(+)-(S,S)-1,3-Bis[(tetra­hydro­furan-2-yl)­meth­yl]thio­urea

Peña, Ú.; Bernès, S.; Gutiérrez, R.

doi:10.1107/S1600536808040373

organic compounds

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Volume 65| Part 1| January 2009| Page o96

doi:10.1107/S1600536808040373

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(+)-(S,S)-1,3-Bis[(tetrahydrofuran-2-yl)methyl]thiourea

Úlises Peña,^a Sylvain Bernès ^b ^* and René Gutiérrez ^a

^aLaboratorio de Síntesis de Complejos, Facultad de Ciencias Químicas, Universidad Autónoma de Puebla, AP 1067, 72001 Puebla, Pue., Mexico, and ^bDEP Facultad de Ciencias Químicas, UANL, Guerrero y Progreso S/N, Col. Treviño, 64570 Monterrey, NL, Mexico
^*Correspondence e-mail: sylvain_bernes@Hotmail.com

(Received 19 November 2008; accepted 1 December 2008; online 13 December 2008)

The title compound, C₁₁H₂₀N₂O₂S, is an enantiomerically pure heterocycle-substituted thiourea synthesized under solvent-free conditions. The thiourea unit adopts a ZZ conformation, with the HN—(C=S)—NH core almost planar and the tetrahydrofurfuryl groups placed below and above this plane. The whole molecule thus approximates to noncrystallographic C₂ symmetry. Unexpectedly, the C=S group is not involved in intermolecular hydrogen bonding, as generally observed in homodisubstituted thioureas. Instead, molecules form a one-dimensional network based on weak N—H⋯O(heterocycle) hydrogen bonding, resulting in a zigzag ribbon-like structure around the crystallographic 2₁ screw axis along [100].

Related literature

For general background about solvent-free synthesis, see: Tanaka & Toda (2000 ); Jeon et al. (2005 ). For C₂ homosubstituted thioureas, see: Bailey et al. (1997 ); Lai & Tiekink (2002 ). For common hydrogen-bonding schemes in thioureas, see: Vázquez et al. (2004 ); Custelcean et al. (2005 ); Shashidhar et al. (2006 ); Sadiq-ur-Rehman et al. (2007 ); Saxena & Pike (2007 ).

Experimental

Crystal data

C₁₁H₂₀N₂O₂S
M_r = 244.35
Orthorhombic, P 2₁ 2₁ 2₁
a = 7.8588 (9) Å
b = 10.8265 (11) Å
c = 15.6196 (16) Å
V = 1329.0 (2) Å³
Z = 4
Mo Kα radiation
μ = 0.23 mm⁻¹
T = 298 (1) K
0.6 × 0.6 × 0.6 mm

Data collection

Siemens P4 diffractometer
Absorption correction: ψ scan (XSCANS; Siemens, 1996 ) T_min = 0.782, T_max = 0.870
4611 measured reflections
3026 independent reflections
2484 reflections with I > 2σ(I)
R_int = 0.031
3 standard reflections every 97 reflections intensity decay: 1%

Refinement

R[F² > 2σ(F²)] = 0.054
wR(F²) = 0.165
S = 1.03
3026 reflections
151 parameters
2 restraints
H atoms treated by a mixture of independent and constrained refinement
Δρ_max = 0.22 e Å⁻³
Δρ_min = −0.16 e Å⁻³
Absolute structure: Flack (1983 ), 1267 Friedel pairs
Flack parameter: −0.01 (14)

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
N2—H2⋯O18ⁱ	0.85 (1)	2.095 (17)	2.897 (3)	157 (3)
N12—H12⋯O8ⁱⁱ	0.86 (1)	2.197 (18)	2.978 (3)	150 (3)
Symmetry codes: (i) $[x-{\script{1\over 2}}, -y+{\script{3\over 2}}, -z+1]$ ; (ii) $[x+{\script{1\over 2}}, -y+{\script{3\over 2}}, -z+1]$ .

Data collection: XSCANS (Siemens, 1996); cell refinement: XSCANS; data reduction: XSCANS; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008 ); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: Mercury (Macrae et al., 2006 ); software used to prepare material for publication: SHELXL97.

Supporting information

Crystal structure: contains datablocks I, global. DOI: 10.1107/S1600536808040373/ci2734sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536808040373/ci2734Isup2.hkl

checkCIF report

Comment top

The development of straightforward and eco-friendly synthetic procedures remains an important aim in organic synthesis. Many organic solvents, particularly chlorinated hydrocarbons, that are used in large quantities in organic reactions are potential threat to human health and environment. Thus, the design of chemical reactions under solvent-free conditions is getting a renewed interest. In this regard, solvent-free organic syntheses have great applied value and expansive prospects considering their advantages such as high efficiency and selectivity, easy separation and purification and environmental acceptability. All these merits are in accord with the green chemistry's requests of energy-saving, high efficiency and environmentally benign features (Tanaka & Toda, 2000; Jeon et al., 2005). On the other hand, N,N'-disubstituted thioureas have recently received much interest due to their diverse applications, such as, inter alia, antiviral, antituberculous, fungicidal, herbicidal activities, as well as tranquilizing and antidiabetic drugs, agrochemical properties, antioxidants in gasoline, corrosion inhibitors, etc. In view of these and in continuation of our earlier work on the synthesis of thioureas (Vázquez et al., 2004), we synthesized the title compound under solvent-free conditions (see experimental).

The asymmetric unit contains one molecule in general position (Fig. 1). As the amine used as starting material was enantiopure, the thiourea is found to be a pure (S,S) isomer. The central core HN—(C═S)—NH unit is close to be planar, the r.m.s. deviation from the mean plane S1/C1/N2/H2/N12/H12 being 0.039 Å. This core adopts a ZZ conformation (i.e. amine H atoms are arranged syn) and tetrahydrofurfuryl groups are placed below and above the central HN—(C═S)—NH plane. The whole molecule thus approximates a local C₂ point symmetry. The observed conformation is identical to that found in other related homosubstituted thioureas (Lai & Tiekink, 2002; Bailey et al., 1997).

The ZZ conformation avoids the formation of intramolecular hydrogen bonds (Saxena & Pike, 2007). Regarding the packing structure, it is clear that the thioketone functionality does not participate in intermolecular contacts. Such a situation is unexpected, since for previously X-ray characterized chiral and non-chiral homosubstituted thioureas, one-dimensional supramolecular structures based on C═S···H—N hydrogen bonds are predominant, providing that the thiourea is in a ZZ conformation (e.g. Vázquez et al., 2004; Custelcean et al., 2005; Shashidhar et al., 2006; Sadiq-ur-Rehman et al., 2007). Instead, the crystal structure of the title compound is determined by weak N—H···O(heterocycle) hydrogen bonds, aggregating molecules in a backbone arrangement (Fig. 2), parallel to the crystallographic 2₁ screw axis along [100].

Related literature top

For general background about solvent-free synthesis, see: Tanaka & Toda (2000); Jeon et al. (2005). For C₂ homosubstituted thioureas, see: Bailey et al. (1997); Lai & Tiekink (2002). For common hydrogen-bonding schemes in thioureas, see: Vázquez et al. (2004); Custelcean et al. (2005); Shashidhar et al. (2006); Sadiq-ur-Rehman et al. (2007); Saxena & Pike (2007).

Experimental top

Under solvent-free conditions, (S)-(+)-tetrahydrofurfurylamine (0.49 g, 4.88 mmol) and CS₂ (0.19 g, 2.44 mmol) were mixed at 298 K, giving a white solid. The crude was recrystallized from EtOH, affording colourless crystals of the title compound. Yield 99%; m.p. 376–378 K; [α]²⁵_D=+28.7 (c=1, CHCl₃). Anal. Calcd for C₁₁H₂₀N₂O₂S: C 54.07, H 8.25, N 11.46, O 13.10, S 13.12%; found: C 53.12, H 8.18, N 11.30, O 12.98, S 13.87%. Spectroscopic data are in agreement with the X-ray formula (see archived CIF).

Computing details top

Data collection: XSCANS (Siemens, 1996); cell refinement: XSCANS (Siemens, 1996); data reduction: XSCANS (Siemens, 1996); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: Mercury (Macrae et al., 2006); software used to prepare material for publication: SHELXL97 (Sheldrick, 2008).

Figures top

Fig. 1. Molecular structure of the title compound, with 30% probability level displacement ellipsoids for non-H atoms.

Fig. 2. Part of the crystal structure of the title compound, showing the network of N—H···O hydrogen bonds (dashed lines). The 2₁ screw axis forming the backbone supramolecular structure is shown with a standard symbol, and each molecule has a color corresponding to symmetry related positions in the crystal: grey: asymmetric unit; red: -1/2 + x, 3/2 - y, 1 - z; green: 1/2 + x, 3/2 - y, 1 - z; blue: 1 + x, y, z; purple: 3/2 + x, 3/2 - y, 1 - z.

(+)-(S,S)-1,3-Bis[(tetrahydrofuran-2-yl)methyl]thiourea top

Crystal data top

C₁₁H₂₀N₂O₂S	D_x = 1.221 Mg m⁻³
M_r = 244.35	Melting point = 376–378 K
Orthorhombic, P2₁2₁2₁	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: P 2ac 2ab	Cell parameters from 78 reflections
a = 7.8588 (9) Å	θ = 4.6–13.7°
b = 10.8265 (11) Å	µ = 0.23 mm⁻¹
c = 15.6196 (16) Å	T = 298 K
V = 1329.0 (2) Å³	Block, colourless
Z = 4	0.6 × 0.6 × 0.6 mm
F(000) = 528

Data collection top

Siemens P4 diffractometer	2484 reflections with I > 2σ(I)
Radiation source: fine-focus sealed tube	R_int = 0.031
Graphite monochromator	θ_max = 27.5°, θ_min = 2.3°
2θ/ω scans	h = −10→10
Absorption correction: ψ scan (XSCANS; Siemens, 1996)	k = −14→14
T_min = 0.782, T_max = 0.870	l = −20→20
4611 measured reflections	3 standard reflections every 97 reflections
3026 independent reflections	intensity decay: 1%

Refinement top

Refinement on F²	Secondary atom site location: difference Fourier map
Least-squares matrix: full	Hydrogen site location: inferred from neighbouring sites
R[F² > 2σ(F²)] = 0.054	H atoms treated by a mixture of independent and constrained refinement
wR(F²) = 0.165	w = 1/[σ²(F_o²) + (0.0916P)² + 0.2186P] where P = (F_o² + 2F_c²)/3
S = 1.03	(Δ/σ)_max = 0.001
3026 reflections	Δρ_max = 0.22 e Å⁻³
151 parameters	Δρ_min = −0.16 e Å⁻³
2 restraints	Absolute structure: Flack (1983), 1267 Friedel pairs
0 constraints	Absolute structure parameter: −0.01 (14)
Primary atom site location: structure-invariant direct methods

Crystal data top

C₁₁H₂₀N₂O₂S	V = 1329.0 (2) Å³
M_r = 244.35	Z = 4
Orthorhombic, P2₁2₁2₁	Mo Kα radiation
a = 7.8588 (9) Å	µ = 0.23 mm⁻¹
b = 10.8265 (11) Å	T = 298 K
c = 15.6196 (16) Å	0.6 × 0.6 × 0.6 mm

Data collection top

Siemens P4 diffractometer	2484 reflections with I > 2σ(I)
Absorption correction: ψ scan (XSCANS; Siemens, 1996)	R_int = 0.031
T_min = 0.782, T_max = 0.870	3 standard reflections every 97 reflections
4611 measured reflections	intensity decay: 1%
3026 independent reflections

Refinement top

R[F² > 2σ(F²)] = 0.054	H atoms treated by a mixture of independent and constrained refinement
wR(F²) = 0.165	Δρ_max = 0.22 e Å⁻³
S = 1.03	Δρ_min = −0.16 e Å⁻³
3026 reflections	Absolute structure: Flack (1983), 1267 Friedel pairs
151 parameters	Absolute structure parameter: −0.01 (14)
2 restraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
S1	0.48959 (12)	0.46401 (7)	0.34087 (6)	0.0889 (3)
C1	0.4852 (3)	0.5875 (2)	0.40545 (16)	0.0582 (5)
N2	0.4051 (3)	0.6930 (2)	0.38410 (17)	0.0688 (6)
H2	0.389 (5)	0.752 (2)	0.4186 (18)	0.083*
C3	0.3199 (4)	0.7142 (4)	0.3037 (2)	0.0845 (9)
H3A	0.3768	0.6668	0.2594	0.101*
H3B	0.3303	0.8009	0.2889	0.101*
C4	0.1361 (4)	0.6803 (3)	0.30448 (19)	0.0750 (8)
H4A	0.1225	0.5952	0.3248	0.090*
C5	0.0534 (7)	0.6947 (6)	0.2172 (3)	0.1303 (19)
H5A	0.1088	0.7591	0.1842	0.156*
H5B	0.0582	0.6180	0.1852	0.156*
C6	−0.1224 (7)	0.7283 (7)	0.2368 (3)	0.146 (2)
H6A	−0.1636	0.7898	0.1967	0.175*
H6B	−0.1957	0.6563	0.2336	0.175*
C7	−0.1204 (6)	0.7774 (6)	0.3230 (3)	0.1225 (16)
H7A	−0.1498	0.8644	0.3220	0.147*
H7B	−0.2033	0.7344	0.3580	0.147*
O8	0.0435 (3)	0.7618 (2)	0.35753 (14)	0.0872 (7)
N12	0.5602 (3)	0.5888 (2)	0.48261 (15)	0.0677 (5)
H12	0.541 (4)	0.651 (2)	0.5153 (17)	0.081*
C13	0.6525 (4)	0.4849 (3)	0.5184 (2)	0.0796 (8)
H13A	0.5944	0.4092	0.5026	0.096*
H13B	0.6503	0.4911	0.5803	0.096*
C14	0.8356 (3)	0.4768 (2)	0.48913 (18)	0.0647 (6)
H14A	0.8394	0.4820	0.4265	0.078*
C15	0.9260 (6)	0.3588 (3)	0.5178 (4)	0.1075 (14)
H15A	0.8723	0.3243	0.5684	0.129*
H15B	0.9252	0.2972	0.4727	0.129*
C16	1.0986 (6)	0.3991 (4)	0.5365 (4)	0.1293 (18)
H16A	1.1421	0.3569	0.5867	0.155*
H16B	1.1731	0.3815	0.4885	0.155*
C17	1.0897 (5)	0.5297 (5)	0.5515 (4)	0.132 (2)
H17A	1.1046	0.5465	0.6121	0.158*
H17B	1.1801	0.5711	0.5205	0.158*
O18	0.9296 (3)	0.57528 (19)	0.52400 (18)	0.0850 (7)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
S1	0.0894 (5)	0.0744 (4)	0.1029 (6)	0.0133 (4)	−0.0215 (5)	−0.0363 (4)
C1	0.0455 (10)	0.0558 (11)	0.0733 (14)	−0.0014 (10)	0.0034 (11)	−0.0067 (10)
N2	0.0585 (12)	0.0615 (12)	0.0864 (15)	0.0100 (10)	−0.0053 (11)	−0.0065 (11)
C3	0.0738 (18)	0.098 (2)	0.0813 (18)	0.0236 (18)	0.0124 (15)	0.0169 (17)
C4	0.0749 (18)	0.0776 (17)	0.0724 (16)	0.0170 (15)	−0.0110 (14)	−0.0098 (14)
C5	0.117 (3)	0.194 (5)	0.081 (2)	0.061 (4)	−0.026 (2)	−0.034 (3)
C6	0.114 (4)	0.217 (6)	0.107 (3)	0.064 (4)	−0.045 (3)	−0.035 (4)
C7	0.075 (2)	0.160 (4)	0.133 (4)	0.036 (3)	−0.009 (2)	−0.030 (3)
O8	0.0645 (11)	0.1192 (17)	0.0778 (12)	0.0116 (11)	0.0004 (10)	−0.0262 (11)
N12	0.0551 (11)	0.0755 (13)	0.0725 (13)	−0.0009 (10)	−0.0071 (10)	−0.0105 (11)
C13	0.0672 (15)	0.0860 (19)	0.0856 (18)	−0.0126 (14)	−0.0081 (14)	0.0201 (16)
C14	0.0654 (13)	0.0590 (13)	0.0698 (14)	0.0061 (11)	−0.0113 (11)	−0.0025 (12)
C15	0.107 (3)	0.0628 (16)	0.153 (4)	0.0133 (17)	−0.039 (3)	0.004 (2)
C16	0.096 (3)	0.104 (3)	0.188 (5)	0.032 (2)	−0.051 (3)	−0.006 (3)
C17	0.0660 (19)	0.128 (4)	0.201 (5)	0.005 (2)	−0.030 (3)	−0.052 (4)
O18	0.0640 (11)	0.0632 (10)	0.1278 (18)	−0.0006 (9)	−0.0082 (12)	−0.0171 (11)

Geometric parameters (Å, º) top

S1—C1	1.675 (2)	C7—H7B	0.97
C1—N12	1.342 (3)	N12—C13	1.450 (4)
C1—N2	1.346 (3)	N12—H12	0.862 (10)
N2—C3	1.442 (4)	C13—C14	1.513 (4)
N2—H2	0.849 (10)	C13—H13A	0.97
C3—C4	1.490 (5)	C13—H13B	0.97
C3—H3A	0.97	C14—O18	1.407 (3)
C3—H3B	0.97	C14—C15	1.529 (4)
C4—O8	1.413 (4)	C14—H14A	0.98
C4—C5	1.519 (5)	C15—C16	1.454 (7)
C4—H4A	0.98	C15—H15A	0.97
C5—C6	1.461 (7)	C15—H15B	0.97
C5—H5A	0.97	C16—C17	1.436 (6)
C5—H5B	0.97	C16—H16A	0.97
C6—C7	1.447 (6)	C16—H16B	0.97
C6—H6A	0.97	C17—O18	1.418 (5)
C6—H6B	0.97	C17—H17A	0.97
C7—O8	1.407 (5)	C17—H17B	0.97
C7—H7A	0.97

N12—C1—N2	114.8 (2)	C7—O8—C4	108.8 (3)
N12—C1—S1	122.69 (19)	C1—N12—C13	123.9 (2)
N2—C1—S1	122.5 (2)	C1—N12—H12	118 (2)
C1—N2—C3	124.6 (3)	C13—N12—H12	118 (2)
C1—N2—H2	124 (2)	N12—C13—C14	113.8 (2)
C3—N2—H2	111 (3)	N12—C13—H13A	108.8
N2—C3—C4	113.8 (3)	C14—C13—H13A	108.8
N2—C3—H3A	108.8	N12—C13—H13B	108.8
C4—C3—H3A	108.8	C14—C13—H13B	108.8
N2—C3—H3B	108.8	H13A—C13—H13B	107.7
C4—C3—H3B	108.8	O18—C14—C13	109.8 (2)
H3A—C3—H3B	107.7	O18—C14—C15	106.0 (2)
O8—C4—C3	110.5 (3)	C13—C14—C15	113.7 (3)
O8—C4—C5	104.0 (3)	O18—C14—H14A	109.1
C3—C4—C5	112.5 (4)	C13—C14—H14A	109.1
O8—C4—H4A	109.9	C15—C14—H14A	109.1
C3—C4—H4A	109.9	C16—C15—C14	104.0 (3)
C5—C4—H4A	109.9	C16—C15—H15A	111.0
C6—C5—C4	104.0 (4)	C14—C15—H15A	111.0
C6—C5—H5A	111.0	C16—C15—H15B	111.0
C4—C5—H5A	111.0	C14—C15—H15B	111.0
C6—C5—H5B	111.0	H15A—C15—H15B	109.0
C4—C5—H5B	111.0	C17—C16—C15	106.4 (4)
H5A—C5—H5B	109.0	C17—C16—H16A	110.4
C7—C6—C5	106.0 (4)	C15—C16—H16A	110.4
C7—C6—H6A	110.5	C17—C16—H16B	110.4
C5—C6—H6A	110.5	C15—C16—H16B	110.4
C7—C6—H6B	110.5	H16A—C16—H16B	108.6
C5—C6—H6B	110.5	O18—C17—C16	109.7 (4)
H6A—C6—H6B	108.7	O18—C17—H17A	109.7
O8—C7—C6	108.8 (3)	C16—C17—H17A	109.7
O8—C7—H7A	109.9	O18—C17—H17B	109.7
C6—C7—H7A	109.9	C16—C17—H17B	109.7
O8—C7—H7B	109.9	H17A—C17—H17B	108.2
C6—C7—H7B	109.9	C14—O18—C17	108.6 (3)
H7A—C7—H7B	108.3

N12—C1—N2—C3	178.2 (3)	N2—C1—N12—C13	179.6 (2)
S1—C1—N2—C3	−2.2 (4)	S1—C1—N12—C13	0.0 (4)
C1—N2—C3—C4	91.0 (4)	C1—N12—C13—C14	84.0 (3)
N2—C3—C4—O8	68.5 (4)	N12—C13—C14—O18	68.7 (3)
N2—C3—C4—C5	−175.7 (3)	N12—C13—C14—C15	−172.8 (3)
O8—C4—C5—C6	−28.9 (6)	O18—C14—C15—C16	−22.6 (5)
C3—C4—C5—C6	−148.5 (5)	C13—C14—C15—C16	−143.3 (4)
C4—C5—C6—C7	21.4 (7)	C14—C15—C16—C17	21.4 (7)
C5—C6—C7—O8	−6.3 (7)	C15—C16—C17—O18	−13.3 (8)
C6—C7—O8—C4	−12.9 (6)	C13—C14—O18—C17	138.2 (4)
C3—C4—O8—C7	146.8 (4)	C15—C14—O18—C17	15.0 (5)
C5—C4—O8—C7	25.9 (5)	C16—C17—O18—C14	−1.6 (7)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
N2—H2···O18ⁱ	0.85 (1)	2.10 (2)	2.897 (3)	157 (3)
N12—H12···O8ⁱⁱ	0.86 (1)	2.20 (2)	2.978 (3)	150 (3)

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

Experimental details

Crystal data
Chemical formula	C₁₁H₂₀N₂O₂S
M_r	244.35
Crystal system, space group	Orthorhombic, P2₁2₁2₁
Temperature (K)	298
a, b, c (Å)	7.8588 (9), 10.8265 (11), 15.6196 (16)
V (Å³)	1329.0 (2)
Z	4
Radiation type	Mo Kα
µ (mm⁻¹)	0.23
Crystal size (mm)	0.6 × 0.6 × 0.6

Data collection
Diffractometer	Siemens P4 diffractometer
Absorption correction	ψ scan (XSCANS; Siemens, 1996)
T_min, T_max	0.782, 0.870
No. of measured, independent and observed [I > 2σ(I)] reflections	4611, 3026, 2484
R_int	0.031
(sin θ/λ)_max (Å⁻¹)	0.650

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.054, 0.165, 1.03
No. of reflections	3026
No. of parameters	151
No. of restraints	2
H-atom treatment	H atoms treated by a mixture of independent and constrained refinement
Δρ_max, Δρ_min (e Å⁻³)	0.22, −0.16
Absolute structure	Flack (1983), 1267 Friedel pairs
Absolute structure parameter	−0.01 (14)

Computer programs: XSCANS (Siemens, 1996), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), Mercury (Macrae et al., 2006).

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
N2—H2···O18ⁱ	0.85 (1)	2.095 (17)	2.897 (3)	157 (3)
N12—H12···O8ⁱⁱ	0.86 (1)	2.197 (18)	2.978 (3)	150 (3)

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

Acknowledgements

Partial support from VIEP-UAP (grant No. GUPJ-NAT08-G) is acknowledged.

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