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Acta Cryst. (2008). E64, m1260-m1261    [ doi:10.1107/S1600536808028675 ]

Poly[tetraaqua-[mu]4-bromido-di-[mu]2-bromido-[mu]2-hydroxido-di-[mu]3-isonicotinato-tetra-[mu]2-isonicotinato-tetracopper(I)dithulium(III)]

G.-M. Wang, Z.-X. Li, Q.-H. Zheng and H.-L. Liu

Abstract top

A new thulium(III)-copper(I) heterometallic coordination polymer, [Cu4Tm2Br3(C6H4NO2)6(OH)(H2O)4]n, has been prepared by a hydrothermal method. The Tm and both Cu atoms lie on mirror planes. The Tm atom is seven-coordinate with a capped distorted trigonal-prismatic coordination geometry, while the Cu atoms adopt trigonal CuBrN2 and tetrahedral CuBr3N coordination modes, respectively. The Cu atom in the trigonal coordination environment is disordered over two sites of equal occupancy. The crystal structure is constructed from two distinct units of dimeric [Tm2([mu]2-OH(IN)6(H2O)4] cores (IN = isonicotinate) and one-dimensional inorganic [Cu4Br3]n chains, which are linked together, forming heterometallic Cu-halide-lanthanide-organic layers.

Comment top

The rational design and synthesis of heterometallic lanthanide(Ln)–transition metal(TM) compounds have attracted increasing attention in recent years, not only because of their intriguing variety of architectures and topologies but also owing to their potential applications in luminescence, magnetism, bimetallic catalysis, and molecular adsorption. (Benelli & Gatteschi, 2002; Shibasaki & Yoshikawa, 2002; Zhao et al., 2004a,b; Guillou et al., 2006; Wang et al., 2006). So far, most work has been focused on the assembly of homometallic Ln and TM compounds, while the construction of heterometallic Ln–TM extented architectures is still a challenge (Ren et al., 2003; Prasad et al., 2007; Cheng et al., 2008). This may be attributed to the different competitive reactions of Ln and TM metals with the same ligand, which often results in the formation of the homometallic compounds rather than heterometallic ones. Generally, the Ln ions prefer O-donors, while TM ions have stronger tendency to coordinate to N-donors. Therefore, isonicotinic acid (HIN) has been chosen here as the multifunctional bridging ligand to construct new hetero-Ln–TM complexes. The title compound [Tm2Cu4Br3(µ2-OH)(C6H4NO2)6(H2O)4]n (1) is reported here and displays novel two-dimensional coordination features.

In the asymmetric unit, the Tm1 atom, occupying a special position on a mirror plane (Fig. 1), is seven-coordinated and has a capped trigonal–prismatic coordination environment comprising two coordinated water molecules, one µ2-OH and four carboxylate oxygen atoms from four IN- ligands. The Tm—O bond lengths range from 2.193 (2) to 2.449 (5) Å. Both Cu1 and Cu2 atoms occupy special positions on two crystallographic mirror planes, one perpendicular (z = 0) and the other parallel to the c axis. The Cu1 atom is three-coordinate with one µ4-Br1 and two N atoms from two bridging IN- moieties, while the Cu2 center is coordinated to one µ4-Br1, two µ2-Br2 atoms and one N atom from one IN- ligand that defines a distorted tetrahedral geometry. The Cu—N and Cu—Br distances are in the range 1.928 (3)–2.020 (4) Å and 2.518 (9)–2.688 (7) Å, respectively. Although copper(II) salts were used as starting materials, the Cu centers in the product are in the +1 oxidation state. This is attributed to a reduction reaction occurring under the hydrothermal conditions used.

The framework of 1 is constructed from two subunits, dimeric [Tm2(µ2-OH)(IN-)6] (Tm2) cores and inorganic [Cu4Br3]n chains. As shown in Fig. 2, two crystallographically identical Tm(III) ions are linked by one bridging hydroxo group and six IN- ligands to form the dimeric Tm2 fragment, in which the IN- ligand adopts two different coordination modes. The Tm···Tm distance is 4.071 (1) Å. The Cu(I) centers, however, are bridged by µ4-Br and µ2-Br atoms to form one-dimensional inorganic [Cu4Br3]n chains (Fig. 3). Interestingly, such one-dimensional copper halide chains appear to be constructed directly from corner- and edge-sharing tetrahedral Cu(2)Br3N units, decorated by the trigonal Cu(1)BrN2 groups protruding outside the chain. In addition, the Cu···Cu distance within the copper(I) halide chains is 2.688 (7) Å, less than twice the van der Waals radius of the Cu(I) ion (1.4) Å), indicating a strong Cu···Cu interaction. The linkages between dimeric Tm2 and inorganic [Cu4Br3]n motifs through N—Cu bonds give rise to a novel two-dimensional hetero-Tm—Cu framework (Fig. 4). Adjacent sheets are further packed to form a three-dimensional supramolecular framework through O—H···O hydrogen bonds.

Related literature top

For background to the structures and applications of heterometallic lanthanide–transition metal polymers, see: Benelli & Gatteschi (2002); Shibasaki & Yoshikawa (2002); Zhao et al. (2004a,b); Guillou et al. (2006); Wang et al. (2006). For some examples of heterometallic lanthanide–transition metal extended architectures, see: Ren et al. (2003); Prasad et al. (2007); Cheng et al. (2008).

Experimental top

The title compound was synthesized under mild hydrothermal conditions. Typically, a mixture of Tm2O3 (0.5 mmol, 0.193 g), CuBr2 (0.089 g, 0.40 mmol), HIN (2.00 mmol, 0.247 g) and H2O (8 ml) was sealed in a 25 ml Teflon-lined steel autoclave and heated under autogenous pressure at 443 K for 8 days. The yellow block-like crystals obtained were recovered by filtration, washed with distilled water and dried in air.

Refinement top

H atoms bound to C atoms were placed in calculated positions, with C—H distances of 0.93 Å. All other H atoms were located in a difference Fourier map and treated as riding, with fixed Uiso(H) = 1.2Ueq(O). The H4 and H6C atoms lie close to a mirror plane, and were treated as disordered with constrained site occupancy factors of 0.25 and 0.5, respectively. Atom Cu1 was refined as disordered over two positions, each with 50% site occupancy.

Computing details top

Data collection: APEX2 (Bruker, 2002); cell refinement: SAINT (Bruker, 2002); data reduction: SAINT (Bruker, 2002); 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 molecular structure of 1, showing the atom-numbering scheme. Displacement ellipsoids are drawn at the 50% probability level. [Symmetry codes: (i) -x, y, z; (ii) x, y, 1/2 - z; (iii) x, 1 - y, 1 - z; (iv) x, 1 - y, -1/2 + z; (v) x, 2 - y, 1 - z; (vi) -x, 1 - y, 1 - z].
[Figure 2] Fig. 2. Dimeric [Tm2(µ2-OH)(IN)6] fragment and the coordination modes of IN- found in 1.
[Figure 3] Fig. 3. One-dimensional infinite [Cu4Br3]n chains along b axis (a), and polyhedral view of the [Cu4Br3N6]n chains. [Symmetry codes: (i) -x, 2 - y, 1 - z; (ii) x, 1 - y, 1 - z; (iii) -x, 1 - y, 1 - z; (iv) x, -1 + y, z].
[Figure 4] Fig. 4. View of the two-dimensional layer structure of 1.
Poly[tetraaqua-µ4-bromido-di-µ2-bromido-µ2-hydroxido-di-µ3- isonicotinato-tetra-µ2-isonicotinato-tetracopper(I)dithulium(III)] top
Crystal data top
[Cu4Tm2Br3(C6H4NO2)6(OH)(H2O)4]F(000) = 3144
Mr = 1653.43Dx = 2.463 Mg m−3
Orthorhombic, CmcmMo Kα radiation, λ = 0.71073 Å
Hall symbol: -C 2c 2Cell parameters from 6070 reflections
a = 19.1815 (2) Åθ = 2.1–26.5°
b = 6.6973 (4) ŵ = 8.58 mm−1
c = 34.7044 (5) ÅT = 295 K
V = 4458.3 (3) Å3Block, yellow
Z = 40.16 × 0.09 × 0.08 mm
Data collection top
Bruker APEXII area-detector
diffractometer		2408 independent reflections
Radiation source: fine-focus sealed tube2090 reflections with I > 2σ(I)
graphiteRint = 0.037
φ and ω scansθmax = 26.5°, θmin = 2.1°
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)		h = −24→24
Tmin = 0.341, Tmax = 0.547k = −8→8
17231 measured reflectionsl = −43→42
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)] = 0.027H-atom parameters constrained
wR(F2) = 0.064 w = 1/[σ2(Fo2) + (0.0288P)2 + 11.1134P]
where P = (Fo2 + 2Fc2)/3
S = 1.11(Δ/σ)max = 0.001
2408 reflectionsΔρmax = 0.83 e Å−3
179 parametersΔρmin = −0.87 e Å−3
6 restraintsExtinction correction: SHELXL97 (Sheldrick, 2008), Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4
Primary atom site location: structure-invariant direct methodsExtinction coefficient: 0.00119 (4)
Crystal data top
[Cu4Tm2Br3(C6H4NO2)6(OH)(H2O)4]V = 4458.3 (3) Å3
Mr = 1653.43Z = 4
Orthorhombic, CmcmMo Kα radiation
a = 19.1815 (2) ŵ = 8.58 mm−1
b = 6.6973 (4) ÅT = 295 K
c = 34.7044 (5) Å0.16 × 0.09 × 0.08 mm
Data collection top
Bruker APEXII area-detector
diffractometer		2408 independent reflections
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)		2090 reflections with I > 2σ(I)
Tmin = 0.341, Tmax = 0.547Rint = 0.037
17231 measured reflectionsθmax = 26.5°
Refinement top
R[F2 > 2σ(F2)] = 0.027 w = 1/[σ2(Fo2) + (0.0288P)2 + 11.1134P]
where P = (Fo2 + 2Fc2)/3
wR(F2) = 0.064Δρmax = 0.83 e Å−3
S = 1.11Δρmin = −0.87 e Å−3
2408 reflectionsAbsolute structure: ?
179 parametersFlack parameter: ?
6 restraintsRogers parameter: ?
H-atom parameters constrained
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)
Tm10.106107 (10)−0.00154 (3)0.25000.01935 (10)
Cu10.1384 (6)1.00000.50000.037 (3)0.50
Cu1'0.1361 (6)1.064 (2)0.4997 (8)0.038 (2)0.25
Cu20.00000.63637 (13)0.47157 (2)0.0465 (2)
Br10.00001.00000.50000.0344 (2)
Br20.11106 (3)0.50000.50000.04244 (17)
O10.16098 (14)1.0892 (4)0.69667 (7)0.0352 (6)
O20.27354 (15)1.1077 (5)0.68216 (7)0.0475 (8)
O30.05791 (13)0.2265 (5)0.29248 (8)0.0390 (7)
O40.0000−0.1250 (8)0.25000.0389 (13)
H40.0000−0.27880.26000.047*0.50
O50.19308 (19)0.2314 (6)0.25000.0365 (9)
H50.20410.28710.26990.044*
O60.1159 (2)−0.3688 (7)0.25000.0647 (15)
H6B0.1529−0.29880.25000.078*
H6C0.1278−0.48590.23800.078*0.50
C10.2117 (2)1.0891 (6)0.67359 (9)0.0278 (8)
C20.19342 (19)1.0649 (5)0.63145 (9)0.0244 (7)
C30.1278 (2)1.0044 (6)0.62013 (11)0.0307 (8)
H3A0.09320.98230.63840.037*
C40.1139 (2)0.9771 (6)0.58161 (12)0.0344 (10)
H4A0.07020.92950.57440.041*
C50.2424 (2)1.0999 (6)0.60318 (10)0.0303 (8)
H5A0.28731.13970.60980.036*
C60.2246 (2)1.0755 (6)0.56502 (10)0.0323 (8)
H6A0.25801.10080.54620.039*
C70.0597 (2)0.5415 (6)0.39568 (11)0.0307 (9)
H7A0.10150.57540.40760.037*
C80.06218 (19)0.4538 (5)0.35956 (11)0.0260 (8)
H8A0.10470.43020.34750.031*
C90.00000.4016 (7)0.34159 (13)0.0216 (10)
C100.00000.2775 (8)0.30518 (13)0.0240 (10)
N10.16080 (18)1.0165 (5)0.55395 (9)0.0340 (8)
N20.00000.5802 (7)0.41440 (12)0.0275 (9)
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Tm10.01730 (14)0.02826 (15)0.01249 (13)0.00208 (9)0.0000.000
Cu10.0466 (19)0.052 (8)0.0131 (15)0.0000.0000.000 (8)
Cu1'0.047 (2)0.051 (7)0.0145 (17)−0.004 (4)0.0008 (16)0.001 (5)
Cu20.0465 (4)0.0686 (6)0.0243 (4)0.0000.000−0.0111 (4)
Br10.0271 (4)0.0500 (5)0.0260 (4)0.0000.000−0.0129 (3)
Br20.0286 (3)0.0619 (4)0.0368 (4)0.0000.0000.0074 (3)
O10.0438 (16)0.0448 (16)0.0171 (13)−0.0065 (14)0.0069 (11)0.0009 (12)
O20.0406 (17)0.081 (2)0.0212 (14)−0.0224 (17)−0.0013 (12)−0.0077 (15)
O30.0250 (13)0.0570 (18)0.0348 (15)0.0037 (13)0.0023 (11)−0.0254 (13)
O40.023 (3)0.033 (3)0.061 (4)0.0000.0000.000
O50.046 (2)0.051 (2)0.0127 (17)−0.021 (2)0.0000.000
O60.053 (3)0.027 (2)0.113 (5)0.008 (2)0.0000.000
C10.039 (2)0.031 (2)0.0138 (16)−0.0086 (17)0.0020 (15)0.0005 (15)
C20.034 (2)0.0235 (16)0.0161 (16)−0.0035 (15)−0.0005 (14)−0.0006 (14)
C30.0290 (18)0.042 (2)0.0207 (18)−0.0027 (17)0.0035 (15)0.0043 (15)
C40.030 (2)0.050 (3)0.024 (2)−0.0011 (18)−0.0018 (16)−0.0005 (17)
C50.0307 (19)0.039 (2)0.0211 (18)−0.0088 (17)0.0013 (15)−0.0008 (17)
C60.033 (2)0.045 (2)0.0188 (18)−0.0045 (18)0.0048 (15)−0.0001 (17)
C70.028 (2)0.040 (2)0.024 (2)−0.0044 (16)−0.0016 (15)−0.0091 (16)
C80.0226 (18)0.032 (2)0.0229 (18)−0.0003 (14)0.0006 (14)−0.0065 (15)
C90.029 (2)0.018 (2)0.018 (2)0.0000.0000.0000 (19)
C100.024 (2)0.028 (3)0.020 (2)0.0000.000−0.003 (2)
N10.0339 (18)0.051 (2)0.0176 (15)0.0022 (15)0.0003 (13)0.0007 (14)
N20.034 (2)0.030 (2)0.018 (2)0.0000.000−0.0048 (19)
Geometric parameters (Å, °) top
Tm1—O42.197 (2)O3—C101.243 (3)
Tm1—O1i2.208 (2)O4—Tm1ix2.197 (2)
Tm1—O1ii2.208 (2)O4—H41.0867
Tm1—O52.284 (4)O5—H50.8127
Tm1—O3iii2.315 (2)O6—H6B0.8504
Tm1—O32.315 (2)O6—H6C0.9163
Tm1—O62.467 (5)C1—C21.512 (5)
Tm1—H6B2.1833C2—C51.379 (5)
Cu1—N11.924 (4)C2—C31.380 (5)
Cu1—N1iv1.924 (4)C3—C41.375 (5)
Cu1—Br12.654 (12)C3—H3A0.9300
Cu1'—N11.97 (3)C4—N11.342 (5)
Cu1'—N1iv2.00 (3)C4—H4A0.9300
Cu1'—Br12.645 (12)C5—C61.377 (5)
Cu2—N22.020 (4)C5—H5A0.9300
Cu2—Br22.5191 (7)C6—N11.342 (5)
Cu2—Br2v2.5191 (7)C6—H6A0.9300
Cu2—Br12.6275 (9)C7—N21.341 (4)
Cu2—Cu2v2.6889 (17)C7—C81.385 (6)
Br1—Cu2vi2.6276 (9)C7—H7A0.9300
Br1—Cu1'vi2.645 (12)C8—C91.391 (4)
Br1—Cu1'vii2.645 (12)C8—H8A0.9300
Br1—Cu1'iv2.645 (12)C9—C8vii1.391 (4)
Br1—Cu1vi2.654 (12)C9—C101.512 (7)
Br2—Cu2v2.5191 (7)C10—O3vii1.243 (3)
O1—C11.260 (4)N1—Cu1'iv2.00 (3)
O1—Tm1viii2.208 (2)N2—C7vii1.341 (4)
O2—C11.230 (5)
O4—Tm1—O1i109.97 (10)Cu1'vi—Br1—Cu1'iv161.3 (7)
O4—Tm1—O1ii109.97 (10)Cu1'vii—Br1—Cu1'iv180.000 (4)
O1i—Tm1—O1ii113.85 (14)Cu2vi—Br1—Cu1vi90.0
O4—Tm1—O5159.04 (17)Cu2—Br1—Cu1vi90.000 (2)
O1i—Tm1—O580.41 (9)Cu1'—Br1—Cu1vi170.7 (3)
O1ii—Tm1—O580.41 (9)Cu1'iv—Br1—Cu1vi170.7 (3)
O4—Tm1—O3iii83.02 (12)Cu2vi—Br1—Cu190.000 (2)
O1i—Tm1—O3iii154.05 (11)Cu2—Br1—Cu190.0
O1ii—Tm1—O3iii80.34 (10)Cu1'vi—Br1—Cu1170.7 (3)
O5—Tm1—O3iii80.85 (11)Cu1'vii—Br1—Cu1170.7 (3)
O4—Tm1—O383.02 (12)Cu1vi—Br1—Cu1180.000 (1)
O1i—Tm1—O380.34 (10)Cu2—Br2—Cu2v64.51 (4)
O1ii—Tm1—O3154.05 (11)C1—O1—Tm1viii154.2 (3)
O5—Tm1—O380.85 (11)C10—O3—Tm1139.8 (3)
O3iii—Tm1—O379.09 (15)Tm1—O4—Tm1ix135.8 (3)
O4—Tm1—O672.25 (17)Tm1—O4—H4110.9
O1i—Tm1—O672.46 (9)Tm1ix—O4—H4110.9
O1ii—Tm1—O672.46 (9)Tm1—O5—H5120.2
O5—Tm1—O6128.71 (15)Tm1—O6—H6B60.9
O3iii—Tm1—O6133.49 (10)Tm1—O6—H6C150.7
O3—Tm1—O6133.49 (10)H6B—O6—H6C105.4
O4—Tm1—H6B92.1O2—C1—O1126.2 (3)
O1i—Tm1—H6B64.0O2—C1—C2117.9 (3)
O1ii—Tm1—H6B64.0O1—C1—C2115.9 (3)
O5—Tm1—H6B108.8C5—C2—C3118.0 (3)
O3iii—Tm1—H6B140.0C5—C2—C1120.8 (3)
O3—Tm1—H6B140.0C3—C2—C1121.2 (3)
O6—Tm1—H6B19.9C4—C3—C2119.5 (4)
Cu1'iv—Cu1—N193 (4)C4—C3—H3A120.2
Cu1'iv—Cu1—N1iv89 (4)C2—C3—H3A120.2
N1—Cu1—N1iv154.2 (7)N1—C4—C3122.6 (4)
Cu1'iv—Cu1—Br184 (3)N1—C4—H4A118.7
N1—Cu1—Br1102.9 (4)C3—C4—H4A118.7
N1iv—Cu1—Br1102.9 (4)C6—C5—C2119.7 (3)
Cu1'iv—Cu1'—N179 (4)C6—C5—H5A120.2
Cu1'iv—Cu1'—N1iv76 (4)C2—C5—H5A120.2
N1—Cu1'—N1iv142.3 (6)N1—C6—C5122.4 (4)
Cu1'iv—Cu1'—Br180.7 (3)N1—C6—H6A118.8
N1—Cu1'—Br1102.0 (9)C5—C6—H6A118.8
N1iv—Cu1'—Br1101.2 (9)N2—C7—C8123.3 (4)
N2—Cu2—Br2108.50 (6)N2—C7—H7A118.3
N2—Cu2—Br2v108.50 (7)C8—C7—H7A118.3
Br2—Cu2—Br2v115.49 (4)C7—C8—C9118.9 (4)
N2—Cu2—Br1122.79 (14)C7—C8—H8A120.6
Br2—Cu2—Br1100.894 (19)C9—C8—H8A120.6
Br2v—Cu2—Br1100.894 (19)C8—C9—C8vii118.1 (5)
N2—Cu2—Cu2v126.47 (15)C8—C9—C10120.8 (2)
Br2—Cu2—Cu2v57.744 (18)C8vii—C9—C10120.8 (2)
Br2v—Cu2—Cu2v57.744 (18)O3—C10—O3vii126.7 (5)
Br1—Cu2—Cu2v110.74 (4)O3—C10—C9116.6 (2)
Cu2vi—Br1—Cu2180.0O3vii—C10—C9116.6 (2)
Cu2vi—Br1—Cu1'vi98.6 (4)C4—N1—C6117.7 (3)
Cu2—Br1—Cu1'vi81.4 (4)C4—N1—Cu1122.3 (4)
Cu2vi—Br1—Cu1'vii81.4 (4)C6—N1—Cu1119.9 (4)
Cu2—Br1—Cu1'vii98.6 (4)C4—N1—Cu1'123.7 (5)
Cu2vi—Br1—Cu1'81.4 (4)C6—N1—Cu1'116.5 (5)
Cu2—Br1—Cu1'98.6 (4)C4—N1—Cu1'iv117.0 (5)
Cu1'vi—Br1—Cu1'180.0 (13)C6—N1—Cu1'iv124.3 (5)
Cu1'vii—Br1—Cu1'161.3 (7)C7—N2—C7vii117.2 (4)
Cu2vi—Br1—Cu1'iv98.6 (4)C7—N2—Cu2120.8 (2)
Cu2—Br1—Cu1'iv81.4 (4)C7vii—N2—Cu2120.8 (2)
Symmetry codes: (i) x, −y+1, −z+1; (ii) x, −y+1, z−1/2; (iii) x, y, −z+1/2; (iv) x, −y+2, −z+1; (v) −x, −y+1, −z+1; (vi) −x, −y+2, −z+1; (vii) −x, y, z; (viii) x, −y+1, z+1/2; (ix) −x, y, −z+1/2.
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O5—H5···O2x0.811.862.667 (3)175.
Symmetry codes: (x) −x+1/2, −y+3/2, −z+1.
Table 1
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
O5—H5···O2i0.811.862.667 (3)175.
Symmetry codes: (i) −x+1/2, −y+3/2, −z+1.
Acknowledgements top

This work was supported by the Qingdao University Research Fund (No. 063-06300522).

references
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