metal-organic compounds
Bis(tetraphenylphosphonium) tetrachloridocobaltate(II)
aUnité de Recherche Chimie de l'Environnement et Moléculaire Structurale (CHEMS), Faculté des Sciences Exactes, Campus Chaabet Ersas, Université Constantine I, 25000 Constantine, Algeria
*Correspondence e-mail: Lamia_bendjeddou@yahoo.fr
The title compound, (C24H20P)2[CoCl4], was prepared under hydrothermal conditions. In the crystal, the tetraphenylphosphonium cations are linked by pairs of weak C—H⋯π interactions into supramolecular dimers; the CoII cations lie on twofold rotation axes and the tetrahedral [CoCl4]2− anions are linked with the tetraphenylphosphonium cations via weak C—H⋯Cl hydrogen bonds.
CCDC reference: 1003287
Related literature
For background and applications of compounds with supramolecular structures, see: Rowsell & Yaghi (2005); Dong et al. (2007); Wu & Lin (2007); Zhao et al. (2003); Neville et al. (2008); Huang et al. (2007). For applications of the tetraphenylphosphonium ion in supramolecular chemistry and numerous coordination polymers, see: Zacharie et al. (1985); Schlueter & Geiser (2007).
Experimental
Crystal data
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Data collection: APEX2 (Bruker, 2006); cell SAINT (Bruker, 2006); data reduction: SAINT; program(s) used to solve structure: SIR2002 (Burla et al., 2003); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEP-3 for Windows (Farrugia, 2012); software used to prepare material for publication: WinGX (Farrugia, 2012), Mercury (Macrae et al., 2006) and POVRay (Persistence of Vision Team, 2004).
Supporting information
CCDC reference: 1003287
10.1107/S1600536814011210/xu5791sup1.cif
contains datablocks global, I. DOI:Structure factors: contains datablock I. DOI: 10.1107/S1600536814011210/xu5791Isup2.hkl
A mixture of CoCl2 (2.50 g, 10 mmol), tetraphenylphosphonium chloride hydrate (3.92 g,10 mmol) was dissolved in a 20 ml EtOH/H2O(v/v,1:2). The mixture was then sealed in a 25 ml stainless steel reactor and heated to 433 K for 3 days. Then the reactant mixture was cooled to room temperature at the rate of 5 degrees per hour. Evaporation of the resulting solution for a few days afforded pink crystals of title compound.
The aromatic H atoms were placed at calculated positions with C—H = 0.93, and refined in riding mode with Uiso(H) = 1.2Ueq(C).
Data collection: APEX2 (Bruker, 2006); cell
SAINT (Bruker, 2006); data reduction: SAINT (Bruker, 2006); program(s) used to solve structure: SIR2002 (Burla et al., 2003); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEP-3 for Windows (Farrugia, 2012); software used to prepare material for publication: WinGX (Farrugia, 2012), Mercury (Macrae et al., 2006) and POVRay (Persistence of Vision Team, 2004).Fig. 1. The asymmetric unit of the title structure with the atom numbering scheme. Displacement ellipsoids are drawn at the 50% probability level. [Symmetry code: (*): -x, y, 1/2 - z]. | |
Fig. 2. Part of the layered packing in the crystal viewed down the b axis. | |
Fig. 3. A view along the c axis of the crystal structure of the title compound, showing C—H···Cl hydrogen-bonds. |
(C24H20P)2[CoCl4] | F(000) = 1812 |
Mr = 879.46 | Dx = 1.378 Mg m−3 |
Monoclinic, C2/c | Mo Kα radiation, λ = 0.71073 Å |
Hall symbol: -C 2yc | Cell parameters from 1536 reflections |
a = 10.9154 (4) Å | θ = 3.2–25.1° |
b = 19.2514 (6) Å | µ = 0.77 mm−1 |
c = 20.1826 (7) Å | T = 293 K |
β = 91.008 (2)° | Prism, pink |
V = 4240.4 (3) Å3 | 0.2 × 0.1 × 0.08 mm |
Z = 4 |
Bruker APEXII diffractometer | 3180 reflections with I > 2σ(I) |
Radiation source: fine-focus sealed tube | Rint = 0.028 |
Graphite monochromator | θmax = 25.1°, θmin = 2.0° |
ϕ scans | h = −12→12 |
12226 measured reflections | k = −22→22 |
3748 independent reflections | l = −24→22 |
Refinement on F2 | 0 restraints |
Least-squares matrix: full | H-atom parameters constrained |
R[F2 > 2σ(F2)] = 0.030 | w = 1/[σ2(Fo2) + (0.0401P)2 + 6.1572P] where P = (Fo2 + 2Fc2)/3 |
wR(F2) = 0.084 | (Δ/σ)max = 0.001 |
S = 1.04 | Δρmax = 1.51 e Å−3 |
3748 reflections | Δρmin = −0.22 e Å−3 |
249 parameters |
(C24H20P)2[CoCl4] | V = 4240.4 (3) Å3 |
Mr = 879.46 | Z = 4 |
Monoclinic, C2/c | Mo Kα radiation |
a = 10.9154 (4) Å | µ = 0.77 mm−1 |
b = 19.2514 (6) Å | T = 293 K |
c = 20.1826 (7) Å | 0.2 × 0.1 × 0.08 mm |
β = 91.008 (2)° |
Bruker APEXII diffractometer | 3180 reflections with I > 2σ(I) |
12226 measured reflections | Rint = 0.028 |
3748 independent reflections |
R[F2 > 2σ(F2)] = 0.030 | 0 restraints |
wR(F2) = 0.084 | H-atom parameters constrained |
S = 1.04 | Δρmax = 1.51 e Å−3 |
3748 reflections | Δρmin = −0.22 e Å−3 |
249 parameters |
Geometry. Bond distances, angles etc. have been calculated using the rounded fractional coordinates. All e.s.d.'s are estimated from the variances of the (full) variance-covariance matrix. The cell e.s.d.'s are taken into account in the estimation of distances, angles and torsion angles |
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. |
x | y | z | Uiso*/Ueq | ||
P1 | 0.41273 (5) | 0.36748 (3) | 0.07765 (3) | 0.0200 (2) | |
C1 | 0.3837 (2) | 0.29974 (11) | 0.13639 (11) | 0.0233 (7) | |
C2 | 0.3155 (2) | 0.31597 (12) | 0.19182 (11) | 0.0308 (7) | |
C3 | 0.2894 (3) | 0.26481 (13) | 0.23755 (13) | 0.0376 (8) | |
C4 | 0.3336 (2) | 0.19823 (12) | 0.22924 (12) | 0.0355 (8) | |
C5 | 0.4026 (2) | 0.18228 (12) | 0.17495 (12) | 0.0335 (8) | |
C6 | 0.4272 (2) | 0.23227 (11) | 0.12800 (12) | 0.0290 (7) | |
C7 | 0.5226 (2) | 0.33873 (11) | 0.01785 (11) | 0.0238 (7) | |
C8 | 0.6408 (2) | 0.36542 (13) | 0.01635 (12) | 0.0291 (7) | |
C9 | 0.7214 (2) | 0.34138 (14) | −0.03052 (13) | 0.0375 (8) | |
C10 | 0.6855 (2) | 0.29065 (13) | −0.07487 (12) | 0.0370 (8) | |
C11 | 0.5686 (3) | 0.26452 (12) | −0.07432 (12) | 0.0363 (8) | |
C12 | 0.4866 (2) | 0.28862 (12) | −0.02847 (12) | 0.0314 (7) | |
C13 | 0.27533 (19) | 0.39129 (10) | 0.03328 (10) | 0.0199 (6) | |
C14 | 0.2858 (2) | 0.42599 (12) | −0.02717 (11) | 0.0272 (7) | |
C15 | 0.1824 (2) | 0.44748 (12) | −0.06129 (11) | 0.0297 (7) | |
C16 | 0.0676 (2) | 0.43485 (12) | −0.03560 (11) | 0.0269 (7) | |
C17 | 0.0570 (2) | 0.40171 (12) | 0.02446 (11) | 0.0271 (7) | |
C18 | 0.1604 (2) | 0.38005 (11) | 0.05921 (11) | 0.0232 (7) | |
C19 | 0.47179 (19) | 0.44088 (11) | 0.12202 (10) | 0.0205 (6) | |
C20 | 0.4177 (2) | 0.50598 (11) | 0.11488 (11) | 0.0226 (6) | |
C21 | 0.4699 (2) | 0.56215 (12) | 0.14713 (11) | 0.0289 (7) | |
C22 | 0.5746 (2) | 0.55408 (13) | 0.18543 (11) | 0.0304 (7) | |
C23 | 0.6265 (2) | 0.48956 (14) | 0.19401 (11) | 0.0315 (8) | |
C24 | 0.5750 (2) | 0.43278 (13) | 0.16272 (11) | 0.0280 (7) | |
Co1 | 0.00000 | 0.45721 (2) | 0.25000 | 0.0202 (1) | |
Cl1 | 0.10790 (5) | 0.38813 (3) | 0.32242 (3) | 0.0338 (2) | |
Cl2 | 0.12506 (5) | 0.52482 (3) | 0.18779 (3) | 0.0267 (2) | |
H2 | 0.28760 | 0.36110 | 0.19810 | 0.0370* | |
H3 | 0.24190 | 0.27530 | 0.27400 | 0.0450* | |
H4 | 0.31670 | 0.16410 | 0.26040 | 0.0430* | |
H5 | 0.43290 | 0.13750 | 0.16980 | 0.0400* | |
H6 | 0.47250 | 0.22100 | 0.09090 | 0.0350* | |
H8 | 0.66550 | 0.39920 | 0.04670 | 0.0350* | |
H9 | 0.80030 | 0.35960 | −0.03210 | 0.0450* | |
H10 | 0.74100 | 0.27400 | −0.10550 | 0.0440* | |
H11 | 0.54480 | 0.23060 | −0.10480 | 0.0440* | |
H12 | 0.40700 | 0.27140 | −0.02840 | 0.0380* | |
H14 | 0.36280 | 0.43460 | −0.04440 | 0.0330* | |
H15 | 0.18950 | 0.47050 | −0.10160 | 0.0360* | |
H16 | −0.00230 | 0.44880 | −0.05900 | 0.0320* | |
H17 | −0.02010 | 0.39380 | 0.04180 | 0.0330* | |
H18 | 0.15280 | 0.35800 | 0.09990 | 0.0280* | |
H20 | 0.34740 | 0.51150 | 0.08870 | 0.0270* | |
H21 | 0.43410 | 0.60580 | 0.14300 | 0.0350* | |
H22 | 0.61050 | 0.59260 | 0.20570 | 0.0360* | |
H23 | 0.69600 | 0.48430 | 0.22080 | 0.0380* | |
H24 | 0.60930 | 0.38900 | 0.16880 | 0.0340* |
U11 | U22 | U33 | U12 | U13 | U23 | |
P1 | 0.0206 (3) | 0.0189 (3) | 0.0205 (3) | 0.0001 (2) | 0.0001 (2) | −0.0013 (2) |
C1 | 0.0266 (12) | 0.0210 (11) | 0.0222 (11) | −0.0004 (9) | −0.0037 (9) | 0.0011 (9) |
C2 | 0.0441 (14) | 0.0228 (12) | 0.0255 (12) | 0.0057 (11) | 0.0025 (11) | 0.0007 (9) |
C3 | 0.0528 (16) | 0.0346 (14) | 0.0256 (13) | 0.0020 (12) | 0.0065 (12) | 0.0037 (11) |
C4 | 0.0484 (15) | 0.0275 (13) | 0.0303 (13) | −0.0040 (11) | −0.0064 (12) | 0.0096 (10) |
C5 | 0.0454 (15) | 0.0187 (11) | 0.0361 (14) | 0.0038 (11) | −0.0070 (12) | 0.0010 (10) |
C6 | 0.0324 (13) | 0.0255 (12) | 0.0291 (12) | 0.0037 (10) | −0.0013 (10) | −0.0025 (10) |
C7 | 0.0272 (12) | 0.0207 (11) | 0.0235 (11) | 0.0042 (9) | 0.0018 (9) | 0.0013 (9) |
C8 | 0.0240 (12) | 0.0355 (13) | 0.0279 (12) | 0.0022 (10) | −0.0009 (10) | 0.0005 (10) |
C9 | 0.0236 (12) | 0.0548 (16) | 0.0343 (14) | 0.0069 (12) | 0.0030 (11) | 0.0031 (12) |
C10 | 0.0404 (15) | 0.0422 (15) | 0.0286 (13) | 0.0170 (12) | 0.0101 (11) | 0.0036 (11) |
C11 | 0.0554 (17) | 0.0254 (12) | 0.0284 (13) | 0.0060 (12) | 0.0080 (12) | −0.0034 (10) |
C12 | 0.0372 (13) | 0.0252 (12) | 0.0319 (13) | −0.0035 (10) | 0.0056 (11) | −0.0036 (10) |
C13 | 0.0212 (11) | 0.0186 (10) | 0.0199 (11) | −0.0011 (9) | −0.0018 (9) | −0.0027 (8) |
C14 | 0.0239 (12) | 0.0340 (13) | 0.0237 (12) | −0.0033 (10) | 0.0031 (9) | 0.0040 (10) |
C15 | 0.0342 (13) | 0.0345 (13) | 0.0203 (11) | −0.0004 (11) | 0.0002 (10) | 0.0061 (10) |
C16 | 0.0260 (12) | 0.0286 (12) | 0.0258 (12) | 0.0038 (10) | −0.0048 (10) | −0.0033 (10) |
C17 | 0.0216 (11) | 0.0317 (12) | 0.0282 (12) | −0.0009 (10) | 0.0035 (9) | −0.0010 (10) |
C18 | 0.0256 (12) | 0.0248 (11) | 0.0192 (11) | −0.0028 (9) | 0.0024 (9) | 0.0018 (9) |
C19 | 0.0200 (11) | 0.0219 (11) | 0.0198 (11) | −0.0015 (9) | 0.0036 (9) | −0.0025 (9) |
C20 | 0.0238 (11) | 0.0233 (11) | 0.0208 (11) | −0.0014 (9) | 0.0035 (9) | 0.0007 (9) |
C21 | 0.0395 (14) | 0.0207 (11) | 0.0268 (12) | −0.0033 (10) | 0.0079 (11) | −0.0002 (9) |
C22 | 0.0354 (13) | 0.0335 (13) | 0.0225 (12) | −0.0165 (11) | 0.0072 (10) | −0.0080 (10) |
C23 | 0.0228 (12) | 0.0483 (15) | 0.0235 (12) | −0.0042 (11) | −0.0001 (10) | −0.0070 (11) |
C24 | 0.0253 (12) | 0.0324 (13) | 0.0262 (12) | 0.0054 (10) | −0.0016 (10) | −0.0036 (10) |
Co1 | 0.0188 (2) | 0.0189 (2) | 0.0231 (2) | 0.0000 | 0.0029 (2) | 0.0000 |
Cl1 | 0.0315 (3) | 0.0335 (3) | 0.0367 (3) | 0.0084 (3) | 0.0074 (3) | 0.0151 (3) |
Cl2 | 0.0234 (3) | 0.0296 (3) | 0.0270 (3) | −0.0035 (2) | 0.0015 (2) | 0.0071 (2) |
Co1—Cl2 | 2.2791 (6) | C19—C24 | 1.391 (3) |
Co1—Cl1i | 2.2873 (6) | C19—C20 | 1.392 (3) |
Co1—Cl1 | 2.2873 (6) | C20—C21 | 1.380 (3) |
Co1—Cl2i | 2.2791 (6) | C21—C22 | 1.377 (3) |
P1—C1 | 1.794 (2) | C22—C23 | 1.375 (4) |
P1—C7 | 1.803 (2) | C23—C24 | 1.377 (3) |
P1—C13 | 1.793 (2) | C2—H2 | 0.9301 |
P1—C19 | 1.787 (2) | C3—H3 | 0.9296 |
C1—C6 | 1.394 (3) | C4—H4 | 0.9302 |
C1—C2 | 1.390 (3) | C5—H5 | 0.9298 |
C2—C3 | 1.383 (3) | C6—H6 | 0.9302 |
C3—C4 | 1.381 (3) | C8—H8 | 0.9301 |
C4—C5 | 1.375 (3) | C9—H9 | 0.9310 |
C5—C6 | 1.380 (3) | C10—H10 | 0.9301 |
C7—C8 | 1.390 (3) | C11—H11 | 0.9310 |
C7—C12 | 1.395 (3) | C12—H12 | 0.9300 |
C8—C9 | 1.383 (3) | C14—H14 | 0.9301 |
C9—C10 | 1.377 (4) | C15—H15 | 0.9309 |
C10—C11 | 1.372 (4) | C16—H16 | 0.9299 |
C11—C12 | 1.379 (4) | C17—H17 | 0.9297 |
C13—C14 | 1.397 (3) | C18—H18 | 0.9295 |
C13—C18 | 1.385 (3) | C20—H20 | 0.9301 |
C14—C15 | 1.375 (3) | C21—H21 | 0.9299 |
C15—C16 | 1.386 (3) | C22—H22 | 0.9302 |
C16—C17 | 1.376 (3) | C23—H23 | 0.9293 |
C17—C18 | 1.383 (3) | C24—H24 | 0.9295 |
Cl1—Co1—Cl2 | 112.17 (2) | C22—C23—C24 | 119.7 (2) |
Cl1—Co1—Cl1i | 108.90 (3) | C19—C24—C23 | 120.1 (2) |
Cl1—Co1—Cl2i | 106.66 (2) | C3—C2—H2 | 120.17 |
Cl1i—Co1—Cl2 | 106.66 (2) | C1—C2—H2 | 120.06 |
Cl2—Co1—Cl2i | 110.35 (3) | C2—C3—H3 | 119.85 |
Cl1i—Co1—Cl2i | 112.17 (2) | C4—C3—H3 | 119.91 |
C13—P1—C19 | 109.84 (10) | C3—C4—H4 | 119.98 |
C1—P1—C19 | 108.05 (10) | C5—C4—H4 | 119.92 |
C1—P1—C7 | 110.27 (10) | C6—C5—H5 | 119.72 |
C1—P1—C13 | 111.10 (10) | C4—C5—H5 | 119.80 |
C7—P1—C13 | 107.73 (10) | C1—C6—H6 | 120.13 |
C7—P1—C19 | 109.86 (10) | C5—C6—H6 | 120.17 |
P1—C1—C6 | 122.16 (17) | C7—C8—H8 | 120.18 |
P1—C1—C2 | 118.14 (17) | C9—C8—H8 | 120.26 |
C2—C1—C6 | 119.7 (2) | C10—C9—H9 | 119.85 |
C1—C2—C3 | 119.8 (2) | C8—C9—H9 | 119.83 |
C2—C3—C4 | 120.2 (2) | C9—C10—H10 | 119.68 |
C3—C4—C5 | 120.1 (2) | C11—C10—H10 | 119.72 |
C4—C5—C6 | 120.5 (2) | C12—C11—H11 | 120.17 |
C1—C6—C5 | 119.7 (2) | C10—C11—H11 | 120.04 |
P1—C7—C8 | 121.99 (17) | C7—C12—H12 | 119.84 |
P1—C7—C12 | 118.55 (17) | C11—C12—H12 | 119.90 |
C8—C7—C12 | 119.5 (2) | C13—C14—H14 | 120.00 |
C7—C8—C9 | 119.6 (2) | C15—C14—H14 | 119.88 |
C8—C9—C10 | 120.3 (2) | C16—C15—H15 | 120.01 |
C9—C10—C11 | 120.6 (2) | C14—C15—H15 | 120.04 |
C10—C11—C12 | 119.8 (2) | C15—C16—H16 | 119.90 |
C7—C12—C11 | 120.3 (2) | C17—C16—H16 | 120.01 |
C14—C13—C18 | 119.55 (19) | C16—C17—H17 | 119.87 |
P1—C13—C18 | 121.79 (16) | C18—C17—H17 | 119.73 |
P1—C13—C14 | 118.54 (16) | C17—C18—H18 | 120.08 |
C13—C14—C15 | 120.1 (2) | C13—C18—H18 | 120.05 |
C14—C15—C16 | 120.0 (2) | C19—C20—H20 | 120.40 |
C15—C16—C17 | 120.1 (2) | C21—C20—H20 | 120.54 |
C16—C17—C18 | 120.4 (2) | C20—C21—H21 | 119.77 |
C13—C18—C17 | 119.9 (2) | C22—C21—H21 | 119.71 |
P1—C19—C24 | 119.24 (17) | C23—C22—H22 | 119.70 |
P1—C19—C20 | 120.76 (16) | C21—C22—H22 | 119.73 |
C20—C19—C24 | 120.0 (2) | C22—C23—H23 | 120.14 |
C19—C20—C21 | 119.1 (2) | C24—C23—H23 | 120.18 |
C20—C21—C22 | 120.5 (2) | C23—C24—H24 | 119.93 |
C21—C22—C23 | 120.6 (2) | C19—C24—H24 | 119.98 |
Symmetry code: (i) −x, y, −z+1/2. |
Cg2 and Cg4 are the centroids of the C19–C24 and C7-C12 benzene rings, respectively. |
D—H···A | D—H | H···A | D···A | D—H···A |
C3—H3···Cl1 | 0.93 | 2.80 | 3.552 (3) | 138 |
C11—H11···Cl1ii | 0.93 | 2.81 | 3.633 (2) | 148 |
C23—H23···Cl2iii | 0.93 | 2.77 | 3.644 (2) | 156 |
C14—H14···Cg4iv | 0.93 | 2.88 | 3.650 (2) | 141 |
C21—H21···Cg2iv | 0.93 | 2.79 | 3.446 (2) | 129 |
Symmetry codes: (ii) x+1/2, −y+1/2, z−1/2; (iii) −x+1, y, −z+1/2; (iv) −x+1, −y+1, −z. |
Cg2 and Cg4 are the centroids of the C19–C24 and C7-C12 benzene rings, respectively. |
D—H···A | D—H | H···A | D···A | D—H···A |
C3—H3···Cl1 | 0.93 | 2.80 | 3.552 (3) | 138 |
C11—H11···Cl1i | 0.93 | 2.81 | 3.633 (2) | 148 |
C23—H23···Cl2ii | 0.93 | 2.77 | 3.644 (2) | 156 |
C14—H14···Cg4iii | 0.93 | 2.88 | 3.650 (2) | 141 |
C21—H21···Cg2iii | 0.93 | 2.79 | 3.446 (2) | 129 |
Symmetry codes: (i) x+1/2, −y+1/2, z−1/2; (ii) −x+1, y, −z+1/2; (iii) −x+1, −y+1, −z. |
Acknowledgements
This work was supported by the Unité de Recherche de Chimie de l'Environnement et Moléculaire Structurale (CHEMS), Université de Constantine 1, Algeria. Thanks are due to MESRS and ATRST (Ministère de l'Enseignement Supérieur et de la Recherche Scientifique et l'Agence Thématique de Recherche en Sciences et Technologie, Algérie) for financial support via the PNR program.
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Research on supramolecular compounds has become popular because of their potential applications in areas such as gas storage (Rowsell & Yaghi, 2005), selective absorption (Dong et al., 2007), catalysis (Wu & Lin, 2007), magnetics (Zhao et al., 2003; Neville et al., 2008) and optics (Huang et al., 2007). P-Ligands are important structural motifs in organic syntheses, coordination chemistry and also in various catalytically active compounds, the tetraphenylphosphonium ion have been widely used in supramolecular chemistry and numerous coordination polymers with versatile structures and potential properties have been reported (Zacharie et al., 1985; Schlueter & Geiser, 2007). Thus, we report here the synthesis of title compound [L2.CoCl4], were L is tetraphenylphosphonium and its crystal structure.
The asymmetric unit of (I) and atomic numbering are illustrated in Fig. 1. The (I) contains tetraphenylphosphonium cations linked by weak C—H···π supramolecular interactions into dimmers. The CoII ion lies on a twofold axis and has a distorted tetrahedral coordination. The [CoCl4]-2 anions are linked with the cation via weak C—H···Cl hydrogen bonds (Fig. 2, 3).
The bond lengths for coordination CoII sphere is ranging from 2.2791 (6) to 2.2873 (6) Å for Co—Cl distances (Table 1). The crystal packing in the title structure can be described by altering CoCl4 tetrahedral of complex along the a axis at b = 1/4 and 3/4 (Fig. 2).