acid have been characterized crystallographically, due to growing interest in supramolecular chemistry (Tombul et al., 2007). These include the calcium (Ptasiewicz-Bak & Leciejewicz, 1997a; Starosta & Leciejewicz, 2005) and magnesium (Ptasiewicz-Bak & Leciejewicz, 1997b) complexes. In this paper, we report the synthesis and crystal structure of the title complex,(I).

The CuII ion displays octahedral coordination formed by the one L ligand and three coordinated water molecules. The ligand L is tridentate with the N atom of the pyridine ring and one O atom of one carboxylate forming a chelate ring whereas one O atom from the second carboxylate is coordinated to another Cu atom (Fig. 1). Then the ligand L links molecules to form an infinite chain parallel to the [1 O 1] direction. The chains are further linked through O—H···O and O—H···N involving the water molecules to build up a three dimensionnal network (Table 1).

For related literature, see: Gokel et al. (2004); Shan et al. (2001); Starosta & Leciejewicz (2005); Takusagawa & Shimada (1973); Tombul et al. (2007); Ptasiewicz-Bak & Leciejewicz (1997a,b).

Computing details top

Data collection: APEX2 (Bruker, 2004); cell refinement: SAINT (Bruker, 2004); data reduction: SAINT (Bruker, 2004); program(s) used to solve structure: SHELXS97 (Sheldrick, 1997); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997); molecular graphics: ORTEPIII (Burnett & Johnson, 1996) and PLATON (Spek, 2003); software used to prepare material for publication: SHELXL97 (Sheldrick, 1997).

Figures top
[Figure 1] Fig. 1. View of compound (I) with the atom-labelling scheme. Displacement ellipsoids are drawn at the 30% probability level. H atoms have been omitted for clarity. [Symmetry codes: (i) x - 1/2, 1/2 - y, z - 1/2; (ii) 1/2 + x, 1/2 - y, 1/2 + z]
catena-Poly[[[triaquacopper(II)]-µ2-pyrazine-2,3-dicarboxylato] monohydrate] top
Crystal data top
[Cu(C6H2N2O4)(H2O)3]·H2OF(000) = 612
Mr = 301.70Dx = 1.949 Mg m3
Monoclinic, CcMo Kα radiation, λ = 0.71073 Å
Hall symbol: C -2ycCell parameters from 1683 reflections
a = 8.4254 (4) Åθ = 2.3–25.2°
b = 18.0692 (8) ŵ = 2.16 mm1
c = 7.4187 (3) ÅT = 298 K
β = 114.412 (1)°Block, blue
V = 1028.45 (8) Å30.28 × 0.25 × 0.17 mm
Z = 4
Data collection top
Bruker APEXII area-detector
diffractometer
1683 independent reflections
Radiation source: fine-focus sealed tube1654 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.067
φ and ω scanθmax = 25.2°, θmin = 2.3°
Absorption correction: multi-scan
(SADABS; Sheldrick, 2004)
h = 109
Tmin = 0.519, Tmax = 0.660k = 2120
3079 measured reflectionsl = 88
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)] = 0.036H-atom parameters constrained
wR(F2) = 0.089 w = 1/[σ2(Fo2) + (0.0632P)2]
where P = (Fo2 + 2Fc2)/3
S = 1.11(Δ/σ)max < 0.001
1683 reflectionsΔρmax = 0.67 e Å3
156 parametersΔρmin = 1.08 e Å3
2 restraintsAbsolute structure: Flack (1983), 758 Friedel pairs
Primary atom site location: structure-invariant direct methodsAbsolute structure parameter: 0.04 (2)
Crystal data top
[Cu(C6H2N2O4)(H2O)3]·H2OV = 1028.45 (8) Å3
Mr = 301.70Z = 4
Monoclinic, CcMo Kα radiation
a = 8.4254 (4) ŵ = 2.16 mm1
b = 18.0692 (8) ÅT = 298 K
c = 7.4187 (3) Å0.28 × 0.25 × 0.17 mm
β = 114.412 (1)°
Data collection top
Bruker APEXII area-detector
diffractometer
1683 independent reflections
Absorption correction: multi-scan
(SADABS; Sheldrick, 2004)
1654 reflections with I > 2σ(I)
Tmin = 0.519, Tmax = 0.660Rint = 0.067
3079 measured reflections
Refinement top
R[F2 > 2σ(F2)] = 0.036H-atom parameters constrained
wR(F2) = 0.089Δρmax = 0.67 e Å3
S = 1.11Δρmin = 1.08 e Å3
1683 reflectionsAbsolute structure: Flack (1983), 758 Friedel pairs
156 parametersAbsolute structure parameter: 0.04 (2)
2 restraints
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
Cu10.89418 (3)0.118643 (19)0.89933 (3)0.02206 (17)
N10.6873 (5)0.18186 (18)0.9306 (5)0.0277 (8)
N20.4368 (5)0.2821 (2)0.9255 (5)0.0289 (10)
O10.9124 (6)0.34477 (15)0.9145 (8)0.0380 (8)
O20.9816 (4)0.22606 (16)0.9101 (5)0.0311 (7)
O30.5382 (5)0.40933 (18)0.6974 (4)0.0347 (8)
O40.6390 (5)0.42851 (17)1.0176 (4)0.0346 (7)
O50.7715 (5)0.01798 (16)0.8841 (6)0.0428 (8)
H5A0.81680.00800.98720.064*
H5B0.66180.01100.82020.064*
O61.1011 (4)0.07381 (17)0.8605 (4)0.0314 (7)
H6A1.11200.02920.90210.047*
H6B1.10000.07800.74710.047*
O70.7584 (5)0.12285 (15)0.5794 (5)0.0302 (8)
H7A0.81410.15100.53790.039 (15)*
H7B0.65010.12500.52490.033 (16)*
C10.8817 (8)0.2773 (2)0.9095 (8)0.0247 (9)
C20.7074 (6)0.2543 (2)0.9108 (6)0.0229 (9)
C30.5804 (6)0.3047 (2)0.9043 (6)0.0232 (9)
C40.4209 (6)0.2097 (3)0.9474 (6)0.0328 (10)
H40.32370.19270.96420.039*
C50.5417 (6)0.1592 (2)0.9461 (6)0.0322 (10)
H50.52270.10890.95610.039*
C70.5890 (6)0.3869 (2)0.8690 (7)0.0226 (9)
O80.4396 (6)0.0183 (3)0.7257 (8)0.0755 (15)
H8A0.35370.00990.66420.113*
H8B0.40430.06010.74630.113*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Cu10.0207 (2)0.0188 (2)0.0264 (2)0.0014 (2)0.00942 (17)0.00095 (19)
N10.028 (2)0.0247 (17)0.0301 (16)0.0003 (15)0.0116 (15)0.0023 (14)
N20.023 (3)0.0329 (18)0.031 (2)0.0011 (14)0.0125 (18)0.0029 (15)
O10.0243 (17)0.0237 (14)0.065 (2)0.0024 (14)0.0179 (14)0.0017 (17)
O20.0231 (17)0.0236 (14)0.0479 (16)0.0006 (13)0.0160 (14)0.0014 (13)
O30.0441 (19)0.0313 (16)0.0264 (15)0.0016 (15)0.0122 (13)0.0034 (12)
O40.049 (2)0.0275 (14)0.0286 (14)0.0017 (14)0.0175 (14)0.0071 (12)
O50.0312 (17)0.0303 (16)0.0567 (19)0.0052 (14)0.0080 (14)0.0110 (15)
O60.0317 (16)0.0314 (14)0.0324 (13)0.0075 (14)0.0145 (12)0.0045 (13)
O70.0225 (18)0.0338 (18)0.0320 (19)0.0006 (11)0.0088 (15)0.0049 (11)
C10.020 (2)0.0256 (18)0.0268 (16)0.004 (2)0.0080 (14)0.0001 (18)
C20.022 (2)0.0225 (19)0.0215 (17)0.0037 (16)0.0063 (15)0.0012 (14)
C30.019 (2)0.027 (2)0.0194 (16)0.0031 (16)0.0039 (14)0.0017 (15)
C40.029 (3)0.035 (2)0.042 (3)0.009 (2)0.023 (2)0.0048 (16)
C50.034 (3)0.0257 (19)0.039 (2)0.0028 (19)0.0162 (18)0.0010 (17)
C70.020 (2)0.025 (2)0.024 (2)0.0004 (15)0.0112 (17)0.0018 (14)
O80.031 (2)0.062 (3)0.102 (3)0.0071 (18)0.004 (2)0.021 (2)
Geometric parameters (Å, º) top
Cu1—O62.048 (3)O5—H5A0.8422
Cu1—O22.066 (3)O5—H5B0.8559
Cu1—O52.072 (3)O6—H6A0.8539
Cu1—O3i2.098 (3)O6—H6B0.8410
Cu1—O72.169 (4)O7—H7A0.8328
Cu1—N12.175 (4)O7—H7B0.8323
N1—C21.335 (5)C1—C21.530 (7)
N1—C51.343 (7)C2—C31.391 (6)
N2—C41.332 (6)C3—C71.515 (5)
N2—C31.347 (6)C4—C51.369 (7)
O1—C11.244 (5)C4—H40.9300
O2—C11.249 (6)C5—H50.9300
O3—C71.232 (6)O8—H8A0.8483
O3—Cu1ii2.098 (3)O8—H8B0.8472
O4—C71.255 (5)
O6—Cu1—O293.81 (13)Cu1—O6—H6B116.5
O6—Cu1—O594.56 (15)H6A—O6—H6B113.7
O2—Cu1—O5171.41 (14)Cu1—O7—H7A107.6
O6—Cu1—O3i84.14 (13)Cu1—O7—H7B120.6
O2—Cu1—O3i98.28 (13)H7A—O7—H7B117.6
O5—Cu1—O3i84.50 (14)O1—C1—O2126.5 (6)
O6—Cu1—O787.38 (13)O1—C1—C2117.0 (5)
O2—Cu1—O791.59 (12)O2—C1—C2116.5 (4)
O5—Cu1—O786.87 (14)N1—C2—C3121.0 (4)
O3i—Cu1—O7167.38 (13)N1—C2—C1115.6 (4)
O6—Cu1—N1171.52 (13)C3—C2—C1123.2 (4)
O2—Cu1—N177.94 (13)N2—C3—C2120.8 (4)
O5—Cu1—N193.62 (15)N2—C3—C7115.3 (4)
O3i—Cu1—N198.85 (14)C2—C3—C7123.9 (4)
O7—Cu1—N190.86 (14)N2—C4—C5122.8 (5)
C2—N1—C5118.1 (4)N2—C4—H4118.6
C2—N1—Cu1111.0 (3)C5—C4—H4118.6
C5—N1—Cu1130.5 (3)N1—C5—C4120.3 (4)
C4—N2—C3116.9 (4)N1—C5—H5119.8
C1—O2—Cu1117.8 (3)C4—C5—H5119.8
C7—O3—Cu1ii144.4 (3)O3—C7—O4123.9 (4)
Cu1—O5—H5A114.2O3—C7—C3118.7 (4)
Cu1—O5—H5B124.0O4—C7—C3117.3 (4)
H5A—O5—H5B113.7H8A—O8—H8B110.3
Cu1—O6—H6A107.0
Symmetry codes: (i) x+1/2, y+1/2, z+1/2; (ii) x1/2, y+1/2, z1/2.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
O5—H5A···O7iii0.842.302.953 (4)134
O5—H5B···O80.861.792.630 (6)168
O6—H6A···O4iv0.851.982.837 (4)176
O6—H6A···O3iv0.852.573.170 (5)128
O6—H6B···O4v0.841.872.691 (4)167
O7—H7A···N2v0.831.982.817 (5)177
O7—H7B···O1ii0.831.902.719 (6)166
O8—H8A···O4ii0.852.022.864 (6)176
O8—H8B···O1vi0.852.112.898 (6)155
Symmetry codes: (ii) x1/2, y+1/2, z1/2; (iii) x, y, z+1/2; (iv) x+1/2, y1/2, z; (v) x+1/2, y+1/2, z1/2; (vi) x1/2, y1/2, z.

Experimental details

Crystal data
Chemical formula[Cu(C6H2N2O4)(H2O)3]·H2O
Mr301.70
Crystal system, space groupMonoclinic, Cc
Temperature (K)298
a, b, c (Å)8.4254 (4), 18.0692 (8), 7.4187 (3)
β (°) 114.412 (1)
V3)1028.45 (8)
Z4
Radiation typeMo Kα
µ (mm1)2.16
Crystal size (mm)0.28 × 0.25 × 0.17
Data collection
DiffractometerBruker APEXII area-detector
diffractometer
Absorption correctionMulti-scan
(SADABS; Sheldrick, 2004)
Tmin, Tmax0.519, 0.660
No. of measured, independent and
observed [I > 2σ(I)] reflections
3079, 1683, 1654
Rint0.067
(sin θ/λ)max1)0.598
Refinement
R[F2 > 2σ(F2)], wR(F2), S 0.036, 0.089, 1.11
No. of reflections1683
No. of parameters156
No. of restraints2
H-atom treatmentH-atom parameters constrained
Δρmax, Δρmin (e Å3)0.67, 1.08
Absolute structureFlack (1983), 758 Friedel pairs
Absolute structure parameter0.04 (2)

Computer programs: APEX2 (Bruker, 2004), SAINT (Bruker, 2004), SHELXS97 (Sheldrick, 1997), SHELXL97 (Sheldrick, 1997), ORTEPIII (Burnett & Johnson, 1996) and PLATON (Spek, 2003).

Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
O5—H5A···O7i0.842.302.953 (4)134.4
O5—H5B···O80.861.792.630 (6)167.8
O6—H6A···O4ii0.851.982.837 (4)175.8
O6—H6A···O3ii0.852.573.170 (5)128.2
O6—H6B···O4iii0.841.872.691 (4)166.8
O7—H7A···N2iii0.831.982.817 (5)177.1
O7—H7B···O1iv0.831.902.719 (6)165.9
O8—H8A···O4iv0.852.022.864 (6)176.2
O8—H8B···O1v0.852.112.898 (6)154.7
Symmetry codes: (i) x, y, z+1/2; (ii) x+1/2, y1/2, z; (iii) x+1/2, y+1/2, z1/2; (iv) x1/2, y+1/2, z1/2; (v) x1/2, y1/2, z.
 

Acknowledgements

The authors are grateful to Sichuan University for financial support.

References

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