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ISSN: 2056-9890

Crystal structure of bis­­(N-tert-butyl­benzamidinium) hexa­chlorido­zirconate(IV) di­chloro­methane disolvate

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aInstitute of Applied Chemistry, Shanxi University, Taiyuan 030006, People's Republic of China
*Correspondence e-mail: sdbai@sxu.edu.cn

Edited by D.-J. Xu, Zhejiang University (Yuquan Campus), China (Received 1 February 2016; accepted 20 February 2016; online 2 March 2016)

In the ZrIV complex anion of the title complex salt, [(C4H9)HNC(C6H5)NH2]2[ZrCl6]·2CH2Cl2, the ZrIV cation, located on an inversion centre, is coordinated by six Cl anions in a distorted octa­hedral geometry with Zr—Cl distances in the range 2.433 (2)–2.4687 (19) Å; in the amidinium cation, the dihedral angle between the aromatic ring and [NCN] plane is 43.3 (4)°. In the crystal, the amidinium cations and [ZrCl6]2− anions are linked by N—H⋯Cl hydrogen bonds, forming a two-dimensional network extending along the b axis; two di­chloro­methane solvent mol­ecules are linked by a pair of weak C—H⋯Cl hydrogen bonds, forming a centrosymmetric [CHCl]2 six-membered ring.

1. Chemical context

Amidinates represent an important class in the array of N-centered ligands comparable to the cyclo­penta­dienyl system (Edelmann, 1994[Edelmann, F. T. (1994). Coord. Chem. Rev. 137, 403-481.]; Barker & Kilner, 1994[Barker, J. & Kilner, M. (1994). Coord. Chem. Rev. 133, 219-300.]; Collins, 2011[Collins, S. (2011). Coord. Chem. Rev. 255, 118-138.]). They are four-electron, monoanionic and N-donor bidentate chelates, demonstrating a great diversity by variation of substituents on the conjugated N–C–N backbone. Their steric and electronic properties are easily tunable to meet the requirements of different metal atoms. In the course of extending amidinate chemistry, we have explored a practical synthetic pathway to the alkyl-ended amidinate and ansa-bis­(amidinate) ligands (Bai et al., 2013[Bai, S.-D., Liu, R.-Q., Guan, F., Wang, T., Tong, H.-B., Guo, J.-P. & Liu, D.-S. (2013). Mendeleev Commun. 23, 265-267.]). They have been applied in the synthesis of Group 4 complexes, which are good catalysts for ethyl­ene polymerization (Bai et al., 2010[Bai, S.-D., Tong, H.-B., Guo, J.-P., Zhou, M.-S., Liu, D.-S. & Yuan, S.-F. (2010). Polyhedron, 29, 262-269.]). Amidines are convenient precursors for both monoanionic amidinate ligands and bianionic ansa-bis­(amidinate) ligands (Coles, 2006[Coles, M. P. (2006). Dalton Trans. pp. 985-1001.]). Some amidines could be prepared by Yb complex-catalysed addition reactions of aromatic amines and nitriles (Wang et al., 2008[Wang, J.-F., Xu, F., Cai, T. & Shen, Q. (2008). Org. Lett. 10, 445-448.]). On the other hand, monoanionic amidinate could be used to prepare amidine and amidinium through acidolysis. Based on the same skeleton, the transformation from amidinate to amidinium will undergo an electrical inversion. Here, we report the synthesis and structural characterization of a bis­(N-tert-butyl-benzamidinium) hexa­chlorido­zirconate complex derived from the monoanionic amidinate.

2. Structural commentary

The anion in the title compound, (I)[link], is centrosymmetric with the ZrIV cation located on an inversion centre (Fig. 1[link]) and is six-coordinated by Cl atoms. The corresponding coordination polyhedron can be described as a distorted octa­hedron where atoms Cl1, Cl2, Cl1i and Cl2i [symmetry code: (i) −x + 2, −y, −z + 1] define the equatorial plane while atoms Cl3 and Cl3i occupy the axial positions. The equatorial Zr—Cl bond lengths are 2.4674 (18) Å and 2.4687 (19) Å while the axial bond length [2.433 (2) Å] is a little shorter. In the amidinium moiety, the terminal tert-butyl group is disposed in the direction opposite to the phenyl group on the ipso-carbon of the N–C–N backbone, which could minimize steric hindrance between the two groups. The dihedral angle between the aromatic ring and [NCN] plane is 43.3 (4)°. The two C—N bond lengths are equivalent [1.300 (8) and 1.299 (9) Å], composing a typical conjugated N—C—N skeleton. The lengths of the C—N bonds in (I)[link] are shorter than those reported for a similar amidinium cation (1.325 Å; Centore et al., 2003[Centore, R., Tuzi, A. & Liguori, D. (2003). Acta Cryst. C59, m241-m242.]).

[Scheme 1]
[Figure 1]
Figure 1
The mol­ecular structure of (I)[link], showing the atom-numbering scheme. Displacement ellipsoids are drawn at the 30% probability level. H atoms are presented as small spheres of arbitrary radius. [Symmetry code: (i) −x + 2, −y, −z + 1.]

3. Supra­molecular features

The extended structure consists of amidinium cations forming an extended hydrogen-bonded network with the chlorine atoms of the hexa­chlorido­zirconate anions. The amidinium cations involving N1 and N2 all serve as hydrogen-bond donors while only the chlorine atoms in the equatorial plane of the hexa­chlorido­zirconate anions act as acceptors (Table 1[link], Fig. 2[link]). With the N—H⋯Cl hydrogen bonds, each amidinium cation connects two adjacent [ZrCl6]2− anions and each [ZrCl6]2− anion links four neighboring amidinium cations. The existence of bifurcated hydrogen bonds enables the formation of a two-dimensional network. Four amidinium cations and four [ZrCl6]2− anions compose a square-like hole. [ZrCl6]2− anions occupy the vertex positions and amidinium cations are on the edge. The corresponding motif obeys the operation of centrosymmetry and the inversion centre is the central point of the square. Moreover, the two-dimensional network extends along the b axis (Fig. 3[link]). In other words, the layered network is parallel to (101) and perpendicular to (010). Besides the N—H⋯Cl hydrogen bonds, a C—H⋯Cl hydrogen bond can be observed between two centrosymmetrically related di­chloro­methane solvent mol­ecules, leading to the formation of a [CHCl]2 six-membered ring geometry. The angle of the C—H⋯Cl hydrogen bond is 171°, suggesting a closely linear arrangement of the related C, H and Cl atoms, also resulting in a long distance between donor and acceptor atoms [3.70 (2) Å].

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A D—H H⋯A DA D—H⋯A
N1—H1⋯Cl2i 0.88 2.64 3.491 (6) 162
N2—H2A⋯Cl2ii 0.88 2.60 3.270 (7) 133
N2—H2B⋯Cl1ii 0.88 2.56 3.353 (7) 150
C12—H12A⋯Cl5iii 0.99 2.72 3.70 (2) 171
Symmetry codes: (i) -x+2, -y+1, -z+1; (ii) [-x+{\script{3\over 2}}, y+{\script{1\over 2}}, -z+{\script{3\over 2}}]; (iii) -x+1, -y+1, -z+1.
[Figure 2]
Figure 2
Crystal packing diagram for (I)[link], showing the two-dimensional hydrogen-bonded network. [Symmetry codes: (ii) −x + 2, −y + 1, −z + 1; (iii) −x + [{3\over 2}], y + [{1\over 2}], −z + [{3\over 2}].]
[Figure 3]
Figure 3
A view of the two-dimensional network along the b axis.

4. Database survey

There are 38 structures that incorporate the zirconate anions, including [ZrCl6]2−, [Zr2Cl10]2− and [Zr2Cl9]. Of those 38 structures, only one has amidinium as the counter-cation (Centore et al., 2003[Centore, R., Tuzi, A. & Liguori, D. (2003). Acta Cryst. C59, m241-m242.]). Its [Zr2Cl10]2− anion has two bridging Cl atoms and its amidinium cation has three substituents attached on the two nitro­gen atoms. In contrast to the title compound, no N—H⋯Cl hydrogen bond is observed due to the hindrance of the N-substituents and the lack of an N-bound hydrogen atom.

5. Synthesis and crystallization

General Procedure: All manipulations and reactions were performed under an inert atmosphere of nitro­gen using standard Schlenk techniques. Solvents were pre-dried over sodium, distilled from sodium-benzo­phenone (diethyl ether and dioxane) and stored over mol­ecular sieves (4 Å). CH2Cl2 was purified by distillation over CaH2. HCl was prepared by treating NaCl with concentrated H2SO4 and dissolved in dioxane.

Synthesis of bis­(N-tert-butyl-benzamidinium) hexa­chlorido­zirconate(IV): The title compound was prepared by a one-pot reaction of tert-butyl­amine, LiBu, PhCN, HCl (3.6 M in dioxane) and ZrCl4. A solution of LiBun (2.2 M, 2.27 ml, 5.0 mmol) in hexane was slowly added into a solution of tert-butyl­amine (0.53 ml, 5.0 mmol) in Et2O (30 ml) by syringe at 273 K. The reaction mixture was warmed to room temperature and kept stirring for 3 h. Then benzo­nitrile (0.51 ml, 5.0 mmol) was added by syringe at 273 K. The reaction mixture was warmed to room temperature and kept stirring for 4 h. HCl (2.78 ml, 10.0 mmol, 3.6 M in dioxane) was added by syringe at 273 K. After stirring at room temperature for 4 h, ZrCl4 (0.583 g, 2.5 mmol) was added at 273 K. The resulting mixture was stirred at room temperature overnight and all volatiles were removed in vacuo. The residue was extracted with di­chloro­methane and the filtrate was concentrated to give colorless crystals (yield 1.325 g, 64%). The inter­mediate process involved an addition reaction of lithium amide and nitrile to yield lithium monoamidinate. Crystals of (I)[link] suitable for single-crystal X-ray investigation were obtained by recrystallization from CH2Cl2.

6. Catalytic activity for ethyl­ene polymerization

The catalytic activity of the title compound for ethyl­ene polymerization was examined. At normal pressure and in the presence of methyl­aluminoxane (MAO), it was found to be an inactive catalyst for ethyl­ene polymerization at 303 K or higher temperature. The reaction was then performed at 10 atm in a 250 mL autoclave. However, only a trace to very small amount of polymer formation could be observed, even when heating the reaction system or changing the ratio of (I)[link] to MAO. Therefore, a conclusion could be drawn that the title compound can not catalyse ethyl­ene polymerization.

7. Refinement

Crystal data, data collection and structure refinement details are summarized in Table 2[link]. Hydrogen atoms were included in geometrically calculated positions. For N-bound H atoms, N—H = 0.88 Å and Uiso(H) = 1.2Ueq(N). For methyl­ene H atoms, C—H = 0.99 Å and Uiso(H) = 1.2Ueq(C) and for phenyl H atoms, C—H = 0.95 Å and Uiso(H) = 1.2Ueq(C). Methyl H atoms were constrained to an ideal geometry, with C—H = 0.98 Å and Uiso(H) = 1.5Ueq(C), but each group was allowed to rotate freely along its C—C bond.

Table 2
Experimental details

Crystal data
Chemical formula (C11H17N2)[ZrCl6]·2CH2Cl2
Mr 828.30
Crystal system, space group Monoclinic, P21/n
Temperature (K) 200
a, b, c (Å) 10.443 (5), 16.154 (9), 10.891 (6)
β (°) 91.259 (10)
V3) 1836.9 (17)
Z 2
Radiation type Mo Kα
μ (mm−1) 1.05
Crystal size (mm) 0.20 × 0.20 × 0.15
 
Data collection
Diffractometer Bruker SMART area-detector
Absorption correction Multi-scan (SADABS; Sheldrick, 1996[Sheldrick, G. M. (1996). SADABS. University of Göttingen, Germany.])
Tmin, Tmax 0.818, 0.859
No. of measured, independent and observed [I > 2σ(I)] reflections 10309, 3410, 2255
Rint 0.061
(sin θ/λ)max−1) 0.606
 
Refinement
R[F2 > 2σ(F2)], wR(F2), S 0.071, 0.213, 1.00
No. of reflections 3410
No. of parameters 181
No. of restraints 1
H-atom treatment H-atom parameters constrained
Δρmax, Δρmin (e Å−3) 1.73, −0.90
Computer programs: SMART and SAINT (Bruker, 2000[Bruker (2000). SMART and SAINT. Bruker AXS Inc., Madison, Wisconsin, USA.]), SHELXS97, SHELXL97 and SHELXTL/PC (Sheldrick, 2008[Sheldrick, G. M. (2008). Acta Cryst. A64, 112-122.]).

Supporting information


Computing details top

Data collection: SMART (Bruker, 2000); cell refinement: SAINT (Bruker, 2000); data reduction: SAINT (Bruker, 2000); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: SHELXTL/PC (Sheldrick, 2008); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Bis(N-tert-butylbenzamidinium) hexachloridozirconate(IV) dichloromethane disolvate top
Crystal data top
(C11H17N2)[ZrCl6]·2CH2Cl2F(000) = 840
Mr = 828.30Dx = 1.498 Mg m3
Monoclinic, P21/nMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ynCell parameters from 2094 reflections
a = 10.443 (5) Åθ = 2.3–22.0°
b = 16.154 (9) ŵ = 1.05 mm1
c = 10.891 (6) ÅT = 200 K
β = 91.259 (10)°Block, colorless
V = 1836.9 (17) Å30.20 × 0.20 × 0.15 mm
Z = 2
Data collection top
Bruker SMART area-detector
diffractometer
3410 independent reflections
Radiation source: fine-focus sealed tube2255 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.061
φ and ω scanθmax = 25.5°, θmin = 2.3°
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)
h = 1012
Tmin = 0.818, Tmax = 0.859k = 1915
10309 measured reflectionsl = 1213
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)] = 0.071Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.213H-atom parameters constrained
S = 1.00 w = 1/[σ2(Fo2) + (0.1176P)2 + 2.835P]
where P = (Fo2 + 2Fc2)/3
3410 reflections(Δ/σ)max = 0.001
181 parametersΔρmax = 1.73 e Å3
1 restraintΔρmin = 0.90 e Å3
Special details top

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds 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 > 2sigma(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
Zr11.00000.00000.50000.0430 (3)
Cl10.79402 (15)0.05446 (10)0.41944 (16)0.0611 (5)
Cl21.08156 (15)0.14274 (9)0.52270 (16)0.0589 (5)
Cl30.91663 (19)0.00651 (11)0.70660 (16)0.0667 (5)
N10.7169 (5)0.7997 (3)0.7141 (5)0.0614 (15)
H10.77420.82320.66730.074*
N20.6913 (6)0.7035 (4)0.8664 (6)0.0803 (19)
H2A0.60970.71590.87310.096*
H2B0.72590.66490.91350.096*
C10.5840 (7)0.8317 (4)0.6987 (8)0.0678 (19)
C20.5393 (11)0.8666 (7)0.8189 (11)0.116 (4)
H2C0.58480.83900.88710.175*
H2D0.44700.85730.82580.175*
H2E0.55710.92610.82190.175*
C30.5923 (8)0.9014 (6)0.6076 (10)0.097 (3)
H3A0.65710.94130.63600.145*
H3B0.50890.92900.60000.145*
H3C0.61610.87930.52750.145*
C40.4977 (8)0.7642 (6)0.6490 (11)0.105 (3)
H4A0.53050.74410.57080.157*
H4B0.41100.78620.63600.157*
H4C0.49540.71840.70790.157*
C50.7605 (6)0.7420 (4)0.7870 (7)0.0595 (17)
C60.8982 (7)0.7194 (4)0.7806 (7)0.0614 (17)
C70.9913 (7)0.7784 (5)0.7678 (7)0.070 (2)
H70.96870.83530.76400.084*
C81.1165 (8)0.7553 (6)0.7606 (8)0.084 (2)
H81.18130.79620.75400.100*
C91.1485 (9)0.6735 (6)0.7630 (9)0.095 (3)
H91.23580.65770.75750.113*
C101.0578 (9)0.6150 (6)0.7731 (11)0.105 (3)
H101.08140.55820.77330.127*
C110.9321 (9)0.6364 (5)0.7831 (9)0.092 (3)
H110.86840.59490.79160.110*
C120.335 (2)0.4738 (14)0.5569 (19)0.248 (13)
H12A0.42200.45330.53910.298*
H12B0.32460.46920.64680.298*
Cl40.2363 (6)0.4119 (4)0.4939 (5)0.232 (3)
Cl50.3316 (6)0.5793 (3)0.5205 (4)0.1913 (19)
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Zr10.0361 (5)0.0397 (4)0.0535 (5)0.0011 (3)0.0091 (3)0.0003 (3)
Cl10.0445 (9)0.0580 (9)0.0805 (11)0.0060 (7)0.0031 (8)0.0021 (8)
Cl20.0490 (9)0.0442 (8)0.0841 (12)0.0066 (6)0.0129 (8)0.0070 (7)
Cl30.0766 (12)0.0676 (11)0.0566 (10)0.0072 (8)0.0201 (8)0.0021 (8)
N10.043 (3)0.058 (3)0.084 (4)0.001 (2)0.016 (3)0.018 (3)
N20.057 (4)0.075 (4)0.110 (5)0.006 (3)0.018 (3)0.037 (4)
C10.045 (4)0.056 (4)0.102 (6)0.002 (3)0.015 (4)0.015 (4)
C20.096 (8)0.115 (8)0.139 (9)0.037 (6)0.027 (7)0.000 (7)
C30.056 (5)0.082 (6)0.154 (9)0.016 (4)0.010 (5)0.048 (6)
C40.057 (5)0.091 (6)0.166 (10)0.006 (5)0.006 (6)0.011 (6)
C50.049 (4)0.050 (4)0.079 (5)0.006 (3)0.006 (3)0.006 (3)
C60.058 (4)0.049 (3)0.078 (5)0.004 (3)0.002 (3)0.013 (3)
C70.053 (4)0.061 (4)0.095 (5)0.008 (3)0.004 (4)0.022 (4)
C80.056 (5)0.096 (6)0.099 (6)0.009 (4)0.004 (4)0.024 (5)
C90.060 (5)0.097 (7)0.127 (8)0.012 (5)0.002 (5)0.027 (6)
C100.077 (6)0.076 (6)0.163 (10)0.019 (5)0.001 (6)0.011 (6)
C110.076 (6)0.059 (5)0.141 (8)0.004 (4)0.001 (5)0.013 (5)
C120.25 (2)0.30 (3)0.191 (18)0.16 (2)0.103 (18)0.103 (18)
Cl40.249 (6)0.293 (7)0.155 (4)0.105 (6)0.044 (4)0.005 (4)
Cl50.234 (5)0.192 (5)0.148 (3)0.020 (4)0.012 (3)0.025 (3)
Geometric parameters (Å, º) top
Zr1—Cl12.4674 (18)C3—H3C0.9800
Zr1—Cl1i2.4674 (18)C4—H4A0.9800
Zr1—Cl2i2.4687 (19)C4—H4B0.9800
Zr1—Cl22.4687 (19)C4—H4C0.9800
Zr1—Cl3i2.433 (2)C5—C61.487 (10)
Zr1—Cl32.433 (2)C6—C71.371 (10)
N1—C51.300 (8)C6—C111.387 (10)
N1—C11.487 (9)C7—C81.363 (11)
N1—H10.8800C7—H70.9500
N2—C51.299 (9)C8—C91.364 (12)
N2—H2A0.8800C8—H80.9500
N2—H2B0.8800C9—C101.345 (13)
C1—C31.505 (11)C9—H90.9500
C1—C41.508 (12)C10—C111.365 (13)
C1—C21.509 (13)C10—H100.9500
C2—H2C0.9800C11—H110.9500
C2—H2D0.9800C12—Cl41.583 (17)
C2—H2E0.9800C12—Cl51.75 (2)
C3—H3A0.9800C12—H12A0.9900
C3—H3B0.9800C12—H12B0.9900
Cl3i—Zr1—Cl3180.000 (1)C1—C3—H3C109.5
Cl3i—Zr1—Cl190.77 (7)H3A—C3—H3C109.5
Cl3—Zr1—Cl189.23 (7)H3B—C3—H3C109.5
Cl3i—Zr1—Cl1i89.23 (7)C1—C4—H4A109.5
Cl3—Zr1—Cl1i90.77 (7)C1—C4—H4B109.5
Cl1—Zr1—Cl1i180.00 (8)H4A—C4—H4B109.5
Cl3i—Zr1—Cl2i89.81 (6)C1—C4—H4C109.5
Cl3—Zr1—Cl2i90.19 (6)H4A—C4—H4C109.5
Cl1—Zr1—Cl2i90.07 (6)H4B—C4—H4C109.5
Cl1i—Zr1—Cl2i89.93 (6)N2—C5—N1123.9 (6)
Cl3i—Zr1—Cl290.19 (6)N2—C5—C6117.8 (6)
Cl3—Zr1—Cl289.81 (6)N1—C5—C6118.3 (6)
Cl1—Zr1—Cl289.93 (6)C7—C6—C11119.5 (7)
Cl1i—Zr1—Cl290.07 (6)C7—C6—C5121.5 (6)
Cl2i—Zr1—Cl2180.00 (8)C11—C6—C5118.9 (7)
C5—N1—C1129.1 (6)C8—C7—C6119.9 (7)
C5—N1—H1115.4C8—C7—H7120.1
C1—N1—H1115.4C6—C7—H7120.1
C5—N2—H2A120.0C7—C8—C9120.0 (8)
C5—N2—H2B120.0C7—C8—H8120.0
H2A—N2—H2B120.0C9—C8—H8120.0
N1—C1—C3105.5 (6)C10—C9—C8120.6 (9)
N1—C1—C4109.8 (6)C10—C9—H9119.7
C3—C1—C4110.3 (8)C8—C9—H9119.7
N1—C1—C2109.7 (7)C9—C10—C11120.6 (9)
C3—C1—C2108.4 (8)C9—C10—H10119.7
C4—C1—C2112.8 (8)C11—C10—H10119.7
C1—C2—H2C109.5C10—C11—C6119.3 (9)
C1—C2—H2D109.5C10—C11—H11120.4
H2C—C2—H2D109.5C6—C11—H11120.4
C1—C2—H2E109.5Cl4—C12—Cl5120.5 (12)
H2C—C2—H2E109.5Cl4—C12—H12A107.2
H2D—C2—H2E109.5Cl5—C12—H12A107.2
C1—C3—H3A109.5Cl4—C12—H12B107.2
C1—C3—H3B109.5Cl5—C12—H12B107.2
H3A—C3—H3B109.5H12A—C12—H12B106.8
Symmetry code: (i) x+2, y, z+1.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
N1—H1···Cl2ii0.882.643.491 (6)162
N2—H2A···Cl2iii0.882.603.270 (7)133
N2—H2B···Cl1iii0.882.563.353 (7)150
C12—H12A···Cl5iv0.992.723.70 (2)171
Symmetry codes: (ii) x+2, y+1, z+1; (iii) x+3/2, y+1/2, z+3/2; (iv) x+1, y+1, z+1.
 

Acknowledgements

This work was supported by grants from the Natural Science Foundation of China (grant No. 20702029) and the Natural Science Foundation of Shanxi Province, China (grant No. 2008011024).

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