3-(Ammonio­methyl)­pyridinium bis­(perchlorate)

Bayar, I.; Kefi, R.; Sidonio Pereira da Silva, P.; Ramos Silva, M.; Ben Nasr, C.

doi:10.1107/S1600536813012038

organic compounds

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ISSN: 2056-9890

Volume 69| Part 6| June 2013| Page o843

doi:10.1107/S1600536813012038

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3-(Ammoniomethyl)pyridinium bis(perchlorate)

Imen Bayar,^a Riadh Kefi,^a Pedro Sidonio Pereira da Silva,^b Manuela Ramos Silva ^b and Cherif Ben Nasr ^a ^*

^aLaboratoire de Chimie des Matériaux, Faculté des Sciences de Bizerte, 7021 Zarzouna, Tunisia, and ^bCEMDRX, Physics Department, University of Coimbra, P-3004-516 Coimbra, Portugal
^*Correspondence e-mail: cherif_bennasr@yahoo.fr

(Received 17 April 2013; accepted 2 May 2013; online 11 May 2013)

In the title molecular salt, C₆H₁₀N₂²⁺·2ClO₄⁻, the Cl—O bond lengths [anion 1: 1.369 (3)–1.415 (3); anion 2: 1.420 (2)–1.441 (2) Å] and the O—Cl—O angles [anion 1: 105.4 (2)–111.8 (4); anion 2: 107.8 (1)–110.3 (1)°] indicate a slight distortion of the perchlorate anions from regular tetrahedral symmetry. In the crystal, the components are linked into columns along the a-axis direction via N—H⋯O and C—H⋯O hydrogen bonds, with stacks of the organic molecules being surrounded by stacks of perchlorate anions.

Related literature

For general background to perchlorate salts with organic cations, see: Czarnecki et al. (1994 ); Czupinski et al. (2002 , 2006 ). For related structures, see: Kapplinger & Keutel (1999 ); Ye et al. (2002 )

Experimental

Crystal data

C₆H₁₀N₂²⁺·2ClO₄⁻
M_r = 309.06
Monoclinic, P 2₁ /c
a = 5.1947 (1) Å
b = 12.1221 (3) Å
c = 18.2724 (5) Å
β = 98.067 (1)°
V = 1139.24 (5) Å³
Z = 4
Mo Kα radiation
μ = 0.61 mm⁻¹
T = 293 K
0.44 × 0.33 × 0.22 mm

Data collection

Bruker APEXII CCD area-detector diffractometer
Absorption correction: multi-scan (SADABS; Sheldrick, 2003 ) T_min = 0.765, T_max = 0.875
26338 measured reflections
3509 independent reflections
3007 reflections with I > 2σ(I)
R_int = 0.032

Refinement

R[F² > 2σ(F²)] = 0.051
wR(F²) = 0.148
S = 1.06
3509 reflections
163 parameters
H-atom parameters constrained
Δρ_max = 0.70 e Å⁻³
Δρ_min = −0.75 e Å⁻³

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
N1—H1⋯O5	0.86	2.16	2.901 (3)	144
N1—H1⋯O7ⁱ	0.86	2.36	2.945 (3)	125
N2—H2A⋯O2ⁱⁱ	0.89	2.09	2.866 (3)	146
N2—H2A⋯O6ⁱⁱⁱ	0.89	2.54	3.166 (3)	128
N2—H2B⋯O6ⁱⁱ	0.89	2.10	2.925 (3)	155
N2—H2B⋯O3^iv	0.89	2.47	2.922 (4)	112
N2—H2C⋯O1^v	0.89	2.08	2.933 (3)	160
C5—H5⋯O1^vi	0.93	2.54	3.350 (4)	145
C6—H6A⋯O8^vii	0.97	2.50	3.137 (3)	123
Symmetry codes: (i) x-1, y, z; (ii) $[x, -y+{\script{1\over 2}}, z+{\script{1\over 2}}]$ ; (iii) $[x-1, -y+{\script{1\over 2}}, z+{\script{1\over 2}}]$ ; (iv) $[-x+1, y-{\script{1\over 2}}, -z+{\script{1\over 2}}]$ ; (v) $[-x, y-{\script{1\over 2}}, -z+{\script{1\over 2}}]$ ; (vi) x+1, y, z; (vii) $[-x+1, y+{\script{1\over 2}}, -z+{\script{1\over 2}}]$ .

Data collection: APEX2 (Bruker, 2003 ); cell refinement: SAINT (Bruker, 2003); data reduction: SAINT; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008 ); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: PLATON (Spek, 2009 ); software used to prepare material for publication: SHELXL97.

Supporting information

Crystal structure: contains datablocks global, I. DOI: 10.1107/S1600536813012038/lr2104sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536813012038/lr2104Isup2.hkl

Supporting information file. DOI: 10.1107/S1600536813012038/lr2104Isup3.cml

checkCIF report

Comment top

Studies of perchlorate salts containing organic cations have had a great deal of attention in recent years, because of their ferroelectric and dielectric behaviour (Czupinski et al., 2002; Czupinski et al., 2006). It was shown that dynamics of pyridinium cations contributes mainly to the mecanism of solid-solid phase transition and leads ferroelectricity in these molecular-ionic crystals (Czarnecki et al., 1994) Here, we report the synthesis and the crystal structure of the title compound (I), (C₆H₁₀N₂)(ClO₄)₂.

The structure consists of one 3-ammoniomethylpyrinidinium dication and two perchlorate anions (Fig. 1). In the crystal, the three H atoms of the ammonium group are involved into N—H···Cl and N—H···O hydrogen bonds: N2—H2C···Cl1(v), N2—H2B···Cl2(ii), N2—H2C···O1(v), N2—H2A··· (O6vi, O2iv) and N2—H2B··· (O6iv, O3vii), the two laters are bifurcated (for the symmetry codes, see Table 1). These hydrogen bonds link the ionic units (NH₃⁺ and [ClO₄]^-) to form columns running along the a-axis direction (Fig.2) and situated at y = z = 1/2 (Fig.3). The organic groups are located between these columns via two kinds of hydrogen bonds: a bifurcated N—H···O and two C—H···O: N1—H1 (O5, O7viii), C6—H6A···O8(iii) and C5—H5···O1(vi) (Fig. 3, Table 1). No π-π stacking interactions between the organic rings or C—H··· π interactions towards them are observed. It is worth noting that the C—N—C angles of pyridine are very sensitive to protonation: a pyridinium cation always has an expanded angle of the C—N—C in comparison with the parent pydidine. The angle C(5)—N(1)—C(1) [123.4 (2)°] is consistent with the type of pyridinium cation. The hydrogen atom HN(1), which is deprived from its parent, attaches the nitrogen atom. As expected, the [ClO₄]^- anion has typical tetrahedral geometry where the Cl—O bond lengths and O—Cl—O angles are not equal to one another but vary with the environment around the O atoms. In the title compound, the Cl—O bond lengths vary from 1.369 (3) to 1.415 (3) Å for [Cl(1)O₄]^- anion and from 1.420 (2) to 1.440 (2) Å for [Cl(2)O₄]^- anion are comparable to that previuosly reported for the perchlorate anions (Kapplinger & Keutel, 1999). The O—Cl—O angles range from 105.4 (2) to 111.8 (4)° for the first anion and from 109.04 (15) to 110.33 (13)° for the second one. These values, which are characteristic of perchlorate anions (Ye et al., 2002), clearly indicate that the coordination geometry of the Cl(2) atom can be regarded as being a less distorted tetrahedron than the one of the Cl(1). However, for the Cl(2)O₄ tetrahedron, all the oxygen atoms are involved in hydrogen bonds, while only three oxygen atoms act as acceptors of hydrogen bonds for the [Cl(1)O₄] tetrahedron.

Related literature top

For general background to perchlorate salts with organic cations, see: Czarnecki et al. (1994); Czupinski et al. (2002, 2006). For related structures, see: Kapplinger & Keutel (1999); Ye et al. (2002)

Experimental top

3-ammoniomethylpyrinidinium (1 mmol, 0.108 g) was dissolved in a mixture of distilled water (10 ml) and perchloric acid (0.5 ml).The resultant solution was evaporated at room temperature. Crystals of the title compound, which remained stable under normal conditions of temperature and humidity, were isolated after several days and subjected to X-ray diffraction analysis (yield 64%).

Computing details top

Data collection: APEX2 (Bruker, 2003); cell refinement: SAINT (Bruker, 2003); data reduction: SAINT (Bruker, 2003); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: PLATON (Spek, 2009); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Figures top

	Fig. 1. A view of the title compound, showing 50% probability displacement ellipsoids and arbitrary spheres for the H atoms.
	Fig. 2. Partial packing of the title compound, viewed down the b axis, showing columns formed between perchlorate anions and ammonium groups.
	Fig. 3. The crystal packing of the title compound viewed along the a axis.

3-(Ammoniomethyl)pyridinium bis(perchlorate) top

Crystal data top

C₆H₁₀N₂²⁺·2ClO₄⁻	F(000) = 632
M_r = 309.06	D_x = 1.802 Mg m⁻³
Monoclinic, P2₁/c	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybc	Cell parameters from 5314 reflections
a = 5.1947 (1) Å	θ = 2.3–30.6°
b = 12.1221 (3) Å	µ = 0.61 mm⁻¹
c = 18.2724 (5) Å	T = 293 K
β = 98.067 (1)°	Block, colourless
V = 1139.24 (5) Å³	0.44 × 0.33 × 0.22 mm
Z = 4

Data collection top

Bruker APEXII CCD area-detector diffractometer	3509 independent reflections
Radiation source: fine-focus sealed tube	3007 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.032
ϕ and ω scans	θ_max = 30.6°, θ_min = 2.0°
Absorption correction: multi-scan (SADABS; Sheldrick, 2003)	h = −7→7
T_min = 0.765, T_max = 0.875	k = −17→17
26338 measured reflections	l = −26→26

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)] = 0.051	Hydrogen site location: inferred from neighbouring sites
wR(F²) = 0.148	H-atom parameters constrained
S = 1.06	w = 1/[σ²(F_o²) + (0.0782P)² + 0.9452P] where P = (F_o² + 2F_c²)/3
3509 reflections	(Δ/σ)_max < 0.001
163 parameters	Δρ_max = 0.70 e Å⁻³
0 restraints	Δρ_min = −0.75 e Å⁻³

Crystal data top

C₆H₁₀N₂²⁺·2ClO₄⁻	V = 1139.24 (5) Å³
M_r = 309.06	Z = 4
Monoclinic, P2₁/c	Mo Kα radiation
a = 5.1947 (1) Å	µ = 0.61 mm⁻¹
b = 12.1221 (3) Å	T = 293 K
c = 18.2724 (5) Å	0.44 × 0.33 × 0.22 mm
β = 98.067 (1)°

Data collection top

Bruker APEXII CCD area-detector diffractometer	3509 independent reflections
Absorption correction: multi-scan (SADABS; Sheldrick, 2003)	3007 reflections with I > 2σ(I)
T_min = 0.765, T_max = 0.875	R_int = 0.032
26338 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.051	0 restraints
wR(F²) = 0.148	H-atom parameters constrained
S = 1.06	Δρ_max = 0.70 e Å⁻³
3509 reflections	Δρ_min = −0.75 e Å⁻³
163 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
Cl1	0.11942 (12)	0.50919 (5)	0.12761 (3)	0.04193 (16)
Cl2	0.82533 (9)	0.14688 (4)	0.09563 (2)	0.03098 (14)
O1	−0.1534 (5)	0.4964 (2)	0.1114 (2)	0.0912 (10)
O2	0.2513 (6)	0.4146 (3)	0.10993 (17)	0.0916 (10)
O3	0.1869 (7)	0.5993 (4)	0.0893 (4)	0.172 (3)
O4	0.1865 (11)	0.5226 (5)	0.2043 (2)	0.169 (2)
O5	0.5608 (3)	0.16338 (19)	0.10859 (10)	0.0499 (5)
O6	0.8768 (4)	0.22306 (18)	0.03930 (11)	0.0523 (5)
O7	0.9987 (4)	0.16608 (18)	0.16220 (10)	0.0530 (5)
O8	0.8562 (5)	0.03757 (18)	0.07037 (15)	0.0673 (6)
N1	0.4671 (4)	0.26538 (17)	0.24604 (10)	0.0400 (4)
H1	0.4241	0.2266	0.2068	0.048*
N2	0.3730 (4)	0.18501 (18)	0.47673 (11)	0.0429 (5)
H2A	0.2868	0.1768	0.5152	0.064*
H2B	0.5421	0.1917	0.4928	0.064*
H2C	0.3466	0.1263	0.4474	0.064*
C1	0.3473 (5)	0.24549 (19)	0.30466 (12)	0.0377 (4)
H1A	0.2182	0.1919	0.3022	0.045*
C2	0.4155 (4)	0.30457 (18)	0.36882 (10)	0.0318 (4)
C3	0.6048 (5)	0.38511 (19)	0.36950 (12)	0.0385 (4)
H3	0.6529	0.4269	0.4119	0.046*
C4	0.7224 (5)	0.4036 (2)	0.30740 (14)	0.0417 (5)
H4	0.8501	0.4575	0.3079	0.050*
C5	0.6497 (5)	0.3420 (2)	0.24510 (12)	0.0392 (5)
H5	0.7267	0.3537	0.2028	0.047*
C6	0.2791 (5)	0.2852 (2)	0.43480 (13)	0.0443 (5)
H6A	0.3051	0.3487	0.4673	0.053*
H6B	0.0939	0.2781	0.4185	0.053*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Cl1	0.0443 (3)	0.0361 (3)	0.0473 (3)	0.0052 (2)	0.0129 (2)	0.0049 (2)
Cl2	0.0309 (2)	0.0361 (3)	0.0261 (2)	0.00110 (17)	0.00438 (16)	−0.00042 (15)
O1	0.0420 (11)	0.0839 (18)	0.152 (3)	0.0086 (11)	0.0271 (14)	0.0586 (18)
O2	0.0762 (17)	0.104 (2)	0.0872 (19)	0.0430 (16)	−0.0138 (14)	−0.0469 (16)
O3	0.078 (2)	0.132 (3)	0.295 (6)	−0.031 (2)	−0.009 (3)	0.148 (4)
O4	0.190 (5)	0.245 (6)	0.066 (2)	0.104 (4)	−0.003 (2)	−0.048 (3)
O5	0.0325 (8)	0.0816 (14)	0.0368 (8)	0.0027 (8)	0.0089 (6)	−0.0020 (8)
O6	0.0490 (10)	0.0643 (12)	0.0455 (9)	−0.0003 (9)	0.0130 (8)	0.0202 (9)
O7	0.0437 (9)	0.0765 (13)	0.0356 (8)	−0.0043 (9)	−0.0061 (7)	−0.0026 (8)
O8	0.0796 (16)	0.0426 (11)	0.0782 (15)	0.0089 (10)	0.0055 (12)	−0.0173 (10)
N1	0.0520 (11)	0.0437 (10)	0.0241 (7)	−0.0097 (8)	0.0048 (7)	−0.0045 (7)
N2	0.0466 (11)	0.0530 (12)	0.0296 (8)	−0.0131 (9)	0.0066 (7)	0.0008 (8)
C1	0.0407 (11)	0.0434 (11)	0.0284 (9)	−0.0099 (9)	0.0028 (8)	−0.0009 (8)
C2	0.0318 (9)	0.0379 (10)	0.0256 (8)	0.0052 (7)	0.0037 (7)	−0.0005 (7)
C3	0.0416 (11)	0.0383 (10)	0.0337 (10)	−0.0002 (9)	−0.0009 (8)	−0.0066 (8)
C4	0.0400 (11)	0.0409 (11)	0.0430 (11)	−0.0087 (9)	0.0016 (9)	0.0012 (9)
C5	0.0428 (11)	0.0441 (12)	0.0318 (9)	−0.0024 (9)	0.0089 (8)	0.0049 (8)
C6	0.0426 (12)	0.0597 (14)	0.0325 (10)	0.0077 (10)	0.0116 (9)	0.0003 (10)

Geometric parameters (Å, º) top

Cl1—O3	1.369 (3)	N2—H2B	0.8900
Cl1—O2	1.397 (2)	N2—H2C	0.8900
Cl1—O4	1.405 (4)	C1—C2	1.377 (3)
Cl1—O1	1.415 (3)	C1—H1A	0.9300
Cl2—O8	1.420 (2)	C2—C3	1.385 (3)
Cl2—O7	1.4272 (18)	C2—C6	1.500 (3)
Cl2—O6	1.4357 (18)	C3—C4	1.380 (3)
Cl2—O5	1.4406 (18)	C3—H3	0.9300
N1—C5	1.329 (3)	C4—C5	1.369 (3)
N1—C1	1.334 (3)	C4—H4	0.9300
N1—H1	0.8600	C5—H5	0.9300
N2—C6	1.482 (3)	C6—H6A	0.9700
N2—H2A	0.8900	C6—H6B	0.9700

O3—Cl1—O2	111.6 (3)	N1—C1—C2	119.7 (2)
O3—Cl1—O4	111.8 (4)	N1—C1—H1A	120.1
O2—Cl1—O4	105.4 (2)	C2—C1—H1A	120.1
O3—Cl1—O1	107.49 (19)	C1—C2—C3	118.16 (19)
O2—Cl1—O1	111.8 (2)	C1—C2—C6	120.6 (2)
O4—Cl1—O1	108.8 (3)	C3—C2—C6	121.2 (2)
O8—Cl2—O7	110.04 (14)	C4—C3—C2	120.2 (2)
O8—Cl2—O6	109.04 (15)	C4—C3—H3	119.9
O7—Cl2—O6	110.33 (13)	C2—C3—H3	119.9
O8—Cl2—O5	109.69 (14)	C5—C4—C3	119.5 (2)
O7—Cl2—O5	109.86 (11)	C5—C4—H4	120.2
O6—Cl2—O5	107.84 (12)	C3—C4—H4	120.2
C5—N1—C1	123.42 (19)	N1—C5—C4	119.0 (2)
C5—N1—H1	118.3	N1—C5—H5	120.5
C1—N1—H1	118.3	C4—C5—H5	120.5
C6—N2—H2A	109.5	N2—C6—C2	112.65 (19)
C6—N2—H2B	109.5	N2—C6—H6A	109.1
H2A—N2—H2B	109.5	C2—C6—H6A	109.1
C6—N2—H2C	109.5	N2—C6—H6B	109.1
H2A—N2—H2C	109.5	C2—C6—H6B	109.1
H2B—N2—H2C	109.5	H6A—C6—H6B	107.8

C5—N1—C1—C2	−1.3 (4)	C2—C3—C4—C5	0.3 (4)
N1—C1—C2—C3	1.3 (3)	C1—N1—C5—C4	0.7 (4)
N1—C1—C2—C6	178.6 (2)	C3—C4—C5—N1	−0.2 (4)
C1—C2—C3—C4	−0.8 (3)	C1—C2—C6—N2	78.7 (3)
C6—C2—C3—C4	−178.1 (2)	C3—C2—C6—N2	−104.1 (3)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
N1—H1···O5	0.86	2.16	2.901 (3)	144
N1—H1···O7ⁱ	0.86	2.36	2.945 (3)	125
N2—H2A···O2ⁱⁱ	0.89	2.09	2.866 (3)	146
N2—H2A···O6ⁱⁱⁱ	0.89	2.54	3.166 (3)	128
N2—H2B···O6ⁱⁱ	0.89	2.10	2.925 (3)	155
N2—H2B···O3^iv	0.89	2.47	2.922 (4)	112
N2—H2B···Cl2ⁱⁱ	0.89	2.96	3.600 (2)	131
N2—H2C···O1^v	0.89	2.08	2.933 (3)	160
N2—H2C···Cl1^v	0.89	2.97	3.654 (2)	135
C5—H5···O1^vi	0.93	2.54	3.350 (4)	145
C6—H6A···O8^vii	0.97	2.50	3.137 (3)	123

Symmetry codes: (i) x−1, y, z; (ii) x, −y+1/2, z+1/2; (iii) x−1, −y+1/2, z+1/2; (iv) −x+1, y−1/2, −z+1/2; (v) −x, y−1/2, −z+1/2; (vi) x+1, y, z; (vii) −x+1, y+1/2, −z+1/2.

Experimental details

Crystal data
Chemical formula	C₆H₁₀N₂²⁺·2ClO₄⁻
M_r	309.06
Crystal system, space group	Monoclinic, P2₁/c
Temperature (K)	293
a, b, c (Å)	5.1947 (1), 12.1221 (3), 18.2724 (5)
β (°)	98.067 (1)
V (Å³)	1139.24 (5)
Z	4
Radiation type	Mo Kα
µ (mm⁻¹)	0.61
Crystal size (mm)	0.44 × 0.33 × 0.22

Data collection
Diffractometer	Bruker APEXII CCD area-detector diffractometer
Absorption correction	Multi-scan (SADABS; Sheldrick, 2003)
T_min, T_max	0.765, 0.875
No. of measured, independent and observed [I > 2σ(I)] reflections	26338, 3509, 3007
R_int	0.032
(sin θ/λ)_max (Å⁻¹)	0.716

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.051, 0.148, 1.06
No. of reflections	3509
No. of parameters	163
H-atom treatment	H-atom parameters constrained
Δρ_max, Δρ_min (e Å⁻³)	0.70, −0.75

Computer programs: APEX2 (Bruker, 2003), SAINT (Bruker, 2003), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), PLATON (Spek, 2009).

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
N1—H1···O5	0.86	2.16	2.901 (3)	143.9
N1—H1···O7ⁱ	0.86	2.36	2.945 (3)	125.4
N2—H2A···O2ⁱⁱ	0.89	2.09	2.866 (3)	145.8
N2—H2A···O6ⁱⁱⁱ	0.89	2.54	3.166 (3)	127.6
N2—H2B···O6ⁱⁱ	0.89	2.10	2.925 (3)	154.5
N2—H2B···O3^iv	0.89	2.47	2.922 (4)	112.3
N2—H2B···Cl2ⁱⁱ	0.89	2.96	3.600 (2)	130.5
N2—H2C···O1^v	0.89	2.08	2.933 (3)	159.7
N2—H2C···Cl1^v	0.89	2.97	3.654 (2)	135.0
C5—H5···O1^vi	0.93	2.54	3.350 (4)	145
C6—H6A···O8^vii	0.97	2.50	3.137 (3)	123

Symmetry codes: (i) x−1, y, z; (ii) x, −y+1/2, z+1/2; (iii) x−1, −y+1/2, z+1/2; (iv) −x+1, y−1/2, −z+1/2; (v) −x, y−1/2, −z+1/2; (vi) x+1, y, z; (vii) −x+1, y+1/2, −z+1/2.

Acknowledgements

We would like to acknowledge the support provided by the Secretary of State for Scientific Research and Technology of Tunisia. PSPS acknowledges support by the Fundação para a Ciência e a Tecnologia (FCT), under scholarship SFRH/BPD/84173/2012.

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