[gamma]-Modification of poly[(N,N-diethyl­dithio­carbamato)silver(I)]

In the title compound, catena-poly[[tri­silver(I)-tri-[mu]₃-N,N-diethyl­dithio­carbamato-3'[kappa]S:1[kappa]S':2[kappa]S;1[kappa]S:2[kappa]S':3[kappa]S;2[kappa]S:3[kappa]²S,S':1'[kappa]S'], [Ag₃(C₅H₁₀NS₂)₃]_n, the trigonally and tetra­hedrally coordinated Ag atoms are [mu]₃-bridged by [kappa]³- and [kappa]⁴-S₂CNEt₂ ligands to form a ribbon structure along the c axis. There is a twofold axis parallel to the b axis and passing through the tetra­hedrally coordinated Ag atom. The S₂CNEt₂ ligands coordinate the Ag atoms in [eta]¹,[eta]²- and [eta]²,[eta]²-fashions, depending on the bridging S atoms. The distances between the trigonal Ag and S atoms are 2.4915 (11)-2.6205 (11) Å, while those between the tetra­hedral Ag and S atoms are 2.5457 (11) and 2.7145 (10) Å. The shortest Ag...Ag distance between trigonal Ag atoms is 2.8336 (7) Å, which indicates a weak Ag...Ag inter­action, whereas the shortest distance between trigonal and tetra­hedral Ag atoms is 3.463 (6) Å, which is considered as non-bonding.

S1. Comment

Dithiocarbamates (R₂NCS₂⁻) are a versatile class of monoanionic 1,1'-dithio ligand and, as they are easily prepared, a wide range of chemistry has been developed around them (Coucouvanis, 1979). The chemistry of transition metal–dithiocarbamate complexes is still of much current interest becasue of their applications in inorganic analysis, such as separation of different metal ions by high-performance liquid chromatography (Liśka et al., 1979) and capillary gas chromatography (Riekkola, 1982), and as rubber vulcanization accelerators, fungicides and pesticides (Hogarth, 2005). A large number of transition metal–dithiocarbamate complexes have been synthesized to date and have been reported to exhibit very abundant reactivities and a variety structures (Hogarth, 2005).

Simple silver(I)–dithiocarbamate complexes, [Ag(S₂CNR₂)]_n, have been known since the 1950 s (Akerström, 1959), yet in the intervening years little further work had been carried out. The low solubility of these complexes probably hindered research in this field. Early crystallographic studies revealed the hexameric nature of Ag—S₂CNR₂ (R = Et and Pr) complexes in the solid state (Hesse, 1963), although in solution they are believed to be in equilibrium with monomeric species. Recently, the silver–N,N-diethyldithiocarbamate complex has been the subject of two further crystallographic studies. The known monoclinic α-modification, (I), characterized by X-ray powder diffraction data, is hexameric and consists of a distorted octahedron of Ag atoms (Hesse & Nilson, 1969). The dithiocarbamate anions cap six of the faces in a η¹,η²-fashion; the remaining two faces with long silver–silver interactions remain uncapped. The β-modification, (II), characterized by single-crystal X-ray diffraction, exhibits a polymeric chain structure in which each Ag atom is coordinated in a distorted fashion by three dithiocarbamate ligands (Anacker-Eickhoff et al., 1982), one acting as a chelate and the others as µ₂-bridges. Although the reaction of the hexameric α-modification, [Ag(S₂CNEt₂)]₆, with (SCN)₂ resulted in the formation of the postulated products [Ag₆(S₂CNEt₂)₅(SCN)] and [Ag₆(S₂CNEt₂)₆(SCN)₄] (Calabro et al., 1981), poor solubility in common organic solvents made the characterization of the complexes possible only on the basis of spectroscopic data; accordingly, no X-ray crystallographic study was reported. Huang et al. (1992) prepared a polynuclear silver(I)–diethyldithiocarbamate cluster, [Ag₁₁S(S₂CNEt₂)₉], with a centered µ₅-S atom. One of the authors also isolated and crystallographically characterized an Se analog, [Ag₁₁(µ₅-Se)(µ₃-S₂CNEt₂)₆(µ₄-S₂CNEt₂)₃], which contains six triply and three quadruply bridging dithiocarbamate ligands (Zhang et al., 1998). We have maintained an interest in silver(I)–dithiocarbamate complexes and we report here the structure of the title compound, (III), which is the γ-modification of [Ag(S₂CNEt₂)]_n.

Compound (III) has a polymeric structure (Fig. 1). There is a twofold axis through atoms Ag2, C2 and N2. The trigonally and tetrahedrally coordinated Ag atoms are bridged by µ₃– and µ₄-S₂CNEt₂ ligands to form a ribbon structure along the c axis (Fig. 2). Similar to (I) (Hesse & Nilson, 1969) and (II) (Anacker-Eickhoff et al., 1982), the formula of (III) is also defined as [Ag(S₂CNEt₂)]_n on the basis of X-ray analysis, together with microanalyses and spectroscopy. We conclude that it is reasonable to name complex (III), which has a new structural mode, as the γ-modification of [Ag(S₂CNEt₂)]_n. The S₂CNEt₂ ligands coordinate the Agr atoms in η¹,η²– and η²,η²-fashions, depending on the bridging S atoms.

The distances between the trigonal Ag (Ag1) and S atoms are 2.4915 (11)–2.6205 (11) Å (Table 1), and the S—Ag1—S bond angles are 109.49 (4)–128.56 (3)°, indicating a highly distorted coordination geometry around atom Ag1. The tetrahedral Ag atom (Ag2) is bound by two µ₃-S atoms from two η¹,η²-S₂CNEt₂ ligands and chelated by two µ₃-S atoms from one η²,η²-S₂CNEt₂ ligand. The coordination geometry of atom Ag2 is severely distorted, as observed in other silver–dithiocarbamate complexes with tetrahedral Ag atoms, such as [Ag(S₂CNC₄H₉)(PPh₃)₂] (Othman et al., 1996) and [Ag(S₂CNEt₂)(PFc₂Ph)] (Fc is ferrocenyl; Gimeno et al., 1998). The distortion arises from the restricted bite angle of S2—Ag2—S2C [67.15 (4)°; symmetry codes A–C are as in Fig. 1]. There is also a large deviation of the S1A—Ag2—S1B angle [125.84 (4)°] from the ideal tetrahedral value. Each tetrahedral Ag atom has a pair of long and short Ag—S bonds [Ag2—S2 = 2.7145 (10) Å and Ag2—S1A = 2.5457 (11) Å]; interatomic Ag···S distances longer than 2.80 Å are considered to be non-bonding. The Ag···Ag distance between two adjacent trigonal Ag atoms [2.8336 (7) Å] is shorter than twice the van der Waals radius (1.7 Å) of Ag atoms, indicating a weak Ag···Ag interaction, which is obviously compatible with the Ag···Ag distance in metallic silver [2.889 (6) Å]. The reported Ag···Ag distances where bonding is considered to occur are in the range 2.729–3.065 Å (Kanatzidis & Huang, 1989; Huang et al., 1992; Yam et al., 1996; Zhang et al., 1998). The separation of adjacent trigonal and tetrahedral Ag atoms is 3.463 (6) Å, indicating that these atoms are non-bonded. The C—S bond lengths [1.714 (4)–1.751 (3) Å] are comparable to those in related silver–dithiocarbamate complexes. The C1—N1 and C2—N2 bond lengths are 1.325 (4) and 1.322 (6) Å, respectively, suggesting considerable partial double-bond character because the N-atom lone pair is involved in delocalized π-bonding over the NCS₂ group (Eisenberg, 1970).

S2. Experimental

Compound (III) was prepared by mixing stoichiometeric quantities of NaS₂CNEt₂·3H₂O (225 mg, 0.10 mmol) and Ag(CF₃SO₃) (257 mg, 0.10 mmol) in absolute ethanol. The mixture was stirred for 15 min at room temperature. The resulting brown precipitate was filtered and further recrystallized from DMF/THF (1:5 v/v). Dark-red crystals of (III) were obtained after a couple of weeks (yield 116 mg, 45%). ¹H NMR (DMSO-d⁶): δ 1.23 (m, CH₃), 3.54 (m, CH₂). MS (FAB): m/z 256 (M⁺ + 1). IR (KBr pepplt, cm⁻¹): ν(C═N), 1479 (s); ν(C—S) 990 (s), 917(m). Analysis calculated for C₉H₃₀Ag₃N₃S₆: C 23.35, H 3.94, N 5.47%; found: C 23.25, H 3.91, N 5.43%.

S3. Refinement

All H atoms were found in difference density maps, but were then placed in calculated positions (C—H = 0.96–0.97%A) and included in the refinement using a riding-model approximation, with U_iso(H) = 1.2U_eq(C). The largest peak in the final difference maps is in the vicinity of the Ag atom.

Computing details

Data collection: SMART (Bruker, 1998); cell refinement: SAINT-Plus (Bruker, 1998); data reduction: SAINT-Plus; 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.

Figures

	Fig. 1. A view of the structure of (III), indicating atom displacement ellipsoids at the 50% probability level. H atoms have been omitted for clarity. [Symmetry codes: (A) 1 − x, 1 − y, 1 − z; (B) x, 1 − y, z + 1/2; (C) 1 − x, y, 3/2 − z.]
	Fig. 2. The crystal structure of (III), projected along the b axis. H atoms have been omitted for clarity.

catena-poly[[trisilver(I)-di-µ₃-N,N-diethyldithiocarbamato- 3'κS:1κS':2κS;1κS:2κS':3κS- µ₄-N,N-diethyldithiocarbamato- 2κS:3κ²S,S':1'κS']

Crystal data

[Ag3(C5H10NS2)3]	F(000) = 1512
Mr = 768.45	Dx = 2.136 Mg m^−3
Monoclinic, C2/c	Mo Kα radiation, λ = 0.71073 Å
a = 18.138 (4) Å	Cell parameters from 2887 reflections
b = 9.5890 (19) Å	θ = 2.3–18.3°
c = 14.245 (3) Å	µ = 2.97 mm^−1
β = 105.30 (3)°	T = 153 K
V = 2389.8 (8) Å^3	Prism, dark red
Z = 4	0.25 × 0.22 × 0.18 mm

Data collection

Bruker SMART CCD area-detector diffractometer	2853 independent reflections
Radiation source: fine-focus sealed tube	2506 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.031
ϕ and ω scans	θmax = 28.3°, θmin = 2.3°
Absorption correction: multi-scan (SADABS; Sheldrick, 1996)	h = −24→23
Tmin = 0.457, Tmax = 0.586	k = −12→12
7241 measured reflections	l = −11→18

Refinement

Refinement on F^2	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F^2 > 2σ(F^2)] = 0.034	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.079	H-atom parameters constrained
S = 1.09	w = 1/[σ^2(Fo^2) + (0.0369P)^2 + 2.307P] where P = (Fo^2 + 2Fc^2)/3
2853 reflections	(Δ/σ)max = 0.001
124 parameters	Δρmax = 1.59 e Å^−3
0 restraints	Δρmin = −0.97 e Å^−3

Special details

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. The data collection covered over a hemisphere of reciprocal space by a combination of three sets of exposures; each set had a different ϕ angle (0, 88 and 180°) for the crystal and each exposure of 20 s covered 0.3° in ω. The crystal-to-detector distance was 4 cm and the detector swing angle was −35°. Coverage of the unique set is over 99% complete. Crystal decay was monitored by repeating fifty initial frames at the end of data collection and analysing the duplicate reflections, and was found to be negligible.

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 (Ų)

	x	y	z	Uiso*/Ueq
Ag1	0.519534 (16)	0.37249 (3)	0.55071 (2)	0.02157 (10)
Ag2	0.5000	0.56129 (4)	0.7500	0.01967 (11)
S1	0.57677 (5)	0.31786 (9)	0.40397 (6)	0.01522 (18)
S2	0.57616 (5)	0.32543 (9)	0.72706 (6)	0.01755 (18)
S3	0.37699 (5)	0.41359 (9)	0.48830 (6)	0.01747 (19)
N1	0.70878 (16)	0.4503 (3)	0.4185 (2)	0.0137 (6)
N2	0.5000	0.0960 (4)	0.7500	0.0150 (8)
C1	0.64259 (19)	0.4536 (3)	0.4414 (2)	0.0149 (7)
C2	0.5000	0.2339 (5)	0.7500	0.0157 (9)
C11	0.76313 (19)	0.5670 (4)	0.4412 (2)	0.0169 (7)
H11A	0.7351	0.6536	0.4396	0.020*
H11B	0.7911	0.5724	0.3920	0.020*
C12	0.81906 (19)	0.5506 (4)	0.5403 (3)	0.0207 (8)
H12A	0.8542	0.6274	0.5520	0.031*
H12B	0.8468	0.4649	0.5422	0.031*
H12C	0.7917	0.5490	0.5895	0.031*
C13	0.7341 (2)	0.3304 (4)	0.3700 (3)	0.0192 (7)
H13A	0.7102	0.2462	0.3856	0.023*
H13B	0.7890	0.3198	0.3948	0.023*
C14	0.7144 (2)	0.3475 (5)	0.2611 (3)	0.0289 (9)
H14A	0.7320	0.2675	0.2326	0.043*
H14B	0.7387	0.4298	0.2452	0.043*
H14C	0.6601	0.3560	0.2360	0.043*
C21	0.56207 (19)	0.0118 (4)	0.7296 (2)	0.0169 (7)
H21A	0.5671	−0.0748	0.7659	0.020*
H21B	0.6100	0.0622	0.7507	0.020*
C22	0.5461 (2)	−0.0203 (4)	0.6220 (3)	0.0213 (8)
H22A	0.5882	−0.0713	0.6101	0.032*
H22B	0.5394	0.0653	0.5858	0.032*
H22C	0.5004	−0.0752	0.6019	0.032*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Ag1	0.02415 (17)	0.02126 (16)	0.02314 (17)	0.00380 (11)	0.01298 (12)	0.00766 (11)
Ag2	0.0214 (2)	0.01563 (19)	0.0194 (2)	0.000	0.00092 (14)	0.000
S1	0.0172 (4)	0.0133 (4)	0.0156 (4)	−0.0013 (3)	0.0050 (3)	−0.0007 (3)
S2	0.0187 (4)	0.0163 (4)	0.0196 (4)	−0.0015 (3)	0.0085 (3)	0.0004 (3)
S3	0.0189 (4)	0.0175 (4)	0.0177 (4)	−0.0013 (3)	0.0077 (3)	−0.0057 (3)
N1	0.0170 (14)	0.0124 (14)	0.0132 (14)	−0.0005 (11)	0.0066 (11)	−0.0003 (11)
N2	0.018 (2)	0.019 (2)	0.0096 (19)	0.000	0.0063 (15)	0.000
C1	0.0158 (16)	0.0144 (16)	0.0131 (15)	0.0008 (13)	0.0016 (12)	0.0023 (13)
C2	0.017 (2)	0.018 (2)	0.012 (2)	0.000	0.0025 (17)	0.000
C11	0.0177 (17)	0.0162 (17)	0.0182 (17)	−0.0040 (13)	0.0070 (13)	0.0029 (14)
C12	0.0177 (18)	0.0196 (18)	0.0250 (19)	−0.0028 (14)	0.0058 (14)	0.0006 (15)
C13	0.0200 (18)	0.0192 (18)	0.0205 (18)	0.0045 (14)	0.0092 (14)	−0.0017 (15)
C14	0.033 (2)	0.038 (2)	0.0184 (19)	0.0036 (17)	0.0111 (16)	−0.0082 (17)
C21	0.0185 (17)	0.0153 (16)	0.0190 (18)	0.0013 (13)	0.0084 (13)	0.0009 (14)
C22	0.0245 (18)	0.0232 (19)	0.0197 (18)	−0.0036 (15)	0.0118 (14)	−0.0058 (15)

Geometric parameters (Å, º)

Ag1—S2	2.4915 (11)	C1—S3^i	1.714 (4)
Ag1—S3	2.5333 (11)	C2—S2^iii	1.739 (3)
Ag1—S1	2.6205 (11)	C11—C12	1.513 (5)
Ag1—Ag1^i	2.8336 (7)	C11—H11A	0.9700
Ag1—S3^i	2.9308 (10)	C11—H11B	0.9700
Ag2—S1^ii	2.5457 (11)	C12—H12A	0.9600
Ag2—S1^i	2.5457 (11)	C12—H12B	0.9600
Ag2—S2^iii	2.7145 (10)	C12—H12C	0.9600
Ag2—S2	2.7145 (10)	C13—C14	1.507 (5)
S1—C1	1.751 (3)	C13—H13A	0.9700
S1—Ag2^i	2.5457 (11)	C13—H13B	0.9700
S2—C2	1.739 (3)	C14—H14A	0.9600
S3—C1^i	1.714 (4)	C14—H14B	0.9600
S3—Ag1^i	2.9308 (10)	C14—H14C	0.9600
N1—C1	1.325 (4)	C21—C22	1.514 (5)
N1—C11	1.470 (4)	C21—H21A	0.9700
N1—C13	1.476 (4)	C21—H21B	0.9700
N2—C2	1.322 (6)	C22—H22A	0.9600
N2—C21^iii	1.476 (4)	C22—H22B	0.9600
N2—C21	1.476 (4)	C22—H22C	0.9600
S2—Ag1—S3	119.54 (4)	N1—C11—C12	111.8 (3)
S2—Ag1—S1	128.56 (3)	N1—C11—H11A	109.3
S3—Ag1—S1	109.49 (4)	C12—C11—H11A	109.3
S2—Ag1—Ag1^i	130.65 (3)	N1—C11—H11B	109.3
S3—Ag1—Ag1^i	65.92 (3)	C12—C11—H11B	109.3
S1—Ag1—Ag1^i	82.20 (2)	H11A—C11—H11B	107.9
S2—Ag1—S3^i	101.75 (3)	C11—C12—H12A	109.5
S3—Ag1—S3^i	118.03 (2)	C11—C12—H12B	109.5
S1—Ag1—S3^i	65.19 (3)	H12A—C12—H12B	109.5
Ag1^i—Ag1—S3^i	52.11 (2)	C11—C12—H12C	109.5
S1^ii—Ag2—S1^i	125.84 (4)	H12A—C12—H12C	109.5
S1^ii—Ag2—S2^iii	117.10 (3)	H12B—C12—H12C	109.5
S1^i—Ag2—S2^iii	107.63 (3)	N1—C13—C14	112.3 (3)
S1^ii—Ag2—S2	107.63 (3)	N1—C13—H13A	109.1
S1^i—Ag2—S2	117.10 (3)	C14—C13—H13A	109.1
S2^iii—Ag2—S2	67.15 (4)	N1—C13—H13B	109.1
C1—S1—Ag2^i	95.63 (11)	C14—C13—H13B	109.1
C1—S1—Ag1	89.44 (12)	H13A—C13—H13B	107.9
Ag2^i—S1—Ag1	110.76 (3)	C13—C14—H14A	109.5
C2—S2—Ag1	98.30 (5)	C13—C14—H14B	109.5
C2—S2—Ag2	86.75 (15)	H14A—C14—H14B	109.5
Ag1—S2—Ag2	83.28 (3)	C13—C14—H14C	109.5
C1^i—S3—Ag1	105.08 (12)	H14A—C14—H14C	109.5
C1^i—S3—Ag1^i	80.35 (12)	H14B—C14—H14C	109.5
Ag1—S3—Ag1^i	61.97 (2)	N2—C21—C22	111.0 (3)
C1—N1—C11	121.8 (3)	N2—C21—H21A	109.4
C1—N1—C13	123.2 (3)	C22—C21—H21A	109.4
C11—N1—C13	115.0 (3)	N2—C21—H21B	109.4
C2—N2—C21^iii	123.19 (19)	C22—C21—H21B	109.4
C2—N2—C21	123.2 (2)	H21A—C21—H21B	108.0
C21^iii—N2—C21	113.6 (4)	C21—C22—H22A	109.5
N1—C1—S3^i	120.1 (3)	C21—C22—H22B	109.5
N1—C1—S1	119.8 (3)	H22A—C22—H22B	109.5
S3^i—C1—S1	120.0 (2)	C21—C22—H22C	109.5
N2—C2—S2	120.32 (14)	H22A—C22—H22C	109.5
N2—C2—S2^iii	120.32 (14)	H22B—C22—H22C	109.5
S2—C2—S2^iii	119.4 (3)
S2—Ag1—S1—C1	−72.66 (11)	S2—Ag1—S3—Ag1^i	124.54 (3)
S3—Ag1—S1—C1	125.33 (11)	S1—Ag1—S3—Ag1^i	−71.58 (3)
Ag1^i—Ag1—S1—C1	64.37 (11)	S3^i—Ag1—S3—Ag1^i	0.0
S3^i—Ag1—S1—C1	12.66 (11)	C11—N1—C1—S3^i	−7.8 (4)
S2—Ag1—S1—Ag2^i	−168.48 (3)	C13—N1—C1—S3^i	171.4 (2)
S3—Ag1—S1—Ag2^i	29.52 (4)	C11—N1—C1—S1	174.9 (2)
Ag1^i—Ag1—S1—Ag2^i	−31.45 (3)	C13—N1—C1—S1	−5.9 (4)
S3^i—Ag1—S1—Ag2^i	−83.16 (4)	Ag2^i—S1—C1—N1	−95.0 (3)
S3—Ag1—S2—C2	33.91 (15)	Ag1—S1—C1—N1	154.2 (3)
S1—Ag1—S2—C2	−126.53 (15)	Ag2^i—S1—C1—S3^i	87.69 (19)
Ag1^i—Ag1—S2—C2	116.36 (15)	Ag1—S1—C1—S3^i	−23.12 (19)
S3^i—Ag1—S2—C2	165.95 (14)	C21^iii—N2—C2—S2	178.98 (16)
S3—Ag1—S2—Ag2	−51.79 (4)	C21—N2—C2—S2	−1.02 (16)
S1—Ag1—S2—Ag2	147.77 (3)	C21^iii—N2—C2—S2^iii	−1.02 (16)
Ag1^i—Ag1—S2—Ag2	30.65 (4)	C21—N2—C2—S2^iii	178.98 (16)
S3^i—Ag1—S2—Ag2	80.24 (3)	Ag1—S2—C2—N2	97.28 (4)
S1^ii—Ag2—S2—C2	112.63 (4)	Ag2—S2—C2—N2	180.0
S1^i—Ag2—S2—C2	−98.84 (4)	Ag1—S2—C2—S2^iii	−82.72 (4)
S2^iii—Ag2—S2—C2	0.0	Ag2—S2—C2—S2^iii	0.000 (1)
S1^ii—Ag2—S2—Ag1	−148.61 (3)	C1—N1—C11—C12	90.4 (4)
S1^i—Ag2—S2—Ag1	−0.08 (4)	C13—N1—C11—C12	−88.9 (4)
S2^iii—Ag2—S2—Ag1	98.76 (4)	C1—N1—C13—C14	94.4 (4)
S2—Ag1—S3—C1^i	54.42 (13)	C11—N1—C13—C14	−86.4 (4)
S1—Ag1—S3—C1^i	−141.70 (12)	C2—N2—C21—C22	−90.5 (3)
Ag1^i—Ag1—S3—C1^i	−70.11 (12)	C21^iii—N2—C21—C22	89.5 (3)
S3^i—Ag1—S3—C1^i	−70.11 (12)

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