supplementary materials


Acta Cryst. (2008). E64, m1318    [ doi:10.1107/S1600536808030286 ]

Tetrakis(1,3,4,6,7,9-hexaaza-1H-phenalen-6-ium) sodium(I) pentakis(tetrafluoridoborate)

Y. Jiang, Y. Hou and Y. Chen

Abstract top

In the title compound, Na+·4C7H5N6+·5BF4-, the Na+ ion lies on a fourfold rotation axis and one of the tetrafluoridoborate ions lies on a site of symmetry \overline{4}. Each Na+ ion is surrounded by four symmetry-related tetrafluoridoborate ions, and is eight-coordinated by F atoms, the Na...F separation being 2.3956 (15) or 2.4347 (17) Å. The hexaazaphenalenium ring system is essentially planar. In the crystal structure, the cations and anions are linked into a three-dimensional network by N-H...N and C-H...F hydrogen bonds.

Comment top

Phenalenyl, as a planar π system with D3h symmetry, has received much attention and still played an important role as a building block for spin-mediated molecular functional materials (Haddon 1975; Koutentis et al., 2001; Goto et al., 1999). Hexaazaphenalene (HAP) is a highly symmetric heterocycle with full nitrogen substitution in all of the α sites of phenalene (Morita et al., 2002; Morita et al., 2003; Suzuki et al., 2005; Tamaki et al., 1997; Zheng et al., 2003; Zheng et al., 2005). Because of the directionality of lone-pair electrons at the nitrogen sites, incorporation of nitrogen atoms into the phenalenyl skeleton was found to give substantial effects on its electronic structure. Suzuki et al. have reported the preparation and single cystal structure of the anion of hexaazaphenalene. Herein, we report the preparation and crystal structure of a hexaazaphenalene fluoroboric salt, the title compound.

In the title compound, the Na+ ion lies on a four-fold rotation axis and one of the tetrafluoridoborate ions lies on a 4 site symmetry. One of the remaining four BF4- ions and one of the four hexaazaphenalenium ions are symmetry independent, and they lie on general positions. Each Na+ ion is surrounded by four symmetry related tetrafluoridoborate ions, with the Na···F separation being 2.3956 (15) or 2.4347 (17) Å (Fig.1). The hexaazaphenalenium ring system is essentially planar, with a maximum deviation of 0.042 (2) Å for atom N6. The C—N distances lie in the range 1.310 (3)-1.363 (3) Å.

Two types of hydrogen bonds viz. N—H···N and C—H···F exist in the solid state of the title compound. These hydrogen bonds link the cationic and anionic units into a three-dimensional network (Fig. 2). There are no face to face π-π staking involving the hexaazaphenalenium units, which is a common packing mode in other phenalene compounds.

Related literature top

For related literature, see: Goto et al. (1999); Haddon (1975); Koutentis et al. (2001); Morita et al. (2002, 2003); Suzuki et al. (2005); Tamaki et al. (1997); Zheng et al. (2003, 2005). It would be much more useful to readers if the "Related literature" section had some kind of simple sub-division, so that, instead of just "For related literature, see···" it said, for example, "For general background, see···. For related structures, see···.? etc. Please revise this section as indicated.

Experimental top

The sodium salt of hexaazaphenalene was prepared according to the literature method (Suzuki et al., 2005). To a water (5 ml) solution of sodium salt of hexaazaphenalene (97 mg, 0.5 mmol) was added 40% HBF4 (0.5 ml, 3.0 mmol). The reaction mixture was evaporated slowly for 7 d to afford colourless crystals of the title compound.

Refinement top

H atoms were initially located in difference maps and then refined using the riding-model approximation, with C-H = 0.95 Å, N-H = 0.88 Å and Uiso(H) = 1.2Ueq(C,N).

Computing details top

Data collection: CrystalClear (Rigaku/MSC, 2005); cell refinement: CrystalClear (Rigaku/MSC, 2005); data reduction: CrystalClear (Rigaku/MSC, 2005); 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: CrystalStructure (Rigaku/MSC, 2005).

Figures top
[Figure 1] Fig. 1. A view of the environment around the Na+ ion in the title compound. Displacement ellipsoids are drawn at the 50% probability level. Atoms labelled with the suffixes A, B, C, D, E and F are generated by the symmetry operations (y, 1/2-x, z), (1/2-x, 1/2-y, z), (1/2-y, x, z), (1-y, x-1/2, -z), (y+1/2, 1-x, -z) and (3/2-x, 1/2-y, z), respectively.
[Figure 2] Fig. 2. The crystal packing of the title compound.
Tetrakis(1,3,4,6,7,9-hexaaza-1H-phenalen-6-ium) sodium(I) pentakis(tetrafluoridoborate) top
Crystal data top
Na+·4C7H5N6+·5BF4Dx = 1.883 Mg m3
Mr = 1149.72Mo Kα radiation, λ = 0.71070 Å
Tetragonal, P4/nCell parameters from 4230 reflections
Hall symbol: -P 4aθ = 1.9–27.9°
a = 15.0665 (6) ŵ = 0.20 mm1
c = 8.9322 (5) ÅT = 113 K
V = 2027.60 (16) Å3Block, colourless
Z = 20.14 × 0.12 × 0.12 mm
F(000) = 1144
Data collection top
Rigaku Saturn CCD area-detector
diffractometer
2423 independent reflections
Radiation source: rotating anode2247 reflections with I > 2σ(I)
confocalRint = 0.044
Detector resolution: 7.31 pixels mm-1θmax = 27.9°, θmin = 1.9°
ω and φ scansh = 1919
Absorption correction: multi-scan
(CrystalClear, Rigaku/MSC, 2005)
k = 1919
Tmin = 0.878, Tmax = 0.977l = 1111
21698 measured reflections
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)] = 0.055Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.152H-atom parameters constrained
S = 1.19 w = 1/[σ2(Fo2) + (0.0759P)2 + 1.4095P]
where P = (Fo2 + 2Fc2)/3
2423 reflections(Δ/σ)max = 0.001
177 parametersΔρmax = 0.74 e Å3
0 restraintsΔρmin = 0.79 e Å3
Crystal data top
Na+·4C7H5N6+·5BF4Z = 2
Mr = 1149.72Mo Kα radiation
Tetragonal, P4/nµ = 0.20 mm1
a = 15.0665 (6) ÅT = 113 K
c = 8.9322 (5) Å0.14 × 0.12 × 0.12 mm
V = 2027.60 (16) Å3
Data collection top
Rigaku Saturn CCD area-detector
diffractometer
2423 independent reflections
Absorption correction: multi-scan
(CrystalClear, Rigaku/MSC, 2005)
2247 reflections with I > 2σ(I)
Tmin = 0.878, Tmax = 0.977Rint = 0.044
21698 measured reflectionsθmax = 27.9°
Refinement top
R[F2 > 2σ(F2)] = 0.055H-atom parameters constrained
wR(F2) = 0.152Δρmax = 0.74 e Å3
S = 1.19Δρmin = 0.79 e Å3
2423 reflectionsAbsolute structure: ?
177 parametersFlack parameter: ?
0 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*/Ueq
Na10.25000.25000.4845 (2)0.0275 (4)
B10.42320 (16)0.33819 (16)0.5211 (3)0.0204 (5)
F10.39625 (9)0.28695 (10)0.39963 (15)0.0293 (4)
F20.35132 (8)0.34700 (9)0.61981 (15)0.0241 (3)
F30.49388 (9)0.29764 (9)0.59536 (16)0.0274 (3)
F40.44983 (10)0.42159 (9)0.47314 (19)0.0346 (4)
F50.73061 (11)0.17735 (10)0.08990 (17)0.0371 (4)
B20.75000.25000.00000.0203 (9)
C10.43924 (15)0.86438 (14)0.7069 (2)0.0202 (4)
H10.43170.92050.66020.024*
C20.50050 (13)0.78212 (13)0.8918 (2)0.0150 (4)
C30.56501 (13)0.69463 (13)1.0679 (2)0.0164 (4)
H30.60340.68861.15170.020*
C40.47115 (12)0.62292 (12)0.8956 (2)0.0126 (4)
C50.38121 (13)0.56354 (13)0.7195 (2)0.0157 (4)
H50.35220.51270.67980.019*
C60.40548 (13)0.71345 (12)0.7050 (2)0.0135 (4)
C70.45850 (12)0.70676 (12)0.8328 (2)0.0124 (4)
N10.48868 (12)0.86262 (11)0.82849 (19)0.0188 (4)
N20.55508 (11)0.77454 (11)1.01274 (19)0.0166 (4)
N30.52553 (11)0.61957 (11)1.01648 (18)0.0138 (3)
H3N0.53530.56861.06170.017*
N40.43264 (11)0.54996 (11)0.83918 (18)0.0146 (4)
N50.36565 (11)0.64098 (11)0.64850 (19)0.0155 (4)
N60.39805 (12)0.79459 (11)0.64176 (19)0.0181 (4)
H6N0.36690.80190.55940.022*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Na10.0214 (6)0.0214 (6)0.0396 (11)0.0000.0000.000
B10.0199 (11)0.0185 (11)0.0227 (12)0.0013 (9)0.0011 (9)0.0025 (9)
F10.0278 (7)0.0369 (8)0.0231 (7)0.0037 (6)0.0034 (5)0.0066 (6)
F20.0218 (7)0.0290 (7)0.0214 (7)0.0028 (5)0.0002 (5)0.0023 (5)
F30.0219 (7)0.0271 (7)0.0332 (8)0.0028 (5)0.0018 (5)0.0047 (6)
F40.0324 (8)0.0248 (7)0.0466 (9)0.0042 (6)0.0024 (7)0.0124 (6)
F50.0538 (10)0.0295 (8)0.0279 (8)0.0065 (7)0.0028 (7)0.0055 (6)
B20.0219 (14)0.0219 (14)0.017 (2)0.0000.0000.000
C10.0289 (11)0.0151 (9)0.0167 (9)0.0048 (8)0.0014 (8)0.0023 (8)
C20.0178 (9)0.0139 (9)0.0134 (9)0.0013 (7)0.0007 (7)0.0003 (7)
C30.0188 (9)0.0164 (9)0.0139 (9)0.0011 (7)0.0035 (7)0.0004 (7)
C40.0115 (8)0.0136 (9)0.0126 (8)0.0013 (7)0.0004 (7)0.0005 (7)
C50.0131 (8)0.0185 (9)0.0155 (9)0.0013 (7)0.0005 (7)0.0013 (7)
C60.0159 (9)0.0147 (9)0.0099 (8)0.0026 (7)0.0003 (7)0.0005 (7)
C70.0141 (8)0.0123 (9)0.0110 (8)0.0005 (7)0.0009 (7)0.0006 (7)
N10.0253 (9)0.0131 (8)0.0181 (8)0.0011 (6)0.0028 (7)0.0016 (6)
N20.0205 (9)0.0139 (8)0.0155 (8)0.0003 (6)0.0044 (6)0.0005 (6)
N30.0165 (8)0.0120 (8)0.0130 (8)0.0010 (6)0.0029 (6)0.0023 (6)
N40.0144 (8)0.0153 (8)0.0141 (8)0.0017 (6)0.0012 (6)0.0001 (6)
N50.0143 (8)0.0177 (8)0.0146 (8)0.0006 (6)0.0016 (6)0.0017 (6)
N60.0249 (9)0.0171 (8)0.0124 (8)0.0045 (7)0.0018 (7)0.0033 (6)
Geometric parameters (Å, °) top
Na1—F1i2.3956 (15)C1—H10.95
Na1—F1ii2.3956 (15)C2—N11.350 (3)
Na1—F1iii2.3956 (15)C2—N21.363 (3)
Na1—F12.3957 (15)C2—C71.403 (3)
Na1—F22.4347 (17)C3—N21.310 (3)
Na1—F2i2.4347 (17)C3—N31.358 (3)
Na1—F2ii2.4347 (17)C3—H30.95
Na1—F2iii2.4347 (17)C4—N41.341 (2)
B1—F41.387 (3)C4—N31.356 (2)
B1—F11.392 (3)C4—C71.395 (3)
B1—F31.395 (3)C5—N41.336 (3)
B1—F21.403 (3)C5—N51.349 (3)
F5—B21.3887 (14)C5—H50.95
B2—F5iv1.3887 (14)C6—N51.344 (3)
B2—F5v1.3887 (14)C6—N61.351 (2)
B2—F5vi1.3887 (14)C6—C71.397 (3)
C1—N11.317 (3)N3—H3N0.88
C1—N61.353 (3)N6—H6N0.88
F1i—Na1—F1ii84.26 (3)F1—Na1—B1iii104.13 (6)
F1i—Na1—F1iii143.12 (11)F2—Na1—B1iii72.30 (6)
F1ii—Na1—F1iii84.26 (3)F2i—Na1—B1iii140.43 (9)
F1i—Na1—F184.26 (3)F2ii—Na1—B1iii101.17 (7)
F1ii—Na1—F1143.12 (11)F2iii—Na1—B1iii28.22 (6)
F1iii—Na1—F184.26 (3)B1—Na1—B1iii89.293 (14)
F1i—Na1—F2124.30 (5)B1i—Na1—B1iii167.24 (13)
F1ii—Na1—F2150.83 (6)B1ii—Na1—B1iii89.292 (14)
F1iii—Na1—F275.56 (5)F4—B1—F1110.24 (19)
F1—Na1—F256.00 (4)F4—B1—F3108.83 (18)
F1i—Na1—F2i56.00 (4)F1—B1—F3110.49 (19)
F1ii—Na1—F2i124.30 (5)F4—B1—F2109.39 (19)
F1iii—Na1—F2i150.83 (6)F1—B1—F2108.49 (18)
F1—Na1—F2i75.56 (5)F3—B1—F2109.39 (18)
F2—Na1—F2i75.73 (5)F4—B1—Na1129.09 (15)
F1i—Na1—F2ii75.56 (5)F1—B1—Na153.49 (10)
F1ii—Na1—F2ii56.00 (4)F3—B1—Na1122.09 (14)
F1iii—Na1—F2ii124.30 (5)F2—B1—Na155.16 (10)
F1—Na1—F2ii150.84 (6)B1—F1—Na198.66 (13)
F2—Na1—F2ii120.46 (11)B1—F2—Na196.62 (12)
F2i—Na1—F2ii75.73 (5)F5—B2—F5iv109.54 (6)
F1i—Na1—F2iii150.84 (6)F5—B2—F5v109.54 (6)
F1ii—Na1—F2iii75.56 (5)F5iv—B2—F5v109.34 (13)
F1iii—Na1—F2iii56.00 (4)F5—B2—F5vi109.34 (13)
F1—Na1—F2iii124.29 (5)F5iv—B2—F5vi109.54 (6)
F2—Na1—F2iii75.73 (5)F5v—B2—F5vi109.54 (6)
F2i—Na1—F2iii120.46 (11)N1—C1—N6126.77 (19)
F2ii—Na1—F2iii75.73 (5)N1—C1—H1116.6
F1i—Na1—B1104.13 (6)N6—C1—H1116.6
F1ii—Na1—B1162.64 (8)N1—C2—N2119.14 (18)
F1iii—Na1—B179.99 (6)N1—C2—C7120.70 (18)
F1—Na1—B127.85 (6)N2—C2—C7120.15 (18)
F2—Na1—B128.22 (6)N2—C3—N3126.03 (18)
F2i—Na1—B172.30 (6)N2—C3—H3117.0
F2ii—Na1—B1140.43 (9)N3—C3—H3117.0
F2iii—Na1—B1101.16 (7)N4—C4—N3122.01 (17)
F1i—Na1—B1i27.85 (6)N4—C4—C7122.11 (17)
F1ii—Na1—B1i104.13 (6)N3—C4—C7115.87 (17)
F1iii—Na1—B1i162.64 (8)N4—C5—N5127.57 (18)
F1—Na1—B1i79.99 (6)N4—C5—H5116.2
F2—Na1—B1i101.16 (7)N5—C5—H5116.2
F2i—Na1—B1i28.22 (6)N5—C6—N6122.77 (18)
F2ii—Na1—B1i72.30 (6)N5—C6—C7120.22 (17)
F2iii—Na1—B1i140.43 (9)N6—C6—C7117.00 (17)
B1—Na1—B1i89.292 (14)C4—C7—C6118.18 (17)
F1i—Na1—B1ii79.99 (6)C4—C7—C2121.34 (18)
F1ii—Na1—B1ii27.85 (6)C6—C7—C2120.43 (17)
F1iii—Na1—B1ii104.13 (6)C1—N1—C2115.98 (18)
F1—Na1—B1ii162.64 (8)C3—N2—C2116.39 (17)
F2—Na1—B1ii140.43 (9)C4—N3—C3120.21 (16)
F2i—Na1—B1ii101.17 (7)C4—N3—H3N119.9
F2ii—Na1—B1ii28.22 (6)C3—N3—H3N119.9
F2iii—Na1—B1ii72.30 (6)C5—N4—C4115.22 (17)
B1—Na1—B1ii167.24 (13)C6—N5—C5116.66 (17)
B1i—Na1—B1ii89.293 (14)C6—N6—C1119.05 (18)
F1i—Na1—B1iii162.64 (8)C6—N6—H6N120.5
F1ii—Na1—B1iii79.99 (6)C1—N6—H6N120.5
F1iii—Na1—B1iii27.85 (6)
F1i—Na1—B1—F4133.1 (2)F2i—Na1—F1—B179.36 (13)
F1ii—Na1—B1—F415.7 (4)F2ii—Na1—F1—B189.7 (2)
F1iii—Na1—B1—F49.4 (2)F2iii—Na1—F1—B137.90 (16)
F1—Na1—B1—F487.3 (2)B1i—Na1—F1—B1107.90 (14)
F2—Na1—B1—F487.5 (2)B1ii—Na1—F1—B1160.5 (3)
F2i—Na1—B1—F4179.8 (2)B1iii—Na1—F1—B159.75 (12)
F2ii—Na1—B1—F4142.58 (17)F4—B1—F2—Na1124.71 (16)
F2iii—Na1—B1—F461.5 (2)F1—B1—F2—Na14.42 (17)
B1i—Na1—B1—F4156.90 (17)F3—B1—F2—Na1116.18 (16)
B1ii—Na1—B1—F4119.4 (2)F1i—Na1—F2—B149.76 (16)
B1iii—Na1—B1—F435.8 (2)F1ii—Na1—F2—B1143.4 (2)
F1i—Na1—B1—F145.78 (13)F1iii—Na1—F2—B195.75 (12)
F1ii—Na1—B1—F171.6 (4)F1—Na1—F2—B12.94 (12)
F1iii—Na1—B1—F196.70 (14)F2i—Na1—F2—B179.09 (14)
F2—Na1—B1—F1174.8 (2)F2ii—Na1—F2—B1142.68 (12)
F2i—Na1—B1—F192.50 (13)F2iii—Na1—F2—B1153.72 (12)
F2ii—Na1—B1—F1130.10 (12)B1i—Na1—F2—B166.77 (10)
F2iii—Na1—B1—F1148.85 (13)B1ii—Na1—F2—B1169.42 (10)
B1i—Na1—B1—F169.58 (15)B1iii—Na1—F2—B1124.56 (13)
B1ii—Na1—B1—F1153.24 (12)N4—C4—C7—C61.5 (3)
B1iii—Na1—B1—F1123.10 (13)N3—C4—C7—C6177.86 (16)
F1i—Na1—B1—F347.1 (2)N4—C4—C7—C2178.91 (18)
F1ii—Na1—B1—F3164.5 (2)N3—C4—C7—C20.4 (3)
F1iii—Na1—B1—F3170.43 (18)N5—C6—C7—C42.3 (3)
F1—Na1—B1—F392.9 (2)N6—C6—C7—C4176.63 (17)
F2—Na1—B1—F392.3 (2)N5—C6—C7—C2179.74 (18)
F2i—Na1—B1—F30.36 (17)N6—C6—C7—C20.8 (3)
F2ii—Na1—B1—F337.2 (2)N1—C2—C7—C4179.01 (18)
F2iii—Na1—B1—F3118.28 (17)N2—C2—C7—C40.1 (3)
B1i—Na1—B1—F323.3 (2)N1—C2—C7—C61.7 (3)
B1ii—Na1—B1—F360.38 (17)N2—C2—C7—C6177.25 (18)
B1iii—Na1—B1—F3144.04 (13)N6—C1—N1—C21.5 (3)
F1i—Na1—B1—F2139.43 (13)N2—C2—N1—C1176.21 (19)
F1ii—Na1—B1—F2103.2 (3)C7—C2—N1—C12.7 (3)
F1iii—Na1—B1—F278.08 (12)N3—C3—N2—C20.6 (3)
F1—Na1—B1—F2174.8 (2)N1—C2—N2—C3179.00 (18)
F2i—Na1—B1—F292.71 (14)C7—C2—N2—C30.1 (3)
F2ii—Na1—B1—F255.11 (18)N4—C4—N3—C3178.34 (18)
F2iii—Na1—B1—F225.93 (12)C7—C4—N3—C31.0 (3)
B1i—Na1—B1—F2115.63 (11)N2—C3—N3—C41.1 (3)
B1ii—Na1—B1—F231.97 (11)N5—C5—N4—C41.6 (3)
B1iii—Na1—B1—F251.69 (14)N3—C4—N4—C5179.66 (17)
F4—B1—F1—Na1124.27 (16)C7—C4—N4—C50.3 (3)
F3—B1—F1—Na1115.40 (16)N6—C6—N5—C5177.64 (18)
F2—B1—F1—Na14.51 (18)C7—C6—N5—C51.2 (3)
F1i—Na1—F1—B1135.69 (13)N4—C5—N5—C60.8 (3)
F1ii—Na1—F1—B1151.86 (13)N5—C6—N6—C1179.05 (18)
F1iii—Na1—F1—B179.41 (15)C7—C6—N6—C12.1 (3)
F2—Na1—F1—B12.97 (12)N1—C1—N6—C61.0 (3)
Symmetry codes: (i) y, −x+1/2, z; (ii) −x+1/2, −y+1/2, z; (iii) −y+1/2, x, z; (iv) −y+1, x−1/2, −z; (v) y+1/2, −x+1, −z; (vi) −x+3/2, −y+1/2, z.
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
N3—H3N···N4vii0.882.052.930 (2)176
N6—H6N···N5viii0.882.092.866 (2)146
C1—H1···F3ix0.952.473.179 (2)132
C1—H1···F4viii0.952.552.940 (3)105
C3—H3···F2vii0.952.223.124 (2)159
C5—H5···F3iii0.952.403.097 (2)130
Symmetry codes: (vii) −x+1, −y+1, −z+2; (viii) −y+1, x+1/2, −z+1; (ix) y, −x+3/2, z; (iii) −y+1/2, x, z.
Table 1
Hydrogen-bond geometry (Å, °)
top
D—H···AD—HH···AD···AD—H···A
N3—H3N···N4i0.882.052.930 (2)176
N6—H6N···N5ii0.882.092.866 (2)146
C1—H1···F3iii0.952.473.179 (2)132
C1—H1···F4ii0.952.552.940 (3)105
C3—H3···F2i0.952.223.124 (2)159
C5—H5···F3iv0.952.403.097 (2)130
Symmetry codes: (i) −x+1, −y+1, −z+2; (ii) −y+1, x+1/2, −z+1; (iii) y, −x+3/2, z; (iv) −y+1/2, x, z.
Acknowledgements top

This work was supported by the National Natural Science Foundation of China (NSFC; grant Nos. 20644004 and 20774047), the Ministry of Science and Technology of China (MOST; grant No. 2006CB932702) and the NSF of Tianjin City (grant No. 07JCYBJC03000).

references
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