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Poly[octa­aquadi-μ-phosphato-trinickel(II)]

aFaculty of Science, ZheJiang Forestry University, Lin'An 311300, People's Republic of China, and bDepartment of Chemistry, Liaocheng University, Shandong 252059, People's Republic of China
*Correspondence e-mail: jinsw@zjfc.edu.cn

(Received 26 October 2007; accepted 14 December 2007; online 21 December 2007)

In the title compound, [Ni3(PO4)2(H2O)8]n, which was synthesized hydro­thermally, all the Ni atoms are located in slightly distorted octa­hedral coordination environments. Two phosphate groups and two Ni atoms share a centrosymmetric four-membered ring and an eight-membered ring such that the four-membered ring is inside the eight-membered ring. The eight-membered rings are connected with the other Ni atoms (lying on centres of symmetry) through phosphate anions, generating a one-dimensional chain structure. Adjacent chains are connected through hydrogen bonds, forming a three-dimensional network.

Related literature

For related literature, see: Chang et al. (2004[Chang, J., Hwang, J., Jhung, S. H., Park, S. E., Ferey, G. & Cheetham, A. K. (2004). Angew. Chem. Int. Ed. 43, 2819-2822.]); Gao et al. (1999[Gao, Q., Guillou, N., Nogues, M., Cheetham, A. K. & Ferey, G. (1999). Chem. Mater. 11, 2937-2947.]); Ke et al. (2001[Ke, Y. X., He, G. F., Li, J. M., Zhang, Y. G. & Lu, S. M. (2001). New J. Chem. 25, 1627-1630.]); Kuratieva et al. (2003[Kuratieva, N. V., Naumova, M. I., Naumov, D. Yu. & Podberezskaya, N. V. (2003). Acta Cryst. C59, i1-i3.]); Nardelli (1999[Nardelli, M. (1999). J. Appl. Cryst. 32, 563-571.]); Sanz et al. (1999[Sanz, F., Parada, C., Rojo, J. M. & Ruiz, V. C. (1999). Chem. Mater. 11, 2673-2679.]); Wang & Gao (2005a[Wang, X. L. & Gao, Q. M. (2005a). J. Inorg. Mater. 20, 699-705.],b[Wang, X. L. & Gao, Q. M. (2005b). Chin. J. Inorg. Chem. 19, 73-76.]).

[Scheme 1]

Experimental

Crystal data
  • [Ni3(PO4)2(H2O)8]

  • Mr = 510.14

  • Monoclinic, C 2/m

  • a = 9.963 (3) Å

  • b = 13.225 (4) Å

  • c = 4.6406 (14) Å

  • β = 104.730 (3)°

  • V = 591.4 (3) Å3

  • Z = 2

  • Mo Kα radiation

  • μ = 5.09 mm−1

  • T = 298 (2) K

  • 0.26 × 0.23 × 0.21 mm

Data collection
  • Bruker SMART APEX CCD Diffractometer

  • Absorption correction: multi-scan (SADABS; Sheldrick, 1996[Sheldrick, G. M. (1996). SADABS. University of Göttingen, Germany.]) Tmin = 0.351, Tmax = 0.414 (expected range = 0.291–0.343)

  • 1516 measured reflections

  • 553 independent reflections

  • 513 reflections with I > 2σ(I)

  • Rint = 0.046

Refinement
  • R[F2 > 2σ(F2)] = 0.030

  • wR(F2) = 0.080

  • S = 1.10

  • 553 reflections

  • 55 parameters

  • H-atom parameters constrained

  • Δρmax = 0.70 e Å−3

  • Δρmin = −0.58 e Å−3

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A D—H H⋯A DA D—H⋯A
O4—H1⋯O1i 0.85 1.87 2.700 (3) 164
O4—H2⋯O3ii 0.85 1.98 2.731 (3) 147
O5—H3⋯O3iii 0.85 2.00 2.761 (3) 149
O5—H4⋯O4iv 0.85 2.07 2.906 (3) 167
Symmetry codes: (i) -x+2, -y, -z+2; (ii) -x+2, -y, -z+3; (iii) -x+1, -y, -z+1; (iv) [x-{\script{1\over 2}}, -y+{\script{1\over 2}}, z-1].

Data collection: SMART (Bruker, 1997[Bruker (1997). SAINT and SMART. Bruker AXS Inc., Madison, Wisconsin, USA.]); cell refinement: SMART; data reduction: SAINT (Bruker, 1997[Bruker (1997). SAINT and SMART. Bruker AXS Inc., Madison, Wisconsin, USA.]); program(s) used to solve structure: SHELXS97 (Sheldrick, 1997a[Sheldrick, G. M. (1997a). SHELXS97 and SHELXL97. University of Göttingen, Germany.]); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997a[Sheldrick, G. M. (1997a). SHELXS97 and SHELXL97. University of Göttingen, Germany.]); molecular graphics: SHELXTL (Sheldrick, 1997b[Sheldrick, G. M. (1997b). SHELXTL. Version 5.10. Bruker AXS Inc., Madison Wisconsin, USA.]); software used to prepare material for publication: SHELXTL.

Supporting information


Comment top

In recent years, nickel phosphates as a rich class of inorganic materials, have undergone significant expansion due to their potential application (Chang et al., 2004; Gao et al., 1999; Sanz et al., 1999).

In general, current work has centered on their composite properties and the possibility of tuning their chemistry, by using effects from a wide variety of templates and of additives (Wang & Gao, 2005a; Wang & Gao, 2005b). However, investigations of new synthetic methods are still comparatively less exploited. For our interest in studying inorganic synthesis, herein we report the hydrothermal synthesis, and the crystal structure of the title compound. which is insoluble in water and in common organic solvents. It crystallizes in the monoclinic space group C2/m. The molecular structure, shown in Scheme 1, consists of three nickel ions, two phosphate ions, and eight water molecules. The PO4 anions and the Ni cations are of normal valences, i.e. their valences are -3, and +2 respectively.

The structure of nickel(II) phosphate octahydrate is shown in Fig. 1.

Both Ni atoms have a slightly distorted octahedral geometry. Ni atoms are located in two different environments, one Ni atom is surrounded by four water molecules and the remaining coordination sites are ocuppied by two oxygen atoms of two phosphates respectively. In this case, the Ni atoms are surrounded by two phosphates. The other Ni atom and its symmetry generated mate are surrounded by two water molecules, respectively. Oxygen atoms of the phosphates occupy the remaining four sites to complete the octahedral coordination. In this condition the two Ni atoms are each surrounded by three phosphates, such that two phosphates are parallel, and antiparallel with the other phosphate.

The P atoms of the phosphates do not participate in forming coordination bonds. All the P atoms make four P—O bonds. Ni—O bond distances are normal. The P—O bond distances are in the range of 1.539 (3)–1.562 (3) Å, the average value is 1.551 Å, and the O—P—O angles are in the range of 106.73 (16)–113.64 (16) degree, the average value is 110.18 degree. These geometry parameters are in good agreement with the reported results (Ke et al., 2001). The Ni—O(water) bond lengths are within the reported range (Kuratieva et al., 2003).

The phosphate anion coordinates to two metal atoms in a chelating form with two oxygen atoms. A further bridge is to the third metal ion with one of the remaining oxygen atoms.

Two phosphate and two Ni atoms share a four-membered ring, and an eight-membered ring such that the four membered ring is inside the eight membered ring. In the four membered rings, the Ni—Ni distance is 2.909 Å, while the distance between the Ni forming the four-membered rings and the Ni adjacent to the four-membered rings is 8.059 Å. The eight membered rings are connected with third symmetry-generated nickel atom linked through phosphate anions to provide an one dimensional chain structure. Adjacent chains were connected through hydrogen bonds to provide three-dimensional network topology, which is shown in Fig. 2.

Related literature top

For related literature, see: Chang et al. (2004); Gao et al. (1999); Ke et al. (2001); Kuratieva et al. (2003); Nardelli (1999); Sanz et al. (1999); Wang & Gao (2005a,b).

Experimental top

All reagents and solvents were used as received.

Blue block crystals of the title compound were synthesized hydrothermally in a 23 ml Teflon-lined autoclave by heating a mixture of 1-(4-(1H-imidazol -1-yl)butyl)-1H-imidazole (0.143 g, 0.75 mmol), nickel acetate dihydrate (0.25 g, 1 mmol), phosphoric acid (0.11 g, 1 mmol), and deionized water (6 ml) at 130 degree for 10 days. Then it was slowly cooled to room temperature, giving blue block crystals. Yield (based on Ni(Ac)2.2H2O): 0.11 g, 64.7%.

Refinement top

Hydrogen atoms bound to water molecules were located in the Fourier difference map, and their distances were fixed, and subject to an O—H = 0.85 Å with deviation of positive and negative 0.01 Å restraint. The coordinates of the water H atoms were localized by applying the HYDROGEN program (Nardelli, 1999).

Structure description top

In recent years, nickel phosphates as a rich class of inorganic materials, have undergone significant expansion due to their potential application (Chang et al., 2004; Gao et al., 1999; Sanz et al., 1999).

In general, current work has centered on their composite properties and the possibility of tuning their chemistry, by using effects from a wide variety of templates and of additives (Wang & Gao, 2005a; Wang & Gao, 2005b). However, investigations of new synthetic methods are still comparatively less exploited. For our interest in studying inorganic synthesis, herein we report the hydrothermal synthesis, and the crystal structure of the title compound. which is insoluble in water and in common organic solvents. It crystallizes in the monoclinic space group C2/m. The molecular structure, shown in Scheme 1, consists of three nickel ions, two phosphate ions, and eight water molecules. The PO4 anions and the Ni cations are of normal valences, i.e. their valences are -3, and +2 respectively.

The structure of nickel(II) phosphate octahydrate is shown in Fig. 1.

Both Ni atoms have a slightly distorted octahedral geometry. Ni atoms are located in two different environments, one Ni atom is surrounded by four water molecules and the remaining coordination sites are ocuppied by two oxygen atoms of two phosphates respectively. In this case, the Ni atoms are surrounded by two phosphates. The other Ni atom and its symmetry generated mate are surrounded by two water molecules, respectively. Oxygen atoms of the phosphates occupy the remaining four sites to complete the octahedral coordination. In this condition the two Ni atoms are each surrounded by three phosphates, such that two phosphates are parallel, and antiparallel with the other phosphate.

The P atoms of the phosphates do not participate in forming coordination bonds. All the P atoms make four P—O bonds. Ni—O bond distances are normal. The P—O bond distances are in the range of 1.539 (3)–1.562 (3) Å, the average value is 1.551 Å, and the O—P—O angles are in the range of 106.73 (16)–113.64 (16) degree, the average value is 110.18 degree. These geometry parameters are in good agreement with the reported results (Ke et al., 2001). The Ni—O(water) bond lengths are within the reported range (Kuratieva et al., 2003).

The phosphate anion coordinates to two metal atoms in a chelating form with two oxygen atoms. A further bridge is to the third metal ion with one of the remaining oxygen atoms.

Two phosphate and two Ni atoms share a four-membered ring, and an eight-membered ring such that the four membered ring is inside the eight membered ring. In the four membered rings, the Ni—Ni distance is 2.909 Å, while the distance between the Ni forming the four-membered rings and the Ni adjacent to the four-membered rings is 8.059 Å. The eight membered rings are connected with third symmetry-generated nickel atom linked through phosphate anions to provide an one dimensional chain structure. Adjacent chains were connected through hydrogen bonds to provide three-dimensional network topology, which is shown in Fig. 2.

For related literature, see: Chang et al. (2004); Gao et al. (1999); Ke et al. (2001); Kuratieva et al. (2003); Nardelli (1999); Sanz et al. (1999); Wang & Gao (2005a,b).

Computing details top

Data collection: SMART (Bruker, 1997); cell refinement: SMART (Bruker, 1997); data reduction: SAINT (Bruker, 1997); program(s) used to solve structure: SHELXS97 (Sheldrick, 1997a); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997a); molecular graphics: SHELXTL (Sheldrick, 1997b); software used to prepare material for publication: SHELXTL (Sheldrick, 1997b).

Figures top
[Figure 1] Fig. 1. The structure of the title compound, showing the atom-numbering scheme. Displacement ellipsoids are drawn at the 50% probability level. The symmetry codes are: (a) -x + 1, y, -z + 1; (b) -x + 2, y, -z + 3; (c) -x + 1, -y, -z + 1; (d) 1 - x, -y, 2 - z; (e) -x + 2, -y, -z + 3; (f) x, -y, -1 + z; (g) x, -y, z; (h) x, -y, 1 + z
[Figure 2] Fig. 2. Three dimensional network structure formed via hydrogen bonds.
Poly[octaaquadi-µ-phosphato-trinickel(II)] top
Crystal data top
[Ni3(PO4)2(H2O)8]Z = 2
Mr = 510.14F(000) = 516
Monoclinic, C2/mDx = 2.865 Mg m3
a = 9.963 (3) ÅMo Kα radiation, λ = 0.71073 Å
b = 13.225 (4) ŵ = 5.09 mm1
c = 4.6406 (14) ÅT = 298 K
β = 104.730 (3)°Block, blue
V = 591.4 (3) Å30.26 × 0.23 × 0.21 mm
Data collection top
Bruker SMART APEX CCD Diffractometer553 independent reflections
Radiation source: fine-focus sealed tube513 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.046
phi and ω scansθmax = 25.1°, θmin = 2.6°
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)
h = 711
Tmin = 0.351, Tmax = 0.414k = 1515
1516 measured reflectionsl = 55
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)] = 0.030Hydrogen site location: inferred from neighbouring sites
wR(F2) = 0.080H-atom parameters constrained
S = 1.10 w = 1/[σ2(Fo2) + (0.0471P)2]
where P = (Fo2 + 2Fc2)/3
553 reflections(Δ/σ)max < 0.001
55 parametersΔρmax = 0.70 e Å3
0 restraintsΔρmin = 0.58 e Å3
Crystal data top
[Ni3(PO4)2(H2O)8]V = 591.4 (3) Å3
Mr = 510.14Z = 2
Monoclinic, C2/mMo Kα radiation
a = 9.963 (3) ŵ = 5.09 mm1
b = 13.225 (4) ÅT = 298 K
c = 4.6406 (14) Å0.26 × 0.23 × 0.21 mm
β = 104.730 (3)°
Data collection top
Bruker SMART APEX CCD Diffractometer553 independent reflections
Absorption correction: multi-scan
(SADABS; Sheldrick, 1996)
513 reflections with I > 2σ(I)
Tmin = 0.351, Tmax = 0.414Rint = 0.046
1516 measured reflections
Refinement top
R[F2 > 2σ(F2)] = 0.0300 restraints
wR(F2) = 0.080H-atom parameters constrained
S = 1.10Δρmax = 0.70 e Å3
553 reflectionsΔρmin = 0.58 e Å3
55 parameters
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
Ni11.00000.00001.50000.0107 (3)
Ni20.50000.10997 (4)0.50000.0105 (3)
P10.68525 (10)0.00001.1161 (2)0.0094 (3)
O10.8428 (3)0.00001.1229 (6)0.0127 (6)
O20.6046 (3)0.00000.7859 (6)0.0111 (6)
O30.6554 (2)0.09749 (16)1.2706 (4)0.0127 (5)
O41.0974 (2)0.11417 (14)1.3104 (5)0.0145 (5)
H11.12080.08901.16180.022*
H21.17220.13441.42930.022*
O50.3964 (2)0.22217 (18)0.2226 (5)0.0170 (5)
H30.37530.20540.03980.025*
H40.44400.27590.23150.025*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Ni10.0091 (4)0.0104 (5)0.0115 (4)0.0000.0007 (3)0.000
Ni20.0103 (4)0.0087 (4)0.0117 (4)0.0000.0015 (3)0.000
P10.0069 (6)0.0100 (7)0.0105 (6)0.0000.0006 (4)0.000
O10.0069 (14)0.0174 (17)0.0127 (14)0.0000.0003 (11)0.000
O20.0099 (14)0.0094 (16)0.0127 (15)0.0000.0006 (11)0.000
O30.0129 (11)0.0122 (12)0.0127 (11)0.0011 (8)0.0026 (9)0.0004 (9)
O40.0119 (11)0.0171 (14)0.0143 (11)0.0009 (9)0.0028 (9)0.0005 (9)
O50.0204 (12)0.0143 (12)0.0146 (11)0.0017 (10)0.0015 (9)0.0025 (10)
Geometric parameters (Å, º) top
Ni1—O12.030 (3)Ni2—Ni2v2.9086 (14)
Ni1—O1i2.030 (3)P1—O21.539 (3)
Ni1—O4i2.105 (2)P1—O3ii1.540 (2)
Ni1—O42.105 (2)P1—O31.540 (2)
Ni1—O4ii2.105 (2)P1—O11.562 (3)
Ni1—O4iii2.105 (2)O2—Ni2v2.063 (2)
Ni2—O5iv2.061 (2)O3—Ni2vi2.097 (2)
Ni2—O52.061 (2)O4—H10.8500
Ni2—O2v2.063 (2)O4—H20.8499
Ni2—O22.063 (2)O5—H30.8499
Ni2—O3vi2.097 (2)O5—H40.8499
Ni2—O3vii2.097 (2)
O1—Ni1—O1i180.0O2v—Ni2—O3vii86.31 (10)
O1—Ni1—O4i91.36 (8)O2—Ni2—O3vii87.33 (9)
O1i—Ni1—O4i88.64 (8)O3vi—Ni2—O3vii170.97 (12)
O1—Ni1—O488.64 (8)O5iv—Ni2—Ni2v136.04 (7)
O1i—Ni1—O491.36 (8)O5—Ni2—Ni2v136.04 (7)
O4i—Ni1—O4180.0O2v—Ni2—Ni2v45.19 (6)
O1—Ni1—O4ii88.64 (8)O2—Ni2—Ni2v45.19 (6)
O1i—Ni1—O4ii91.36 (8)O3vi—Ni2—Ni2v85.49 (6)
O4i—Ni1—O4ii88.32 (12)O3vii—Ni2—Ni2v85.49 (6)
O4—Ni1—O4ii91.68 (12)O2—P1—O3ii110.48 (10)
O1—Ni1—O4iii91.36 (8)O2—P1—O3110.48 (10)
O1i—Ni1—O4iii88.64 (8)O3ii—P1—O3113.64 (16)
O4i—Ni1—O4iii91.68 (12)O2—P1—O1106.73 (16)
O4—Ni1—O4iii88.32 (12)O3ii—P1—O1107.60 (10)
O4ii—Ni1—O4iii180.0O3—P1—O1107.60 (10)
O5iv—Ni2—O587.92 (14)P1—O1—Ni1124.65 (17)
O5iv—Ni2—O2v178.67 (9)P1—O2—Ni2v133.69 (7)
O5—Ni2—O2v90.86 (9)P1—O2—Ni2133.69 (7)
O5iv—Ni2—O290.86 (9)Ni2v—O2—Ni289.63 (12)
O5—Ni2—O2178.67 (9)P1—O3—Ni2vi123.84 (12)
O2v—Ni2—O290.37 (12)Ni1—O4—H1108.0
O5iv—Ni2—O3vi93.26 (8)Ni1—O4—H2112.1
O5—Ni2—O3vi93.23 (9)H1—O4—H2105.9
O2v—Ni2—O3vi87.33 (9)Ni2—O5—H3113.1
O2—Ni2—O3vi86.31 (10)Ni2—O5—H4112.8
O5iv—Ni2—O3vii93.23 (9)H3—O5—H4105.1
O5—Ni2—O3vii93.26 (8)
O2—P1—O1—Ni1180.0O5iv—Ni2—O2—P118.9 (2)
O3ii—P1—O1—Ni161.41 (10)O5—Ni2—O2—P14 (4)
O3—P1—O1—Ni161.41 (10)O2v—Ni2—O2—P1161.6 (3)
O1i—Ni1—O1—P198 (100)O3vi—Ni2—O2—P174.3 (2)
O4i—Ni1—O1—P145.86 (6)O3vii—Ni2—O2—P1112.1 (2)
O4—Ni1—O1—P1134.14 (6)Ni2v—Ni2—O2—P1161.6 (3)
O4ii—Ni1—O1—P1134.14 (6)O5iv—Ni2—O2—Ni2v179.48 (8)
O4iii—Ni1—O1—P145.86 (6)O5—Ni2—O2—Ni2v157 (4)
O3ii—P1—O2—Ni2v140.39 (17)O2v—Ni2—O2—Ni2v0.0
O3—P1—O2—Ni2v13.8 (3)O3vi—Ni2—O2—Ni2v87.30 (9)
O1—P1—O2—Ni2v102.92 (19)O3vii—Ni2—O2—Ni2v86.29 (9)
O3ii—P1—O2—Ni213.8 (3)O2—P1—O3—Ni2vi93.29 (16)
O3—P1—O2—Ni2140.39 (17)O3ii—P1—O3—Ni2vi31.5 (2)
O1—P1—O2—Ni2102.92 (19)O1—P1—O3—Ni2vi150.56 (14)
Symmetry codes: (i) x+2, y, z+3; (ii) x, y, z; (iii) x+2, y, z+3; (iv) x+1, y, z+1; (v) x+1, y, z+1; (vi) x+1, y, z+2; (vii) x, y, z1.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
O4—H1···O1viii0.851.872.700 (3)164
O4—H2···O3i0.851.982.731 (3)147
O5—H3···O3v0.852.002.761 (3)149
O5—H4···O4ix0.852.072.906 (3)167
Symmetry codes: (i) x+2, y, z+3; (v) x+1, y, z+1; (viii) x+2, y, z+2; (ix) x1/2, y+1/2, z1.

Experimental details

Crystal data
Chemical formula[Ni3(PO4)2(H2O)8]
Mr510.14
Crystal system, space groupMonoclinic, C2/m
Temperature (K)298
a, b, c (Å)9.963 (3), 13.225 (4), 4.6406 (14)
β (°) 104.730 (3)
V3)591.4 (3)
Z2
Radiation typeMo Kα
µ (mm1)5.09
Crystal size (mm)0.26 × 0.23 × 0.21
Data collection
DiffractometerBruker SMART APEX CCD Diffractometer
Absorption correctionMulti-scan
(SADABS; Sheldrick, 1996)
Tmin, Tmax0.351, 0.414
No. of measured, independent and
observed [I > 2σ(I)] reflections
1516, 553, 513
Rint0.046
(sin θ/λ)max1)0.597
Refinement
R[F2 > 2σ(F2)], wR(F2), S 0.030, 0.080, 1.10
No. of reflections553
No. of parameters55
H-atom treatmentH-atom parameters constrained
Δρmax, Δρmin (e Å3)0.70, 0.58

Computer programs: SMART (Bruker, 1997), SAINT (Bruker, 1997), SHELXS97 (Sheldrick, 1997a), SHELXL97 (Sheldrick, 1997a), SHELXTL (Sheldrick, 1997b).

Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
O4—H1···O1i0.851.872.700 (3)163.9
O4—H2···O3ii0.851.982.731 (3)147.2
O5—H3···O3iii0.852.002.761 (3)149.4
O5—H4···O4iv0.852.072.906 (3)166.5
Symmetry codes: (i) x+2, y, z+2; (ii) x+2, y, z+3; (iii) x+1, y, z+1; (iv) x1/2, y+1/2, z1.
 

Acknowledgements

The authors thank the Zhejiang Forestry University Science Foundation for financial support.

References

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