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Acta Cryst. (2009). E65, m1619    [ doi:10.1107/S1600536809046601 ]

Poly[[[mu]2-acetato-aquadi-[mu]3-isonicotinato-holmium(III)silver(I)] perchlorate]

S. Feng

Abstract top

In the title three-dimensional heterometallic complex, {[AgHo(C6H4NO2)2(C2H3O2)(H2O)]ClO4}n, the HoIII ion is eight-coordinated by four O atoms from four different isonicotinate ligands, three O atoms from two different acetate ligands and one O atom of a water molecule. The two-coordinate AgI ion is bonded to two N atoms from two different isonicotinate anions. These metal coordination units are connected by bridging isonicotinate and acetate ligands, generating a three-dimensional network. The coordinated water molecules link the carboxylate group of the acetate ligand and the nitrate ligand by O-H...O hydrogen bonding. The crystal structure is further stabilized by hydrogen bonds. The perchlorate ion is disordered over two sites with site-occupancy factors 0.539 (12) and 0.461 (12), while the methyl group of the acetate ligand is disordered over two sites with site-occupancy factors 0.51 (4) and 0.49 (4).

Comment top

In the past few years, lanthanide-transition metal heterometallic complexs with bridging multifunctionnal organic ligands are of increasing interest, not only because of their impressive topological structures, but also due to their versatile applications in ion exchange, magnetism, bimetallic catalysis and luminescent probe(Cheng et al., 2006; Kuang et al., 2007; Peng et al., 2008; Zhu et al., 2009). As an extension of this research, the structure of the title compound, a new heterometallic coordination polymer, (I), has been determined which is presented in this artcle.

The asymmetric unite of the title compound (Fig. 1), contains one of each of the HoIII and AgI ions, two halves of acetate ligand, two isonicotinate ligands, and one coordinated water molecule. The HoIII ion is eight-coordinated by four O atoms from four different isonicotinate ligands, and three O atoms from two different acetate ligands, and one O atom of water molecule; the Ho center can be described as adopting a bicapped trigonal prism coordination geometry. The two-coordinate AgI ion is bonded to two N atoms from two different isonicotinate anions. Thus the AgI ion is in a somewhat liner coformation with N1—Ag1—N2 angle 165.55 (17) °. These metal coordination units are connected by bridging isonicotinate and acetate ligands, generating a three-dimensional network (Fig. 2). The coordinated water molecules link the carboxylate group and acetate ligand by O—H···O hydrogen bonding (Table 1). The crystal structure is further stabilized by hydrogen bonds.

Related literature top

For the applications of lanthanide–transition metal heterometallic complexes in ion exchange, magnetism, bimetallic catalysis and as luminescent probes, see: Cheng et al. (2006); Kuang et al. (2007); Peng et al. (2008); Zhu et al. (2009).

Experimental top

A mixture of AgNO3(0.057 g, 0.33 mmol), Ho2O3(0.116 g, 0.33 mmol), isonicotinic acid(0.164 g, 1.33 mmol), CH3COONa(0.057 g, 0.7 mmol), H2O(7 ml), and HClO4(0.257 mmol)(pH 2) was sealed in a 20 ml Teflon-lined reaction vessel at 443 K for 6 days then slowly cooled to room temperature. The product was collected by filtration, washed with water and air-dried. Colorless block crystals suitable for X-ray analysis were obtained.

Refinement top

H atoms bonded to C atoms were positioned geometrically and refined as riding, with C—H = 0.93 or 0.96 Å and Uiso(H) = 1.2 or 1.5 Ueq(C). H atoms of water molecules were found from difference Fourier maps and refined isotropically with a restraint of O—H = 0.81 - 0.82 Å. The perchlorate ion was disordered over two sites with site occupancy factors 0.539 (12) and 0.461 (12), whereas the methyl group of the acetate ligand was disordered over two sites with site occupancy factors 0.51 (4) and 0.49 (4).

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, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: XP in SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Figures top
[Figure 1] Fig. 1. The molecular structure showing the atomic-numbering scheme and displacement ellipsoids drawn at the 30% probability level. Symmetry codes: (A) 1 - x, 1 - y, -z; (B) x, 0.5 - y, 1/2 + z; (C) -x, 1/2 + y, 0.5 - z; (D) 1 + x, 0.5 - y, -1/2 + z.
[Figure 2] Fig. 2. A view of the three-dimensional structure of the title compound.
Poly[[µ2-acetato-aquadi-µ3-isonicotinato-holmium(III)silver(I)] perchlorate] top
Crystal data top
[AgHo(C6H4NO2)2(C2H3O2)(H2O)]ClO4F(000) = 1320
Mr = 693.51Dx = 2.412 Mg m−3
Monoclinic, P21/cMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybcCell parameters from 5514 reflections
a = 16.2158 (10) Åθ = 2.5–27.8°
b = 14.9024 (9) ŵ = 5.34 mm−1
c = 7.9068 (5) ÅT = 296 K
β = 91.826 (1)°Block, colorless
V = 1909.7 (2) Å30.20 × 0.18 × 0.15 mm
Z = 4
Data collection top
Bruker APEXII area-detector
diffractometer		3439 independent reflections
Radiation source: fine-focus sealed tube3085 reflections with I > 2σ(I)
graphiteRint = 0.027
φ and ω scanθmax = 25.2°, θmin = 1.9°
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)		h = −18→19
Tmin = 0.357, Tmax = 0.449k = −17→17
9735 measured reflectionsl = −7→9
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)] = 0.023Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.054H atoms treated by a mixture of independent and constrained refinement
S = 1.04 w = 1/[σ2(Fo2) + (0.0229P)2 + 2.5125P]
where P = (Fo2 + 2Fc2)/3
3439 reflections(Δ/σ)max = 0.001
320 parametersΔρmax = 0.64 e Å−3
158 restraintsΔρmin = −0.71 e Å−3
Crystal data top
[AgHo(C6H4NO2)2(C2H3O2)(H2O)]ClO4V = 1909.7 (2) Å3
Mr = 693.51Z = 4
Monoclinic, P21/cMo Kα radiation
a = 16.2158 (10) ŵ = 5.34 mm−1
b = 14.9024 (9) ÅT = 296 K
c = 7.9068 (5) Å0.20 × 0.18 × 0.15 mm
β = 91.826 (1)°
Data collection top
Bruker APEXII area-detector
diffractometer		3439 independent reflections
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)		3085 reflections with I > 2σ(I)
Tmin = 0.357, Tmax = 0.449Rint = 0.027
9735 measured reflectionsθmax = 25.2°
Refinement top
R[F2 > 2σ(F2)] = 0.023H atoms treated by a mixture of independent and constrained refinement
wR(F2) = 0.054Δρmax = 0.64 e Å−3
S = 1.04Δρmin = −0.71 e Å−3
3439 reflectionsAbsolute structure: ?
320 parametersFlack parameter: ?
158 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.

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*/UeqOcc. (<1)
Ho10.454984 (11)0.383812 (12)0.04996 (2)0.01844 (7)
Ag1−0.02677 (3)0.24033 (4)0.10033 (7)0.06287 (16)
C1−0.3677 (3)0.0808 (3)0.3622 (6)0.0313 (10)
C2−0.2892 (3)0.1236 (3)0.3041 (6)0.0289 (10)
C3−0.2191 (3)0.0720 (3)0.2964 (7)0.0410 (12)
H10−0.21990.01170.32680.049*
C4−0.1474 (3)0.1114 (4)0.2427 (8)0.0519 (15)
H8−0.10020.07620.23800.062*
C5−0.2111 (3)0.2470 (3)0.2064 (8)0.0481 (14)
H7−0.20860.30720.17610.058*
C6−0.2852 (3)0.2128 (3)0.2588 (7)0.0409 (12)
H9−0.33150.24930.26340.049*
C70.1336 (3)0.1719 (4)−0.0405 (7)0.0442 (13)
H50.10660.1176−0.02510.053*
C80.2120 (3)0.1707 (3)−0.1043 (6)0.0351 (11)
H40.23720.1166−0.13020.042*
C90.2521 (3)0.2510 (3)−0.1289 (5)0.0230 (9)
C100.3375 (2)0.2510 (3)−0.1983 (5)0.0206 (9)
C110.2125 (3)0.3295 (3)−0.0872 (6)0.0325 (11)
H30.23820.3846−0.10170.039*
C120.1343 (3)0.3254 (3)−0.0238 (6)0.0376 (12)
H60.10800.37870.00340.045*
N1−0.1427 (2)0.1969 (3)0.1974 (6)0.0458 (11)
N20.0950 (2)0.2480 (3)0.0001 (5)0.0394 (10)
O1−0.4242 (2)0.1332 (2)0.4072 (5)0.0387 (9)
O2−0.3699 (2)−0.0029 (2)0.3616 (4)0.0397 (8)
O30.35906 (18)0.18441 (19)−0.2810 (4)0.0303 (7)
O40.38257 (17)0.31880 (19)−0.1691 (4)0.0260 (7)
O50.54736 (19)0.5035 (2)0.1583 (4)0.0312 (7)
O60.5344 (2)0.3847 (2)0.3129 (4)0.0352 (8)
O1W0.4985 (2)0.2318 (2)0.0763 (4)0.0353 (8)
H1W0.478 (3)0.197 (2)0.143 (5)0.053*
H2W0.516 (3)0.200 (3)0.002 (4)0.053*
C130.5691 (3)0.4589 (3)0.2876 (5)0.0272 (10)
C140.6256 (13)0.4991 (15)0.423 (2)0.040 (3)0.49 (4)
H14A0.59560.50740.52470.060*0.49 (4)
H14B0.64570.55600.38520.060*0.49 (4)
H14C0.67130.45940.44550.060*0.49 (4)
C14'0.6450 (11)0.4886 (16)0.388 (3)0.040 (3)0.51 (4)
H14D0.65450.44860.48210.060*0.51 (4)
H14E0.63680.54840.42970.060*0.51 (4)
H14F0.69180.48770.31710.060*0.51 (4)
Cl10.91931 (10)0.45924 (11)0.2560 (2)0.0625 (4)0.539 (12)
O70.8512 (8)0.4643 (8)0.3717 (18)0.126 (5)0.539 (12)
O80.8847 (9)0.4462 (8)0.0955 (12)0.111 (5)0.539 (12)
O90.9703 (9)0.3896 (8)0.300 (2)0.130 (6)0.539 (12)
O100.9589 (10)0.5442 (8)0.259 (2)0.086 (6)0.539 (12)
Cl1'0.91931 (10)0.45924 (11)0.2560 (2)0.0625 (4)0.461 (12)
O7'0.8366 (6)0.4707 (8)0.210 (2)0.094 (5)0.461 (12)
O8'0.9624 (10)0.4225 (10)0.1137 (18)0.132 (6)0.461 (12)
O9'0.9275 (11)0.3943 (10)0.3831 (18)0.131 (7)0.461 (12)
O10'0.9581 (11)0.5392 (9)0.304 (2)0.082 (6)0.461 (12)
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Ho10.01607 (11)0.01592 (11)0.02369 (12)−0.00070 (7)0.00617 (7)−0.00010 (7)
Ag10.0259 (2)0.0859 (4)0.0783 (3)−0.0142 (2)0.0260 (2)0.0008 (3)
C10.027 (2)0.023 (2)0.044 (3)−0.0049 (19)0.016 (2)−0.004 (2)
C20.027 (2)0.027 (2)0.033 (2)−0.0057 (19)0.0118 (19)−0.0029 (19)
C30.027 (3)0.028 (3)0.069 (4)0.002 (2)0.017 (2)0.004 (2)
C40.027 (3)0.046 (4)0.083 (4)0.000 (2)0.017 (3)−0.006 (3)
C50.033 (3)0.034 (3)0.078 (4)−0.009 (2)0.020 (3)0.008 (3)
C60.026 (3)0.033 (3)0.064 (4)−0.001 (2)0.017 (2)0.005 (2)
C70.032 (3)0.041 (3)0.060 (3)−0.013 (2)0.014 (2)0.003 (3)
C80.028 (3)0.028 (3)0.050 (3)−0.0040 (19)0.012 (2)−0.001 (2)
C90.019 (2)0.027 (2)0.023 (2)−0.0016 (17)0.0029 (17)−0.0019 (17)
C100.018 (2)0.022 (2)0.023 (2)0.0006 (16)0.0015 (17)0.0003 (17)
C110.025 (2)0.028 (3)0.045 (3)−0.0027 (19)0.012 (2)−0.002 (2)
C120.028 (3)0.035 (3)0.051 (3)0.005 (2)0.015 (2)−0.001 (2)
N10.025 (2)0.046 (3)0.067 (3)−0.0090 (19)0.018 (2)−0.001 (2)
N20.025 (2)0.047 (3)0.047 (3)−0.0028 (18)0.0136 (18)−0.001 (2)
O10.0304 (18)0.0221 (17)0.065 (2)−0.0017 (13)0.0281 (17)−0.0023 (15)
O20.0346 (18)0.0219 (17)0.064 (2)−0.0025 (14)0.0279 (17)−0.0039 (15)
O30.0263 (17)0.0225 (16)0.0431 (19)−0.0017 (13)0.0144 (14)−0.0104 (14)
O40.0185 (15)0.0259 (16)0.0338 (17)−0.0060 (12)0.0058 (12)−0.0054 (13)
O50.0382 (18)0.0269 (17)0.0279 (16)−0.0076 (14)−0.0067 (14)0.0043 (13)
O60.044 (2)0.0298 (18)0.0315 (18)−0.0093 (15)−0.0022 (15)0.0102 (14)
O1W0.049 (2)0.0215 (16)0.037 (2)0.0061 (15)0.0189 (16)0.0013 (14)
C130.028 (2)0.029 (2)0.025 (2)−0.0030 (19)−0.0014 (18)0.0027 (19)
C140.039 (6)0.044 (5)0.036 (6)−0.003 (5)−0.005 (5)−0.003 (4)
C14'0.039 (6)0.044 (5)0.036 (6)−0.003 (5)−0.005 (5)−0.003 (4)
Cl10.0599 (10)0.0510 (9)0.0761 (11)0.0018 (7)−0.0046 (8)−0.0019 (8)
O70.119 (9)0.127 (8)0.133 (9)−0.027 (7)0.053 (7)−0.013 (7)
O80.143 (10)0.114 (8)0.072 (6)−0.050 (7)−0.042 (6)−0.005 (6)
O90.142 (9)0.098 (7)0.148 (10)0.072 (7)−0.037 (7)−0.016 (7)
O100.077 (8)0.084 (9)0.097 (9)−0.030 (6)0.007 (6)−0.010 (6)
Cl1'0.0599 (10)0.0510 (9)0.0761 (11)0.0018 (7)−0.0046 (8)−0.0019 (8)
O7'0.057 (6)0.077 (7)0.145 (10)0.001 (5)−0.021 (6)0.011 (7)
O8'0.147 (11)0.131 (9)0.124 (9)0.016 (8)0.068 (8)−0.035 (7)
O9'0.143 (11)0.133 (10)0.117 (9)−0.011 (8)−0.019 (8)0.065 (8)
O10'0.069 (9)0.076 (9)0.103 (9)−0.026 (7)0.022 (7)−0.045 (7)
Geometric parameters (Å, °) top
Ho1—O42.278 (3)C9—C111.380 (6)
Ho1—O2i2.303 (3)C9—C101.504 (5)
Ho1—O1ii2.306 (3)C10—O31.245 (5)
Ho1—O3iii2.318 (3)C10—O41.264 (5)
Ho1—O5iv2.352 (3)C11—C121.379 (6)
Ho1—O1W2.380 (3)C11—H30.9300
Ho1—O62.410 (3)C12—N21.335 (6)
Ho1—O52.465 (3)C12—H60.9300
Ag1—N12.152 (4)O1—Ho1v2.306 (3)
Ag1—N22.154 (4)O2—Ho1vi2.303 (3)
C1—O21.248 (5)O3—Ho1vii2.318 (3)
C1—O11.264 (5)O5—C131.260 (5)
C1—C21.509 (6)O5—Ho1iv2.352 (3)
C2—C31.376 (6)O6—C131.259 (5)
C2—C61.379 (7)O1W—H1W0.82 (4)
C3—C41.380 (7)O1W—H2W0.81 (4)
C3—H100.9300C13—C14'1.510 (9)
C4—N11.327 (7)C13—C141.511 (9)
C4—H80.9300C14—H14A0.9600
C5—N11.341 (7)C14—H14B0.9600
C5—C61.380 (6)C14—H14C0.9600
C5—H70.9300C14'—H14D0.9600
C6—H90.9300C14'—H14E0.9600
C7—N21.340 (7)C14'—H14F0.9600
C7—C81.382 (6)Cl1—O91.364 (8)
C7—H50.9300Cl1—O81.385 (8)
C8—C91.380 (6)Cl1—O101.419 (9)
C8—H40.9300Cl1—O71.458 (8)
O4—Ho1—O2i104.08 (12)C2—C6—H9120.6
O4—Ho1—O1ii90.36 (11)C5—C6—H9120.6
O2i—Ho1—O1ii139.11 (10)N2—C7—C8122.7 (5)
O4—Ho1—O3iii84.99 (11)N2—C7—H5118.7
O2i—Ho1—O3iii74.14 (10)C8—C7—H5118.7
O1ii—Ho1—O3iii146.11 (10)C9—C8—C7119.0 (4)
O4—Ho1—O5iv76.99 (10)C9—C8—H4120.5
O2i—Ho1—O5iv72.10 (11)C7—C8—H4120.5
O1ii—Ho1—O5iv74.39 (11)C11—C9—C8118.4 (4)
O3iii—Ho1—O5iv136.13 (11)C11—C9—C10121.9 (4)
O4—Ho1—O1W78.79 (12)C8—C9—C10119.7 (4)
O2i—Ho1—O1W148.20 (11)O3—C10—O4124.2 (4)
O1ii—Ho1—O1W71.57 (11)O3—C10—C9118.0 (3)
O3iii—Ho1—O1W74.60 (10)O4—C10—C9117.8 (3)
O5iv—Ho1—O1W137.66 (11)C12—C11—C9119.4 (4)
O4—Ho1—O6155.03 (10)C12—C11—H3120.3
O2i—Ho1—O692.45 (12)C9—C11—H3120.3
O1ii—Ho1—O689.10 (12)N2—C12—C11122.5 (4)
O3iii—Ho1—O681.64 (11)N2—C12—H6118.7
O5iv—Ho1—O6126.62 (10)C11—C12—H6118.7
O1W—Ho1—O677.37 (12)C4—N1—C5117.7 (4)
O4—Ho1—O5149.88 (10)C4—N1—Ag1116.4 (3)
O2i—Ho1—O574.19 (11)C5—N1—Ag1125.8 (4)
O1ii—Ho1—O574.55 (11)C12—N2—C7118.0 (4)
O3iii—Ho1—O5121.89 (11)C12—N2—Ag1123.1 (3)
O5iv—Ho1—O573.91 (11)C7—N2—Ag1118.9 (3)
O1W—Ho1—O5118.90 (12)C1—O1—Ho1v134.8 (3)
O6—Ho1—O552.72 (10)C1—O2—Ho1vi138.5 (3)
O4—Ho1—C13170.02 (11)C10—O3—Ho1vii149.1 (3)
O2i—Ho1—C1383.92 (13)C10—O4—Ho1139.5 (3)
O1ii—Ho1—C1379.67 (13)C13—O5—Ho1iv160.5 (3)
O3iii—Ho1—C13103.15 (12)C13—O5—Ho192.9 (2)
O5iv—Ho1—C13100.29 (11)Ho1iv—O5—Ho1106.09 (11)
O1W—Ho1—C1397.63 (13)C13—O6—Ho195.6 (3)
O6—Ho1—C1326.34 (11)Ho1—O1W—H1W122 (3)
O5—Ho1—C1326.45 (11)Ho1—O1W—H2W127 (3)
O4—Ho1—Ho1iv114.60 (7)H1W—O1W—H2W105 (4)
O2i—Ho1—Ho1iv68.77 (7)O6—C13—O5118.5 (4)
O1ii—Ho1—Ho1iv70.43 (7)O6—C13—C14'122.3 (10)
O3iii—Ho1—Ho1iv141.14 (7)O5—C13—C14'118.3 (10)
O5iv—Ho1—Ho1iv37.97 (7)O6—C13—C14119.9 (10)
O1W—Ho1—Ho1iv139.55 (8)O5—C13—C14120.9 (10)
O6—Ho1—Ho1iv88.65 (7)C14'—C13—C1417.2 (11)
O5—Ho1—Ho1iv35.94 (7)O6—C13—Ho158.1 (2)
C13—Ho1—Ho1iv62.34 (9)O5—C13—Ho160.6 (2)
N1—Ag1—N2165.55 (17)C14'—C13—Ho1166.4 (11)
O2—C1—O1126.7 (4)C14—C13—Ho1176.2 (10)
O2—C1—C2116.5 (4)C13—C14—H14A109.5
O1—C1—C2116.8 (4)C13—C14—H14B109.5
C3—C2—C6118.8 (4)C13—C14—H14C109.5
C3—C2—C1118.9 (4)C13—C14'—H14D109.5
C6—C2—C1122.3 (4)C13—C14'—H14E109.5
C2—C3—C4118.7 (5)H14D—C14'—H14E109.5
C2—C3—H10120.6C13—C14'—H14F109.5
C4—C3—H10120.6H14D—C14'—H14F109.5
N1—C4—C3123.3 (5)H14E—C14'—H14F109.5
N1—C4—H8118.4O9—Cl1—O8110.5 (7)
C3—C4—H8118.4O9—Cl1—O10113.8 (8)
N1—C5—C6122.7 (5)O8—Cl1—O10108.2 (8)
N1—C5—H7118.7O9—Cl1—O7110.2 (8)
C6—C5—H7118.7O8—Cl1—O7106.8 (7)
C2—C6—C5118.9 (5)O10—Cl1—O7107.0 (7)
Symmetry codes: (i) −x, y+1/2, −z+1/2; (ii) x+1, −y+1/2, z−1/2; (iii) x, −y+1/2, z+1/2; (iv) −x+1, −y+1, −z; (v) x−1, −y+1/2, z+1/2; (vi) −x, y−1/2, −z+1/2; (vii) x, −y+1/2, z−1/2.
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O1W—H1W···O4iii0.82 (4)2.19 (3)2.898 (4)145 (4)
O1W—H2W···O6vii0.81 (4)1.99 (4)2.787 (4)168 (5)
Symmetry codes: (iii) x, −y+1/2, z+1/2; (vii) x, −y+1/2, z−1/2.
Table 1
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O1W—H1W···O4i0.82 (4)2.19 (3)2.898 (4)145 (4)
O1W—H2W···O6ii0.81 (4)1.99 (4)2.787 (4)168 (5)
Symmetry codes: (i) x, −y+1/2, z+1/2; (ii) x, −y+1/2, z−1/2.
Acknowledgements top

The authors acknowledge South China Normal University for supporting this work.

references
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