A single-strand helical coordination polymer: catena-poly­[[[tri­aqua­nickel(II)]-μ-2,2′-bi­pyridine-3,3′-di­carboxyl­ato-κ³N,N′:O] monohydrate]

The asymmetric unit of the title compound, {[Ni(C₁₂H₆N₂O₄)(H₂O)₃]·H₂O}_n, is composed of a lattice water mol­ecule and a nickel(II) ion that is coordinated by three water mol­ecules and the two N atoms of a 2,2′-bi­pyridine-3,3′-di­carboxyl­ate ligand. The twist of the 2,2′-bi­pyridine-3,3′-di­carboxyl­ate unit and the coordination of one carboxyl­ate group to a symmetry-related Ni^II atom generate a helical chain that runs along the b axis. Intrahelical hydrogen bonds participate in controlling the orientation of the helices, and both right-handed and left-handed helices are connected by interhelical hydrogen bonds into two-dimensional sheets.

S1. Comment

Helical structures have attracted considerable attention in diverse fields, including biochemistry and materials science (Engelkamp et al., 1999; Seo et al., 2000; Gangopadhyay et al., 2001). In artificial supramolecular architectures, helicity can be introduced by conformational restrictions of inter- or intramolecular hydrogen bonds and coordination to metal ions (Piguet et al., 1997). For supramolecular coordination compounds, the formation of helices depends on the nature of the metal and ligands. It is well known that a chiral structure can often lead to the formation of a helical structure (Albrecht, 2001). Some 2,2',6,6'-substituted biphenyl compounds and 1,1'-binaphthalene derivatives can be twisted; in a chiral structure, because of the steric effect of substituting groups, these compounds can also form a helical structure when the molecules are connected in an appropriate way (Minuti et al., 1999; Hamblin et al., 2002). The same is true of some 2,2'-bipyridine derivatives when they are coordinated to metal ions (Tynan et al., 2003; Kaes et al., 2000). We report here the structure of a one-dimensional helical coordination polymer, the title compound, (I).

As shown in Fig. 1, the asymmetric unit of (I) consists of an Ni^II ion, a 2,2'-bipyridine-3,3'-dicarboxylate dianion, three coordinated water molecules and a non-coordinated water molecule. The coordination sphere of atom Ni1 can be described as a slightly distorted octahedron, involving three water O atoms (O1W, O2W and O3W), a fourth oxygen atom, O3ⁱ [symmetry code: (i) 1.5 − x, 0.5 + y, 1.5 − z], from a coordinated carboxyl group, and two N atoms (N1 and N2) from 2,2'-bipyridine unit, which acts as a chelating ligand. The Ni1—O bond distances are in the range 2.037 (3)–2.092 (3) Å. The 2,2-bipyridine unit is twisted with an N1—C5—C6—N2 torsion angle of 29.0 (4)°, such that the two carboxylate groups are located trans to each other. The C12/O3/O4 and C11/O1/O2 carboxyl groups are also twisted from the pyridyl plane, with dihedral angles of 37.4 (5) and 42.3 (3)°, respectively, between the N1/C1/C2/C3/C4/C5 and N2/C6/C7/C8/C9/C10 pyridyl and the carboxyl planes. Similar distortions of the ligand molecule are observed in some transition metal complexes (Ravixumar et al., 1995; Zhang et al., 2002).

One of the two carboxylate groups is bound to the nickel(II) ion in a monodentate fashion in the axial direction of the distorted octahedron, while the base plane defined by atoms O1W/O3W/N1/N2 is an equatorial plane. Furthermore, the twisted 2,2'-bipyridine-Ni chelating units (C5/C1/C2/C3/C4/N1/Ni1/N2/C6/C7/C8/C9/C10) pack in an alternate reverse order with one another. Therefore, the connections of the coordinated carboxyl O atoms generate a helix that propagates along the b axis. There are two types of helices, viz. right-handed and left-handed, which are related by an inversion center.

All uncoordinated carboxylate O atoms are linked to water molecules. The same types of helices are connected by five hydrogen bonds, viz. O1W—H1WA···O4Wⁱ, O4W—H4WA···O1ⁱⁱ, O4W—H4WB···O4ⁱⁱ, O3W—H3WA···O2ⁱⁱ and O2W_H2WA···O4ⁱⁱ [symmetry codes: (i) x, y, z; (ii) x − 1, y, z]. Thus, the lattice water molecule serves as a bridge, together with another two direct interhelical hydrogen bonds, that joins adjacent helices of the same type, so forming a two-dimensional layer that runs along the a axis. Moreover, there are three intrahelical hydrogen bonds, viz. O3W–H3WB···O4ⁱⁱⁱ, O1W—H1WB···O2ⁱⁱⁱ and O2W—H2WB···O1^iv [symmetry codes: (iii) 1.5 − x, 0.5 + y, 1.5 − z; (iv) 1.5 − x, −0.5 + y, 1.5 − z], which connect the main asymmetric units (without the lattice water molecule) and particpate in controlling the orientation of the helix. In the lattice, the right-handed layers and the left-handed layers alternate along the c axis via the intermolecular interactions, as shown in Fig. 2.

S2. Experimental

Ni(ClO₄)₂·6H₂O (83.9 mg, 0.23 mmol), 2,2'-bipryidine-3,3'-dicarboxylic acid (24.3 mg, 0.10 mmol) and pyridine (0.2 ml) were dissolved in a mixture of water (7 ml) and ethanol (2.5 ml), and the mixture was placed in a Teflon-lined stainless steel vessel (25 ml). The vessel was sealed and heated at 413 K for 72 h under autogeneous pressure and then cooled to room temperature. Large blue rod-like crystals were collected by filtration, followed by washing with water and ethanol (yield ca 46%).

S3. Refinement

H atoms bonded to C atoms were introduced at calculated positions and treated as riding, with U_iso(H) values equal to 1.2 U_eq(C) and C—H distances of 0.93 Å. All water H atoms were located from difference maps at the final stages of the refinement and were refined freely.

Computing details

Data collection: CAD-4 Software (Enraf–Nonius, 1989); cell refinement: CAD-4 Software; data reduction: XCAD4 (Harms & Wocadlo, 1995); program(s) used to solve structure: SHELXS97 (Sheldrick, 1997); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997); molecular graphics: Bruker SHELXTL (Sheldrick, 2000); software used to prepare material for publication: Bruker SHELXTL.

Figures

	Fig. 1. A view of the molecular structure of (I), showing the asymmetric unit (solid line portion), with displacement ellipsoids at the 30% probability level. [Symmetry code: (i) 1.5 − x, 0.5 + y, 1.5 − z]
	Fig. 2. The crystal packing of the two types of two-dimensional sheet formed by the hydrogen-bond connections to the helices. H atoms have been omitted for clarity.

catena-Poly[[[triaquanickel(II)]-µ-2,2'-bipyridine-3,3'- dicarboxylato-κ³N,N':O)] monohydrate]

Crystal data

[Ni(C12H6N2O4)(H2O)3]·H2O	F(000) = 768
Mr = 372.96	Dx = 1.714 Mg m^−3
Monoclinic, P21/n	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2yn	Cell parameters from 25 reflections
a = 9.931 (2) Å	θ = 2.2–14.5°
b = 9.1780 (18) Å	µ = 1.39 mm^−1
c = 15.970 (3) Å	T = 298 K
β = 96.71 (3)°	Rod, blue
V = 1445.6 (5) Å^3	0.32 × 0.15 × 0.10 mm
Z = 4

Data collection

Enraf–Nonius CAD-4 diffractometer	1795 reflections with I > 2σ(I)
Radiation source: fine-focus sealed tube	Rint = 0.047
Graphite monochromator	θmax = 25.0°, θmin = 2.3°
ω scans	h = 0→11
Absorption correction: ψ scan XCAD4 (Harms & Wocadlo, 1995)	k = 0→10
Tmin = 0.583, Tmax = 0.870	l = −18→18
2672 measured reflections	3 standard reflections every 200 reflections
2518 independent reflections	intensity decay: 1.0%

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)] = 0.037	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.090	H atoms treated by a mixture of independent and constrained refinement
S = 1.03	w = 1/[σ^2(Fo^2) + (0.0201P)^2 + 2.1522P] where P = (Fo^2 + 2Fc^2)/3
2518 reflections	(Δ/σ)max < 0.001
240 parameters	Δρmax = 0.61 e Å^−3
0 restraints	Δρmin = −0.37 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. 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
Ni1	0.46848 (4)	0.67859 (5)	0.74223 (3)	0.02550 (15)
O1	0.9869 (3)	0.7579 (4)	0.7542 (2)	0.0524 (9)
O2	1.1053 (2)	0.5967 (3)	0.8365 (2)	0.0468 (8)
O3	0.9243 (2)	0.3620 (3)	0.71758 (15)	0.0284 (6)
O4	1.0402 (3)	0.4612 (4)	0.6212 (2)	0.0668 (11)
N1	0.6030 (3)	0.6493 (3)	0.65512 (17)	0.0243 (7)
N2	0.6256 (3)	0.5668 (3)	0.81322 (18)	0.0237 (7)
C1	0.5721 (4)	0.6659 (4)	0.5721 (2)	0.0337 (9)
H1	0.4874	0.7030	0.5517	0.040*
C2	0.6611 (5)	0.6299 (5)	0.5156 (3)	0.0461 (11)
H2	0.6396	0.6473	0.4582	0.055*
C3	0.7832 (4)	0.5675 (5)	0.5465 (3)	0.0416 (11)
H3	0.8444	0.5407	0.5094	0.050*
C4	0.8152 (4)	0.5444 (4)	0.6316 (2)	0.0286 (9)
C5	0.7236 (3)	0.5930 (4)	0.6860 (2)	0.0220 (8)
C6	0.7440 (3)	0.5836 (4)	0.7796 (2)	0.0215 (8)
C7	0.8660 (3)	0.5948 (4)	0.8309 (2)	0.0254 (8)
C8	0.8661 (4)	0.5670 (4)	0.9163 (2)	0.0350 (10)
H8	0.9474	0.5692	0.9517	0.042*
C9	0.7472 (4)	0.5361 (5)	0.9493 (2)	0.0404 (10)
H9	0.7474	0.5117	1.0058	0.048*
C10	0.6284 (4)	0.5427 (4)	0.8960 (2)	0.0342 (9)
H10	0.5468	0.5298	0.9183	0.041*
C11	0.9963 (4)	0.6525 (4)	0.8029 (3)	0.0327 (9)
C12	0.9375 (4)	0.4481 (4)	0.6590 (3)	0.0332 (9)
O1W	0.3435 (3)	0.8071 (4)	0.6634 (2)	0.0513 (9)
H1WA	0.266 (5)	0.794 (5)	0.646 (3)	0.044 (14)*
H1WB	0.363 (5)	0.897 (6)	0.663 (3)	0.069 (18)*
O2W	0.3574 (3)	0.4942 (4)	0.6993 (2)	0.0397 (7)
H2WA	0.276 (5)	0.479 (6)	0.708 (3)	0.068 (18)*
H2WB	0.400 (5)	0.423 (6)	0.719 (3)	0.063 (18)*
O3W	0.3568 (3)	0.6930 (4)	0.8421 (2)	0.0400 (8)
H3WA	0.279 (5)	0.676 (6)	0.832 (3)	0.062 (17)*
H3WB	0.373 (6)	0.777 (7)	0.862 (4)	0.09 (2)*
O4W	0.0888 (4)	0.7629 (5)	0.5948 (3)	0.0615 (11)
H4WA	0.049 (6)	0.766 (7)	0.642 (4)	0.10 (3)*
H4WB	0.092 (7)	0.680 (8)	0.579 (5)	0.12 (3)*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Ni1	0.0181 (2)	0.0243 (3)	0.0342 (3)	0.0004 (2)	0.00356 (18)	−0.0029 (2)
O1	0.0380 (17)	0.0426 (18)	0.077 (2)	−0.0069 (14)	0.0075 (17)	0.0191 (18)
O2	0.0209 (14)	0.0366 (17)	0.080 (2)	−0.0014 (13)	−0.0053 (14)	0.0018 (17)
O3	0.0282 (13)	0.0266 (15)	0.0321 (14)	0.0053 (11)	0.0110 (11)	0.0055 (12)
O4	0.060 (2)	0.061 (2)	0.090 (3)	0.0338 (18)	0.054 (2)	0.041 (2)
N1	0.0243 (15)	0.0254 (18)	0.0225 (15)	0.0024 (13)	0.0000 (12)	−0.0003 (13)
N2	0.0218 (15)	0.0241 (17)	0.0260 (17)	0.0006 (13)	0.0059 (13)	0.0016 (13)
C1	0.037 (2)	0.035 (2)	0.027 (2)	0.0047 (19)	−0.0024 (17)	0.0041 (19)
C2	0.062 (3)	0.052 (3)	0.024 (2)	0.010 (2)	0.005 (2)	0.006 (2)
C3	0.054 (3)	0.043 (3)	0.032 (2)	0.014 (2)	0.019 (2)	0.003 (2)
C4	0.033 (2)	0.023 (2)	0.031 (2)	0.0040 (16)	0.0106 (17)	0.0073 (17)
C5	0.0233 (18)	0.0202 (19)	0.0231 (18)	0.0006 (15)	0.0058 (15)	0.0024 (15)
C6	0.0212 (17)	0.0166 (17)	0.0271 (19)	0.0010 (14)	0.0038 (15)	−0.0011 (15)
C7	0.0244 (18)	0.0203 (19)	0.031 (2)	0.0027 (15)	−0.0003 (16)	0.0009 (16)
C8	0.029 (2)	0.042 (2)	0.031 (2)	0.0013 (18)	−0.0075 (17)	−0.0011 (19)
C9	0.048 (3)	0.053 (3)	0.021 (2)	0.006 (2)	0.0069 (18)	0.006 (2)
C10	0.034 (2)	0.042 (3)	0.029 (2)	−0.0019 (19)	0.0114 (17)	0.0010 (19)
C11	0.026 (2)	0.026 (2)	0.046 (2)	−0.0057 (17)	0.0054 (18)	−0.0051 (19)
C12	0.036 (2)	0.026 (2)	0.040 (2)	0.0078 (18)	0.0169 (19)	−0.0004 (19)
O1W	0.0316 (18)	0.036 (2)	0.080 (3)	−0.0014 (16)	−0.0180 (16)	0.0047 (19)
O2W	0.0295 (17)	0.0335 (18)	0.055 (2)	−0.0044 (15)	0.0003 (15)	−0.0047 (15)
O3W	0.0233 (16)	0.0378 (18)	0.061 (2)	−0.0074 (14)	0.0152 (14)	−0.0147 (17)
O4W	0.042 (2)	0.057 (3)	0.085 (3)	−0.0080 (18)	0.003 (2)	0.015 (2)

Geometric parameters (Å, º)

Ni1—O1W	2.037 (3)	C3—H3	0.9300
Ni1—O3W	2.051 (3)	C4—C5	1.403 (5)
Ni1—O3^i	2.055 (3)	C4—C12	1.524 (5)
Ni1—N1	2.056 (3)	C5—C6	1.487 (5)
Ni1—N2	2.088 (3)	C6—C7	1.385 (5)
Ni1—O2W	2.092 (3)	C7—C8	1.388 (5)
O1—C11	1.239 (5)	C7—C11	1.513 (5)
O2—C11	1.259 (4)	C8—C9	1.377 (6)
O3—C12	1.244 (4)	C8—H8	0.9300
O3—Ni1^ii	2.055 (2)	C9—C10	1.373 (5)
O4—C12	1.250 (4)	C9—H9	0.9300
N1—C1	1.334 (5)	C10—H10	0.9300
N1—C5	1.344 (4)	O1W—H1WA	0.80 (4)
N2—C10	1.338 (5)	O1W—H1WB	0.85 (6)
N2—C6	1.357 (4)	O2W—H2WA	0.85 (5)
C1—C2	1.375 (6)	O2W—H2WB	0.83 (5)
C1—H1	0.9300	O3W—H3WA	0.78 (5)
C2—C3	1.379 (6)	O3W—H3WB	0.84 (6)
C2—H2	0.9300	O4W—H4WA	0.89 (7)
C3—C4	1.377 (5)	O4W—H4WB	0.81 (7)
O1W—Ni1—O3W	95.58 (14)	N1—C5—C4	120.7 (3)
O1W—Ni1—O3^i	88.56 (13)	N1—C5—C6	113.7 (3)
O3W—Ni1—O3^i	90.69 (12)	C4—C5—C6	125.6 (3)
O1W—Ni1—N1	93.15 (13)	N2—C6—C7	120.9 (3)
O3W—Ni1—N1	171.13 (12)	N2—C6—C5	112.4 (3)
O3^i—Ni1—N1	87.96 (11)	C7—C6—C5	126.7 (3)
O1W—Ni1—N2	169.32 (13)	C6—C7—C8	117.8 (3)
O3W—Ni1—N2	92.71 (12)	C6—C7—C11	124.8 (3)
O3^i—Ni1—N2	84.60 (11)	C8—C7—C11	116.9 (3)
N1—Ni1—N2	78.44 (11)	C9—C8—C7	120.8 (4)
O1W—Ni1—O2W	90.33 (15)	C9—C8—H8	119.6
O3W—Ni1—O2W	89.70 (14)	C7—C8—H8	119.6
O3^i—Ni1—O2W	178.85 (12)	C10—C9—C8	117.9 (4)
N1—Ni1—O2W	91.82 (12)	C10—C9—H9	121.1
N2—Ni1—O2W	96.46 (13)	C8—C9—H9	121.1
C12—O3—Ni1^ii	131.2 (2)	N2—C10—C9	122.5 (4)
C1—N1—C5	120.0 (3)	N2—C10—H10	118.8
C1—N1—Ni1	124.4 (2)	C9—C10—H10	118.8
C5—N1—Ni1	115.0 (2)	O1—C11—O2	125.7 (4)
C10—N2—C6	119.4 (3)	O1—C11—C7	117.0 (3)
C10—N2—Ni1	123.1 (2)	O2—C11—C7	117.0 (4)
C6—N2—Ni1	110.9 (2)	O3—C12—O4	126.1 (4)
N1—C1—C2	122.2 (4)	O3—C12—C4	115.7 (3)
N1—C1—H1	118.9	O4—C12—C4	118.3 (4)
C2—C1—H1	118.9	Ni1—O1W—H1WA	129 (3)
C1—C2—C3	118.1 (4)	Ni1—O1W—H1WB	116 (3)
C1—C2—H2	121.0	H1WA—O1W—H1WB	111 (5)
C3—C2—H2	121.0	Ni1—O2W—H2WA	124 (4)
C4—C3—C2	120.6 (4)	Ni1—O2W—H2WB	107 (4)
C4—C3—H3	119.7	H2WA—O2W—H2WB	105 (5)
C2—C3—H3	119.7	Ni1—O3W—H3WA	115 (4)
C3—C4—C5	118.1 (3)	Ni1—O3W—H3WB	105 (4)
C3—C4—C12	117.2 (3)	H3WA—O3W—H3WB	113 (5)
C5—C4—C12	124.1 (3)	H4WA—O4W—H4WB	109 (7)
O1W—Ni1—N1—C1	−22.0 (3)	C3—C4—C5—C6	−177.1 (4)
O3^i—Ni1—N1—C1	−110.4 (3)	C12—C4—C5—C6	12.4 (6)
N2—Ni1—N1—C1	164.7 (3)	C10—N2—C6—C7	−8.3 (5)
O2W—Ni1—N1—C1	68.5 (3)	Ni1—N2—C6—C7	144.0 (3)
O1W—Ni1—N1—C5	166.2 (3)	C10—N2—C6—C5	173.6 (3)
O3^i—Ni1—N1—C5	77.8 (2)	Ni1—N2—C6—C5	−34.1 (3)
N2—Ni1—N1—C5	−7.1 (2)	N1—C5—C6—N2	29.0 (4)
O2W—Ni1—N1—C5	−103.4 (3)	C4—C5—C6—N2	−148.6 (4)
O1W—Ni1—N2—C10	135.5 (7)	N1—C5—C6—C7	−149.0 (3)
O3W—Ni1—N2—C10	−5.4 (3)	C4—C5—C6—C7	33.5 (6)
O3^i—Ni1—N2—C10	85.1 (3)	N2—C6—C7—C8	9.2 (5)
N1—Ni1—N2—C10	174.1 (3)	C5—C6—C7—C8	−173.0 (4)
O2W—Ni1—N2—C10	−95.4 (3)	N2—C6—C7—C11	−163.0 (3)
O1W—Ni1—N2—C6	−15.6 (8)	C5—C6—C7—C11	14.8 (6)
O3W—Ni1—N2—C6	−156.5 (2)	C6—C7—C8—C9	−3.0 (6)
O3^i—Ni1—N2—C6	−66.1 (2)	C11—C7—C8—C9	169.9 (4)
N1—Ni1—N2—C6	23.0 (2)	C7—C8—C9—C10	−3.9 (6)
O2W—Ni1—N2—C6	113.5 (2)	C6—N2—C10—C9	0.9 (6)
C5—N1—C1—C2	−1.4 (6)	Ni1—N2—C10—C9	−147.8 (3)
Ni1—N1—C1—C2	−172.8 (3)	C8—C9—C10—N2	5.1 (6)
N1—C1—C2—C3	3.6 (7)	C6—C7—C11—O1	38.7 (6)
C1—C2—C3—C4	−1.1 (7)	C8—C7—C11—O1	−133.5 (4)
C2—C3—C4—C5	−3.2 (6)	C6—C7—C11—O2	−146.8 (4)
C2—C3—C4—C12	168.0 (4)	C8—C7—C11—O2	40.9 (5)
C1—N1—C5—C4	−3.2 (5)	Ni1^ii—O3—C12—O4	−17.2 (7)
Ni1—N1—C5—C4	168.9 (3)	Ni1^ii—O3—C12—C4	162.3 (2)
C1—N1—C5—C6	179.1 (3)	C3—C4—C12—O3	−139.6 (4)
Ni1—N1—C5—C6	−8.8 (4)	C5—C4—C12—O3	31.0 (6)
C3—C4—C5—N1	5.5 (6)	C3—C4—C12—O4	39.9 (6)
C12—C4—C5—N1	−165.1 (3)	C5—C4—C12—O4	−149.4 (4)

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

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
O4W—H4WB···O4^iii	0.81 (7)	2.20 (8)	2.850 (5)	139 (7)
O4W—H4WA···O1^iii	0.89 (7)	1.97 (7)	2.846 (6)	171 (6)
O3W—H3WB···O4^i	0.84 (6)	1.91 (6)	2.704 (5)	159 (6)
O3W—H3WA···O2^iii	0.78 (5)	1.89 (5)	2.641 (4)	162 (5)
O2W—H2WA···O4^iii	0.85 (5)	2.58 (5)	3.263 (5)	138 (4)
O2W—H2WB···O1^ii	0.83 (5)	1.90 (5)	2.716 (5)	169 (5)
O1W—H1WA···O4W	0.80 (4)	1.87 (5)	2.668 (5)	175 (5)
O1W—H1WB···O2^i	0.85 (6)	1.86 (6)	2.707 (5)	176 (5)

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