trans-5,6-Diphenyl­perhydro­pyran-2,4-dione

Souza, L.C. de; Imbroisi, D. de O.; De Simone, C.A.; Pereira, M.A.; Malta, V.R.S.

doi:10.1107/S1600536809000087

organic compounds

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ISSN: 2056-9890

Volume 65| Part 2| February 2009| Page o250

doi:10.1107/S1600536809000087

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trans-5,6-Diphenylperhydropyran-2,4-dione

Laura C. de Souza,^a Dennis de O. Imbroisi,^a Carlos A. De Simone,^a ^* Mariano A. Pereira ^a and Valéria R. S. Malta ^a

^aInstituto de Química e Biotecnologia, Universidade Federal de Alagoas, 57072-970 Maceió, AL, Brazil
^*Correspondence e-mail: cas@qui.ufal.br

(Received 21 November 2008; accepted 2 January 2009; online 8 January 2009)

In the title compound, C₁₇H₁₄O₃, the pyran ring adopts a boat conformation and the dihedral angle between the aromatic ring planes is 59.1 (1)°. In the crystal structure intermolecular C—H⋯O hydrogen bonds and C—H⋯π interactions link the molecules.

Related literature

For general background, see: Yen & Chen (1995 ); Soler-Rivas et al. (2000 ). For related structures and biological activity, see: Brand-William et al. (1995 ); Sánchez-Moreno et al. (1998 ); Souza et al. (2004 ). For the synthesis, see: Souza (2008 ). For geometric analysis, see: Cremer & Pople (1975 ). For bond-length data, see: Allen et al. (1987 ).

Experimental

Crystal data

C₁₇H₁₄O₃
M_r = 266.28
Monoclinic, P 2₁ /c
a = 8.9940 (2) Å
b = 8.2310 (4) Å
c = 18.9040 (8) Å
β = 101.412 (2)°
V = 1371.79 (9) Å³
Z = 4
Mo Kα radiation
μ = 0.09 mm⁻¹
T = 295 K
0.30 × 0.30 × 0.18 mm

Data collection

Nonius KappaCCD diffractometer
Absorption correction: none
5298 measured reflections
3113 independent reflections
2459 reflections with I > 2σ(I)
R_int = 0.017

Refinement

R[F² > 2σ(F²)] = 0.050
wR(F²) = 0.131
S = 1.05
3113 reflections
181 parameters
H-atom parameters constrained
Δρ_max = 0.19 e Å⁻³
Δρ_min = −0.20 e Å⁻³

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
C6—H6⋯O2ⁱ	0.98	2.44	3.380 (2)	161
C17—H17⋯O3ⁱⁱ	0.93	2.46	3.351 (3)	160
C3—H3B⋯Cg1ⁱ	0.97	2.97	3.681 (2)	131
C5—H5⋯Cg2ⁱⁱⁱ	0.98	2.96	3.830 (2)	149
Symmetry codes: (i) $[-x+2, y-{\script{1\over 2}}, -z+{\script{3\over 2}}]$ ; (ii) $[x, -y-{\script{1\over 2}}, z+{\script{1\over 2}}]$ ; (iii) $[-x+1, y+{\script{1\over 2}}, -z+{\script{3\over 2}}]$ . Cg1 and Cg2 are the centroids of the C7–C12 and C13–C18 rings, respectively.

Data collection: COLLECT (Nonius, 2000 ); cell refinement: SCALEPACK (Otwinowski & Minor, 1997 ); data reduction: DENZO (Otwinowski & Minor, 1997) and SCALEPACK; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008 ); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEP-3 for Windows (Farrugia, 1997 ); software used to prepare material for publication: WinGX (Farrugia, 1999 ).

Supporting information

Crystal structure: contains datablocks I, global. DOI: 10.1107/S1600536809000087/bq2110sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536809000087/bq2110Isup2.hkl

checkCIF report

Comment top

The free radicals generated in bioorganic redoxi processes induce oxidative damage in various components of the cells (e.g., lipids, proteins and nucleic acids) and their play a significant role in the development of life-limiting chronic diseases such as cancer, hypertension, arteriosclerosis, rheumatism, cataracts and other (Yen & Chen, 1995; Soler-Rivas et al.,2000). The dihydropyran-2,4-diones exhibit structural features present bin many biologically active natural products possessing important pharmacological activities (Brand-William et al.,1995; Sánchez-Moreno et al.,1998). As part of our continuing studies aimed at ascertaining the biological activity of this class, the title compound was synthetized (Souza, 2008) and its antioxidant activity analyzed in vitro, by measuring the decrease in absorbance at 515 nm that occurred when the 2,2-diphenyl-1-picryl-hydrazyl radical (DPPH) was reduced by the antioxidant. The spectrophotometric assay was used to determine the radical scavenging activity (Souza et al.,2004).

The ORTEP-3 (Farrugia, 1997) representation of the title compound (5,6-DPDP) is showing in (Fig. 1). Bond lengths and angles are in good agreement with the expected values reported in the literature (Allen et al., 1987). The pirane ring adopts a boat conformation and the calculated puckering parameters are: q₂ = 0.624 (1) Å, q₃ = 0.121 (1) Å, Q_T = 0.636 (1) Å, θ = 79.0 (1)° and ϕ = 287.5 (1)° (Cremer & Pople, 1975). The dihedral angle between planes passing through atoms C7—C12 and C13—C18 of the aromatic rings is 59.1 (1) °. In the crystal packing, molecules interact through two intermolecular C–H···O hydrogen bonds and two C—H···.π interactions, Fig. 2 and Table 1.

Related literature top

For general background, see: Yen & Chen (1995); Soler-Rivas et al. (2000). For related structures and biological activity, see: Brand-William et al. (1995); Sánchez-Moreno et al. (1998); Souza et al. (2004). For the synthesis, see: Souza (2008). For geometric analysis, see: Cremer & Pople (1975). For bond-length data, see: Allen et al. (1987).

Experimental top

The trans-5, 6-diphenyltetradehydropyran-2,4-dione has showed similar antioxidant activity at the positive control, the synthetic antioxidant BHT (2,6-di-tert-butyl-4-methylphenol) used as food conserving. The reduction percentage after 60 minutes to a solution of 20 nM of sample were 88% to 5,6-DPDP and 82% to BHT (Souza, 2008).The 5,6-DPDP was synthesized in one pot by preparation of the dianion of the ethyl 3-oxo-4-phenylbutanoate (NaH, n-butillithium, THF, -10° C), and alkylation reaction with benzaldehyde followed by ester hydrolysis (NaOH, H₂O, 12 h, RT) and lactonization in acidic medium (HCl, H₂O, 2 h, 0°C). The compound was purified by silica gel chromatography and the crystals for x-ray diffraction studies were grown by slow evaporation from a CHCl₃ solution.

Computing details top

Data collection: COLLECT (Nonius, 2000); cell refinement: SCALEPACK (Otwinowski & Minor, 1997); data reduction: DENZO and SCALEPACK (Otwinowski & Minor, 1997); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: ORTEP-3 for Windows (Farrugia, 1997); software used to prepare material for publication: WinGX (Farrugia, 1999).

Figures top

	Fig. 1. Projection of C₁₇H₁₄O₃, showing the atom labelling with 50% probability displacement.
	Fig. 2. Hydrogen interactions.

trans-5,6-Diphenylperhydropyran-2,4-dione top

Crystal data top

C₁₇H₁₄O₃	F(000) = 560
M_r = 266.28	D_x = 1.289 Mg m⁻³
Monoclinic, P2₁/c	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P2ybc	Cell parameters from 2880 reflections
a = 8.9940 (2) Å	θ = 1.0–27.5°
b = 8.2310 (4) Å	µ = 0.09 mm⁻¹
c = 18.9040 (8) Å	T = 295 K
β = 101.412 (2)°	Prism, yellow
V = 1371.79 (9) Å³	0.30 × 0.30 × 0.18 mm
Z = 4

Data collection top

Nonius KappaCCD diffractometer	2459 reflections with I > 2σ(I)
Radiation source: Enraf Nonius FR590	R_int = 0.017
Horizonally mounted graphite crystal monochromator	θ_max = 27.5°, θ_min = 2.3°
Detector resolution: 9 pixels mm^-1	h = −11→11
CCD rotation images, thick slices scans	k = −9→10
5298 measured reflections	l = −24→24
3113 independent reflections

Refinement top

Refinement on F²	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F² > 2σ(F²)] = 0.050	Hydrogen site location: inferred from neighbouring sites
wR(F²) = 0.131	H-atom parameters constrained
S = 1.05	w = 1/[σ²(F_o²) + (0.0508P)² + 0.3831P] where P = (F_o² + 2F_c²)/3
3113 reflections	(Δ/σ)_max < 0.001
181 parameters	Δρ_max = 0.19 e Å⁻³
0 restraints	Δρ_min = −0.20 e Å⁻³

Crystal data top

C₁₇H₁₄O₃	V = 1371.79 (9) Å³
M_r = 266.28	Z = 4
Monoclinic, P2₁/c	Mo Kα radiation
a = 8.9940 (2) Å	µ = 0.09 mm⁻¹
b = 8.2310 (4) Å	T = 295 K
c = 18.9040 (8) Å	0.30 × 0.30 × 0.18 mm
β = 101.412 (2)°

Data collection top

Nonius KappaCCD diffractometer	2459 reflections with I > 2σ(I)
5298 measured reflections	R_int = 0.017
3113 independent reflections

Refinement top

R[F² > 2σ(F²)] = 0.050	0 restraints
wR(F²) = 0.131	H-atom parameters constrained
S = 1.05	Δρ_max = 0.19 e Å⁻³
3113 reflections	Δρ_min = −0.20 e Å⁻³
181 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

	x	y	z	U_iso*/U_eq
C2	0.69397 (19)	−0.0595 (2)	0.63794 (8)	0.0521 (4)
C3	0.84834 (19)	0.0134 (2)	0.65634 (8)	0.0560 (4)
H3A	0.8717	0.0620	0.6131	0.067*
H3B	0.9211	−0.0730	0.6712	0.067*
C4	0.86937 (16)	0.1403 (2)	0.71484 (8)	0.0476 (4)
C5	0.76480 (14)	0.12492 (17)	0.76852 (7)	0.0383 (3)
H5	0.6771	0.1953	0.7515	0.046*
C6	0.70604 (15)	−0.05049 (17)	0.76642 (7)	0.0385 (3)
H6	0.7925	−0.1246	0.7790	0.046*
C7	0.83633 (14)	0.17992 (16)	0.84390 (7)	0.0385 (3)
C8	0.76586 (17)	0.2946 (2)	0.87924 (9)	0.0520 (4)
H8	0.6745	0.3402	0.8562	0.062*
C9	0.8303 (2)	0.3421 (2)	0.94867 (10)	0.0652 (5)
H9	0.7817	0.4189	0.9722	0.078*
C10	0.9657 (2)	0.2763 (2)	0.98306 (9)	0.0631 (5)
H10	1.0094	0.3097	1.0295	0.076*
C11	1.03660 (18)	0.1616 (2)	0.94891 (9)	0.0565 (4)
H11	1.1281	0.1166	0.9723	0.068*
C12	0.97190 (16)	0.11267 (19)	0.87959 (8)	0.0477 (4)
H12	1.0198	0.0340	0.8567	0.057*
C13	0.59831 (15)	−0.07985 (17)	0.81644 (7)	0.0412 (3)
C14	0.46566 (19)	0.0086 (2)	0.80971 (12)	0.0644 (5)
H14	0.4401	0.0834	0.7724	0.077*
C15	0.3711 (3)	−0.0137 (3)	0.85803 (16)	0.0935 (8)
H15	0.2827	0.0473	0.8535	0.112*
C16	0.4061 (3)	−0.1242 (4)	0.91217 (15)	0.1019 (10)
H16	0.3420	−0.1378	0.9447	0.122*
C17	0.5352 (3)	−0.2153 (4)	0.91893 (11)	0.0967 (9)
H17	0.5583	−0.2915	0.9558	0.116*
C18	0.6323 (2)	−0.1938 (2)	0.87041 (9)	0.0643 (5)
H18	0.7197	−0.2564	0.8746	0.077*
O1	0.62369 (12)	−0.08786 (13)	0.69346 (5)	0.0487 (3)
O2	0.96257 (15)	0.24604 (19)	0.71798 (7)	0.0767 (4)
O3	0.63076 (17)	−0.0935 (2)	0.57795 (6)	0.0792 (4)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
C2	0.0651 (10)	0.0484 (9)	0.0431 (8)	0.0015 (7)	0.0113 (7)	−0.0051 (7)
C3	0.0617 (9)	0.0655 (10)	0.0462 (8)	−0.0027 (8)	0.0239 (7)	−0.0015 (7)
C4	0.0423 (7)	0.0550 (9)	0.0467 (8)	−0.0069 (7)	0.0113 (6)	0.0061 (7)
C5	0.0347 (6)	0.0387 (7)	0.0422 (7)	−0.0028 (5)	0.0091 (5)	0.0005 (6)
C6	0.0368 (6)	0.0390 (7)	0.0393 (7)	−0.0005 (5)	0.0065 (5)	−0.0010 (5)
C7	0.0355 (6)	0.0375 (7)	0.0436 (7)	−0.0054 (5)	0.0105 (5)	−0.0013 (6)
C8	0.0456 (8)	0.0520 (9)	0.0578 (9)	0.0054 (7)	0.0089 (7)	−0.0089 (7)
C9	0.0683 (11)	0.0677 (11)	0.0606 (10)	0.0065 (9)	0.0158 (8)	−0.0212 (9)
C10	0.0674 (11)	0.0741 (12)	0.0456 (9)	−0.0053 (9)	0.0054 (7)	−0.0149 (8)
C11	0.0462 (8)	0.0640 (10)	0.0558 (9)	0.0005 (7)	0.0015 (7)	−0.0007 (8)
C12	0.0400 (7)	0.0506 (8)	0.0524 (8)	0.0012 (6)	0.0091 (6)	−0.0066 (7)
C13	0.0426 (7)	0.0381 (7)	0.0435 (7)	−0.0100 (5)	0.0100 (6)	−0.0046 (6)
C14	0.0544 (9)	0.0477 (9)	0.0998 (14)	−0.0010 (7)	0.0365 (9)	0.0020 (9)
C15	0.0823 (14)	0.0728 (14)	0.147 (2)	−0.0182 (11)	0.0759 (15)	−0.0221 (15)
C16	0.113 (2)	0.118 (2)	0.0933 (17)	−0.0592 (18)	0.0672 (16)	−0.0339 (16)
C17	0.1149 (19)	0.122 (2)	0.0517 (11)	−0.0542 (18)	0.0137 (12)	0.0191 (12)
C18	0.0620 (10)	0.0742 (12)	0.0529 (9)	−0.0146 (9)	0.0023 (8)	0.0170 (9)
O1	0.0522 (6)	0.0502 (6)	0.0433 (6)	−0.0104 (5)	0.0084 (4)	−0.0078 (5)
O2	0.0722 (8)	0.0922 (10)	0.0712 (8)	−0.0402 (7)	0.0275 (6)	−0.0053 (7)
O3	0.0974 (10)	0.0914 (10)	0.0454 (7)	−0.0122 (8)	0.0061 (6)	−0.0174 (7)

Geometric parameters (Å, º) top

C2—O3	1.1971 (19)	C9—C10	1.374 (3)
C2—O1	1.3484 (19)	C9—H9	0.9300
C2—C3	1.489 (2)	C10—C11	1.370 (2)
C3—C4	1.506 (2)	C10—H10	0.9300
C3—H3A	0.9700	C11—C12	1.384 (2)
C3—H3B	0.9700	C11—H11	0.9300
C4—O2	1.2013 (19)	C12—H12	0.9300
C4—C5	1.5191 (19)	C13—C18	1.375 (2)
C5—C7	1.5122 (18)	C13—C14	1.382 (2)
C5—C6	1.5353 (19)	C14—C15	1.378 (3)
C5—H5	0.9800	C14—H14	0.9300
C6—O1	1.4633 (16)	C15—C16	1.358 (4)
C6—C13	1.5013 (18)	C15—H15	0.9300
C6—H6	0.9800	C16—C17	1.367 (4)
C7—C8	1.381 (2)	C16—H16	0.9300
C7—C12	1.387 (2)	C17—C18	1.398 (3)
C8—C9	1.382 (2)	C17—H17	0.9300
C8—H8	0.9300	C18—H18	0.9300

O3—C2—O1	119.27 (16)	C10—C9—H9	119.9
O3—C2—C3	124.18 (16)	C8—C9—H9	119.9
O1—C2—C3	116.55 (13)	C11—C10—C9	120.03 (16)
C2—C3—C4	115.28 (13)	C11—C10—H10	120.0
C2—C3—H3A	108.5	C9—C10—H10	120.0
C4—C3—H3A	108.5	C10—C11—C12	119.91 (15)
C2—C3—H3B	108.5	C10—C11—H11	120.0
C4—C3—H3B	108.5	C12—C11—H11	120.0
H3A—C3—H3B	107.5	C11—C12—C7	120.60 (14)
O2—C4—C3	121.58 (14)	C11—C12—H12	119.7
O2—C4—C5	123.07 (15)	C7—C12—H12	119.7
C3—C4—C5	115.35 (12)	C18—C13—C14	119.27 (15)
C7—C5—C4	113.59 (11)	C18—C13—C6	120.03 (14)
C7—C5—C6	112.70 (11)	C14—C13—C6	120.69 (14)
C4—C5—C6	108.47 (11)	C15—C14—C13	120.3 (2)
C7—C5—H5	107.2	C15—C14—H14	119.9
C4—C5—H5	107.2	C13—C14—H14	119.9
C6—C5—H5	107.2	C16—C15—C14	120.5 (2)
O1—C6—C13	106.88 (10)	C16—C15—H15	119.8
O1—C6—C5	109.24 (11)	C14—C15—H15	119.8
C13—C6—C5	113.39 (11)	C15—C16—C17	120.2 (2)
O1—C6—H6	109.1	C15—C16—H16	119.9
C13—C6—H6	109.1	C17—C16—H16	119.9
C5—C6—H6	109.1	C16—C17—C18	119.9 (2)
C8—C7—C12	118.76 (13)	C16—C17—H17	120.0
C8—C7—C5	120.68 (13)	C18—C17—H17	120.0
C12—C7—C5	120.54 (13)	C13—C18—C17	119.8 (2)
C7—C8—C9	120.43 (15)	C13—C18—H18	120.1
C7—C8—H8	119.8	C17—C18—H18	120.1
C9—C8—H8	119.8	C2—O1—C6	118.00 (11)
C10—C9—C8	120.25 (16)

O3—C2—C3—C4	141.51 (18)	C9—C10—C11—C12	0.3 (3)
O1—C2—C3—C4	−38.6 (2)	C10—C11—C12—C7	0.6 (3)
C2—C3—C4—O2	−153.41 (17)	C8—C7—C12—C11	−1.0 (2)
C2—C3—C4—C5	26.3 (2)	C5—C7—C12—C11	−179.25 (14)
O2—C4—C5—C7	−33.0 (2)	O1—C6—C13—C18	119.45 (14)
C3—C4—C5—C7	147.27 (13)	C5—C6—C13—C18	−120.13 (15)
O2—C4—C5—C6	−159.12 (16)	O1—C6—C13—C14	−61.63 (17)
C3—C4—C5—C6	21.12 (17)	C5—C6—C13—C14	58.80 (18)
C7—C5—C6—O1	173.61 (10)	C18—C13—C14—C15	2.1 (3)
C4—C5—C6—O1	−59.72 (13)	C6—C13—C14—C15	−176.85 (17)
C7—C5—C6—C13	54.54 (15)	C13—C14—C15—C16	−0.9 (3)
C4—C5—C6—C13	−178.80 (11)	C14—C15—C16—C17	−0.5 (4)
C4—C5—C7—C8	126.23 (15)	C15—C16—C17—C18	0.6 (4)
C6—C5—C7—C8	−109.89 (15)	C14—C13—C18—C17	−1.9 (3)
C4—C5—C7—C12	−55.59 (18)	C6—C13—C18—C17	176.99 (16)
C6—C5—C7—C12	68.29 (16)	C16—C17—C18—C13	0.6 (3)
C12—C7—C8—C9	0.5 (2)	O3—C2—O1—C6	177.79 (15)
C5—C7—C8—C9	178.72 (15)	C3—C2—O1—C6	−2.1 (2)
C7—C8—C9—C10	0.4 (3)	C13—C6—O1—C2	175.37 (12)
C8—C9—C10—C11	−0.9 (3)	C5—C6—O1—C2	52.33 (16)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
C6—H6···O2ⁱ	0.98	2.44	3.380 (2)	161
C17—H17···O3ⁱⁱ	0.93	2.46	3.351 (3)	160
C3—H3B···Cg1ⁱ	0.97	2.98	3.681 (2)	131
C5—H5···Cg2ⁱⁱⁱ	0.98	2.96	3.830 (2)	149

Symmetry codes: (i) −x+2, y−1/2, −z+3/2; (ii) x, −y−1/2, z+1/2; (iii) −x+1, y+1/2, −z+3/2.

Experimental details

Crystal data
Chemical formula	C₁₇H₁₄O₃
M_r	266.28
Crystal system, space group	Monoclinic, P2₁/c
Temperature (K)	295
a, b, c (Å)	8.9940 (2), 8.2310 (4), 18.9040 (8)
β (°)	101.412 (2)
V (Å³)	1371.79 (9)
Z	4
Radiation type	Mo Kα
µ (mm⁻¹)	0.09
Crystal size (mm)	0.30 × 0.30 × 0.18

Data collection
Diffractometer	Nonius KappaCCD diffractometer
Absorption correction	–
No. of measured, independent and observed [I > 2σ(I)] reflections	5298, 3113, 2459
R_int	0.017
(sin θ/λ)_max (Å⁻¹)	0.649

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.050, 0.131, 1.05
No. of reflections	3113
No. of parameters	181
H-atom treatment	H-atom parameters constrained
Δρ_max, Δρ_min (e Å⁻³)	0.19, −0.20

Computer programs: COLLECT (Nonius, 2000), DENZO and SCALEPACK (Otwinowski & Minor, 1997), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), ORTEP-3 for Windows (Farrugia, 1997), WinGX (Farrugia, 1999).

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
C6—H6···O2ⁱ	0.98	2.44	3.380 (2)	161
C17—H17···O3ⁱⁱ	0.93	2.46	3.351 (3)	160
C3—H3B···Cg1ⁱ	0.97	2.9747	3.681 (2)	131
C5—H5···Cg2ⁱⁱⁱ	0.98	2.9552	3.830 (2)	149

Symmetry codes: (i) −x+2, y−1/2, −z+3/2; (ii) x, −y−1/2, z+1/2; (iii) −x+1, y+1/2, −z+3/2.

Acknowledgements

This work has received partial support from CNPq, CAPES, FAPEAL, IM-INOFAR and FINEP.

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