catena-Poly[[aqua­(piperazin-1-ium-κN⁴)copper(II)]-μ-5-sulfonatoisophthalato-κO¹:κO³]

Blue crystals of the title compound, [Cu(C₈H₃O₇S)(C₄H₁₁N₂)(H₂O)]_n, were obtained from the slow diffusion of a methanol solution of piperazine into an aqueous solution of sodium 5-sulfo­isophthalate and copper nitrate. The Cu^II ion coordination is composed of two O atoms from two trianionic 5-sulfoisophthalic acid ligands, one N atom from the piperazinium cation and one water O atom in a distorted square-planar geometry. The Cu²⁺ ions are bridged by the sulfonate–carboxyl­ate ligands to form a zigzag chain. Hydrogen bonds and π–π inter­actions among the chains produce a three-dimensional hydrogen-bonded architecture.

Key indicators

• Single-crystal X-ray study
• T = 293 K
• Mean [sigma](C-C) = 0.007 Å
• R factor = 0.065
• wR factor = 0.145
• Data-to-parameter ratio = 14.2

checkCIF/PLATON results

No syntax errors found



Alert level C
PLAT199_ALERT_1_C Check the Reported _cell_measurement_temperature        293 K
PLAT200_ALERT_1_C Check the Reported _diffrn_ambient_temperature .        293 K
PLAT222_ALERT_3_C Large Non-Solvent    H     Ueq(max)/Ueq(min) ...       3.75 Ratio
PLAT341_ALERT_3_C Low Bond Precision on C-C bonds (x 1000) Ang ...          7

   0 ALERT level A = In general: serious problem
   0 ALERT level B = Potentially serious problem
   4 ALERT level C = Check and explain
   0 ALERT level G = General alerts; check

   2 ALERT type 1 CIF construction/syntax error, inconsistent or missing data
   0 ALERT type 2 Indicator that the structure model may be wrong or deficient
   2 ALERT type 3 Indicator that the structure quality may be low
   0 ALERT type 4 Improvement, methodology, query or suggestion

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S1. Comment

In the past few years, efforts have focused on the study of carboxylate-based (such as malonate, oxalate and benzene-1,3,5-tricarboxylate) coordination polymers because of their interesting network topologies (Moulton & Zaworotko, 2001; Virovets et al., 1993; Yaghi et al., 1996). In extending our own studies on benzene-1,3,5-tricarboxylates, we have also investigated the 5-sulfoisophthalate, and recently reported a series of lanthanide complexes of 5-sulfoisophthalic acid that demonstrates a lanthanide contraction effect (Liu & Xu, 2005a,b,c). Several Cd^II and Cu^II complexes of 5-sulfoisophthalic acid have also been reported by others (Sun, Cao, Sun, Bi et al., 2003 or Sun, Cao, Sun, Li et al., 2003; Liu & Xu, 2005a,b,c?).

The self-assembly of the Cu²⁺ ion with the 5-sulfoisophthalate ion (sip) and piperazine (pip) yields the title complex, (I). The asymmetric unit of (I) consists of one copper(II) ion, one sip³⁻ anion, one piperazinium cation (Hpip⁺) and one coordinated water molecule. As depicted in Fig. 1, the compound features a zigzag chain structure, which is similar to the structure of the Cd compound (Liu & Xu, 2005a,b,c?). The Cu atom is four-coordinated by two O atoms from two symmetry-related sip³⁻ ligands, one N atom from Hpip⁺ and one water O atom. The bond dimensions involving Cu are normal (Table 1), and are comparable to the values in related copper complexes (Sun, Cao, Sun, Bi et al., 2003 or Sun, Cao, Sun, Li et al., 2003). The [CuNO₃] group assumes a square planar configuration, with a mean deviation of 0.1922 Å; the maximum deviation of 0.2439 (s.u.?) Å is for atoms O5, which is involved in hydrogen bonding. Piperazine should be protonated to achieve charge balance, which is borne out by the existence of the two hydrogen bonds to atom N2. The Hpip⁺ ion coordinates to the metal center, and is different from the unit found in the copper complex [Cu(Hsip)(H₂O)₂]·2H₂O^.0.5pip, which has a free pip molecule (Sun, Cao, Sun, Bi et al., 2003 or Sun, Cao, Sun, Li et al., 2003).

Each sip³⁻ anion bridges two neighboring Cu²⁺ ions through its two monodentate carboxylate groups (Fig. 1). The sulfonate group is uncoordinated and engages in hydrogen bonding to the coordinated water and Hpip⁺ ion. The three identical S—O bond distances imply extensive conjugation. The connection of the Cu²⁺ ions through the carboxylate groups of sip³⁻ anions leads to the formation of a zigzag chain propagating along b. The phenyl rings in the chain are twisted with respect to each other with a dihedral angle of ca 11.6°.

The extensive hydrogen bonds and π–π stacking interactions among the chains are responsible for the three-dimensional supramolecular framework structure (Figs. 2 and 3.) Four types of hydrogen bonds are observed: hydrogen bonds between (i) the coordinated water molecules and carboxylate O atoms; (ii) the coordinated water molecules and sulfonate O atoms; (iii) the coordinated N atom of Hpip⁺ and carboxylate O atoms; (iv) the non-coordinated N atom of Hpip⁺ and sulfonate O atoms. The Hpip⁺ cations appear to fill up the void of the framework. Additionally, π–π stacking interactions between the parallel aromatic rings of the adjacent sip³⁻ anions in an offset fashion with a face-to-face distance of ca 3.614 Å are observed. Similar hydrogen bonds and π-π interactions are also observed in the related Cd^II and Cu^II complexes (Liu & Xu, 2005a,b,c,; Sun, Cao, Sun, Bi et al., 2003 or Sun, Cao, Sun, Li et al., 2003).

S2. Experimental

An aqueous solution (15 ml) of 5-sulfoisophthalic acid sodium salt (0.134 g, 0.5 mmol) and copper(II) nitrate hemipentahydrate (0.116 g, 0.5 mmol) was layered on top of a methanol solution (10 ml) of piperazine (0.047 g, 0.5 mmol). Blue prism-like crystals were obtained after several days (yield 59%). Analysis calculated for C₁₂H₁₆N₂O₈SCu: C 34.96, H 3.88, N 6.80%; found: C 34.93, H 3.79, N 6.79%.

S3. Refinement

H atoms attached to C atoms were placed in calculated positions and treated using a riding-model approximation (C—H = 0.93 Å for aromatic H atoms and C—H = 0.97 Å for methylene H). The H atoms bonded to O atoms and N atoms were visible in the difference map and were included in the refinement with an O—H distance restraint of 0.90 (s.u.?) Å and an N—H distance restraint of 0.93 (s.u.?) Å.

Computing details

Data collection: CrystalClear (Rigaku, 2000); cell refinement: CrystalClear; data reduction: CrystalClear; 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. The title compound, with displacement ellipsoids drawn at the 35% probability level. H atoms have been omitted.
	Fig. 2. View of the hydrogen bonds (dashed lines) in the title compound. H atoms not involved in hydrogen bonding have been omitted. Symmetry codes are given in Table 2.
	Fig. 3. View of the three-dimensional hydrogen-bonded supramolecular architecture along a. H atoms have been omitted and hydrogen bonds are shown as dashed lines.

catena-Poly[[aqua(piperazin-1-ium-κN⁴)copper(II)]-µ-5- sulfonatoisophthalato-κO¹:κO³]

Crystal data

[Cu(C8H3O7S)(C4H11N2)(H2O)]	F(000) = 844
Mr = 411.87	Dx = 1.859 Mg m^−3
Monoclinic, P21/c	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P2ybc	Cell parameters from 2385 reflections
a = 7.4257 (12) Å	θ = 3.1–27.5°
b = 17.214 (2) Å	µ = 1.67 mm^−1
c = 12.028 (2) Å	T = 293 K
β = 106.863 (7)°	Prism, blue
V = 1471.4 (4) Å^3	0.18 × 0.12 × 0.10 mm
Z = 4

Data collection

Rigaku Mercury70 (2 × 2 bin mode) diffractometer	3374 independent reflections
Radiation source: fine-focus sealed tube	2493 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.069
ω scans	θmax = 27.5°, θmin = 3.1°
Absorption correction: multi-scan (CrystalClear; Rigaku, 2000)	h = −9→8
Tmin = 0.788, Tmax = 0.848	k = −22→14
11319 measured reflections	l = −15→15

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)] = 0.065	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.145	H atoms treated by a mixture of independent and constrained refinement
S = 1.00	w = 1/[σ^2(Fo^2) + (0.0382P)^2 + 8.9617P] where P = (Fo^2 + 2Fc^2)/3
3374 reflections	(Δ/σ)max < 0.001
237 parameters	Δρmax = 0.51 e Å^−3
5 restraints	Δρmin = −0.67 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
Cu1	0.59406 (10)	0.22835 (4)	0.29368 (5)	0.0242 (2)
O1	0.6099 (6)	0.3378 (2)	0.3369 (3)	0.0282 (9)
O2	0.7936 (6)	0.2976 (2)	0.5056 (3)	0.0341 (10)
O3	0.4577 (5)	0.6231 (2)	0.2756 (3)	0.0267 (9)
O4	0.6204 (6)	0.7053 (2)	0.4074 (3)	0.0357 (10)
O5	0.8912 (6)	0.5068 (3)	0.8484 (3)	0.0346 (10)
O6	1.1800 (5)	0.5114 (3)	0.7996 (3)	0.0349 (10)
O7	0.9962 (6)	0.6293 (2)	0.7930 (3)	0.0377 (10)
O8	0.7992 (6)	0.2436 (2)	0.2220 (4)	0.0324 (9)
H8A	0.850 (8)	0.2908 (16)	0.219 (6)	0.042 (19)*
H8B	0.785 (12)	0.221 (4)	0.153 (3)	0.08 (3)*
N1	0.4458 (7)	0.1999 (3)	0.4032 (4)	0.0249 (10)
H1A	0.433 (8)	0.240 (2)	0.452 (4)	0.032 (17)*
N2	0.2286 (7)	0.0898 (3)	0.4926 (4)	0.0339 (12)
H2B	0.157 (7)	0.073 (4)	0.540 (4)	0.040 (18)*
H2C	0.214 (12)	0.050 (3)	0.439 (6)	0.09 (3)*
C1	0.7365 (7)	0.4341 (3)	0.4804 (4)	0.0191 (10)
C2	0.6451 (7)	0.4941 (3)	0.4071 (4)	0.0199 (10)
H2A	0.5696	0.4833	0.3323	0.024*
C3	0.6691 (7)	0.5702 (3)	0.4482 (4)	0.0195 (10)
C4	0.7807 (7)	0.5862 (3)	0.5598 (4)	0.0207 (11)
H4A	0.7968	0.6372	0.5862	0.025*
C5	0.8681 (7)	0.5263 (3)	0.6320 (4)	0.0199 (11)
C6	0.8481 (7)	0.4500 (3)	0.5922 (4)	0.0211 (11)
H6A	0.9091	0.4099	0.6403	0.025*
C7	0.7155 (7)	0.3504 (3)	0.4397 (4)	0.0206 (11)
C8	0.5771 (7)	0.6383 (3)	0.3733 (4)	0.0203 (10)
C9	0.5327 (8)	0.1304 (3)	0.4724 (5)	0.0304 (13)
H9A	0.6619	0.1424	0.5153	0.036*
H9B	0.5349	0.0879	0.4199	0.036*
C10	0.4278 (8)	0.1049 (4)	0.5568 (5)	0.0322 (13)
H10A	0.4847	0.0581	0.5968	0.039*
H10B	0.4352	0.1451	0.6145	0.039*
C11	0.1353 (9)	0.1612 (4)	0.4294 (5)	0.0347 (14)
H11A	0.1373	0.2026	0.4844	0.042*
H11B	0.0052	0.1500	0.3877	0.042*
C12	0.2413 (8)	0.1859 (3)	0.3449 (5)	0.0302 (13)
H12A	0.2285	0.1458	0.2864	0.036*
H12B	0.1852	0.2331	0.3059	0.036*
S1	0.99452 (19)	0.54539 (8)	0.77891 (11)	0.0235 (3)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Cu1	0.0316 (4)	0.0180 (3)	0.0204 (3)	−0.0030 (3)	0.0032 (3)	−0.0032 (3)
O1	0.044 (2)	0.0171 (19)	0.0196 (17)	−0.0015 (17)	0.0028 (17)	−0.0039 (15)
O2	0.048 (3)	0.0156 (19)	0.030 (2)	0.0021 (18)	−0.0013 (19)	0.0020 (17)
O3	0.037 (2)	0.023 (2)	0.0157 (17)	0.0010 (17)	0.0005 (16)	−0.0004 (15)
O4	0.051 (3)	0.016 (2)	0.033 (2)	0.0036 (18)	0.0009 (19)	−0.0007 (17)
O5	0.042 (2)	0.038 (2)	0.027 (2)	−0.003 (2)	0.0139 (18)	−0.0062 (19)
O6	0.026 (2)	0.052 (3)	0.0220 (19)	0.0034 (19)	−0.0002 (16)	0.0034 (19)
O7	0.051 (3)	0.022 (2)	0.031 (2)	−0.0065 (19)	−0.0019 (19)	−0.0093 (18)
O8	0.037 (2)	0.029 (2)	0.031 (2)	−0.0095 (18)	0.0100 (18)	−0.0088 (19)
N1	0.031 (3)	0.016 (2)	0.025 (2)	−0.0024 (19)	0.0033 (19)	−0.0018 (19)
N2	0.042 (3)	0.036 (3)	0.023 (2)	−0.014 (2)	0.008 (2)	−0.002 (2)
C1	0.023 (3)	0.015 (2)	0.019 (2)	0.001 (2)	0.006 (2)	0.005 (2)
C2	0.028 (3)	0.017 (2)	0.017 (2)	0.002 (2)	0.009 (2)	0.000 (2)
C3	0.023 (3)	0.017 (3)	0.018 (2)	0.002 (2)	0.007 (2)	0.004 (2)
C4	0.027 (3)	0.014 (2)	0.021 (2)	0.001 (2)	0.007 (2)	−0.001 (2)
C5	0.024 (3)	0.017 (3)	0.018 (2)	−0.002 (2)	0.006 (2)	0.001 (2)
C6	0.027 (3)	0.018 (3)	0.016 (2)	0.002 (2)	0.004 (2)	0.003 (2)
C7	0.030 (3)	0.018 (3)	0.016 (2)	−0.002 (2)	0.009 (2)	−0.003 (2)
C8	0.023 (3)	0.020 (3)	0.020 (2)	0.004 (2)	0.009 (2)	0.004 (2)
C9	0.032 (3)	0.027 (3)	0.028 (3)	0.000 (2)	0.002 (2)	0.006 (2)
C10	0.043 (4)	0.026 (3)	0.022 (3)	−0.007 (3)	0.001 (2)	0.003 (2)
C11	0.035 (3)	0.034 (3)	0.034 (3)	−0.002 (3)	0.009 (3)	0.001 (3)
C12	0.030 (3)	0.029 (3)	0.028 (3)	−0.002 (2)	0.002 (2)	0.003 (3)
S1	0.0287 (7)	0.0217 (7)	0.0168 (6)	−0.0007 (5)	0.0014 (5)	−0.0023 (5)

Geometric parameters (Å, º)

Cu1—O1	1.949 (4)	C1—C2	1.399 (7)
Cu1—O8	1.974 (4)	C1—C7	1.515 (7)
Cu1—O3^i	1.986 (4)	C2—C3	1.393 (7)
Cu1—N1	2.007 (5)	C2—H2A	0.9300
O1—C7	1.276 (6)	C3—C4	1.386 (7)
O2—C7	1.234 (6)	C3—C8	1.515 (7)
O3—C8	1.277 (6)	C4—C5	1.382 (7)
O3—Cu1^ii	1.986 (4)	C4—H4A	0.9300
O4—C8	1.234 (6)	C5—C6	1.391 (7)
O5—S1	1.449 (4)	C5—S1	1.774 (5)
O6—S1	1.450 (4)	C6—H6A	0.9300
O7—S1	1.453 (4)	C9—C10	1.513 (8)
O8—H8A	0.90 (4)	C9—H9A	0.9700
O8—H8B	0.90 (5)	C9—H9B	0.9700
N1—C9	1.493 (7)	C10—H10A	0.9700
N1—C12	1.496 (7)	C10—H10B	0.9700
N1—H1A	0.93 (4)	C11—C12	1.516 (8)
N2—C10	1.480 (8)	C11—H11A	0.9700
N2—C11	1.504 (8)	C11—H11B	0.9700
N2—H2B	0.93 (5)	C12—H12A	0.9700
N2—H2C	0.93 (6)	C12—H12B	0.9700
C1—C6	1.387 (7)
O1—Cu1—O8	89.63 (17)	C1—C6—H6A	120.2
O1—Cu1—O3^i	169.70 (15)	C5—C6—H6A	120.2
O8—Cu1—O3^i	91.04 (17)	O2—C7—O1	122.6 (5)
O1—Cu1—N1	93.61 (17)	O2—C7—C1	120.6 (4)
O8—Cu1—N1	163.55 (18)	O1—C7—C1	116.8 (4)
O3^i—Cu1—N1	88.65 (17)	O4—C8—O3	122.8 (5)
C7—O1—Cu1	113.3 (3)	O4—C8—C3	119.8 (5)
C8—O3—Cu1^ii	101.3 (3)	O3—C8—C3	117.4 (5)
Cu1—O8—H8A	122 (4)	N1—C9—C10	112.5 (5)
Cu1—O8—H8B	117 (5)	N1—C9—H9A	109.1
H8A—O8—H8B	107 (7)	C10—C9—H9A	109.1
C9—N1—C12	110.4 (4)	N1—C9—H9B	109.1
C9—N1—Cu1	109.3 (3)	C10—C9—H9B	109.1
C12—N1—Cu1	114.0 (3)	H9A—C9—H9B	107.8
C9—N1—H1A	111 (4)	N2—C10—C9	109.4 (4)
C12—N1—H1A	98 (4)	N2—C10—H10A	109.8
Cu1—N1—H1A	114 (4)	C9—C10—H10A	109.8
C10—N2—C11	111.4 (5)	N2—C10—H10B	109.8
C10—N2—H2B	113 (4)	C9—C10—H10B	109.8
C11—N2—H2B	107 (4)	H10A—C10—H10B	108.2
C10—N2—H2C	113 (6)	N2—C11—C12	108.4 (5)
C11—N2—H2C	108 (6)	N2—C11—H11A	110.0
H2B—N2—H2C	103 (7)	C12—C11—H11A	110.0
C6—C1—C2	120.6 (5)	N2—C11—H11B	110.0
C6—C1—C7	118.5 (4)	C12—C11—H11B	110.0
C2—C1—C7	120.9 (4)	H11A—C11—H11B	108.4
C3—C2—C1	118.8 (5)	N1—C12—C11	112.7 (5)
C3—C2—H2A	120.6	N1—C12—H12A	109.1
C1—C2—H2A	120.6	C11—C12—H12A	109.1
C4—C3—C2	120.6 (5)	N1—C12—H12B	109.1
C4—C3—C8	117.5 (5)	C11—C12—H12B	109.1
C2—C3—C8	121.9 (4)	H12A—C12—H12B	107.8
C5—C4—C3	120.0 (5)	O5—S1—O6	110.6 (3)
C5—C4—H4A	120.0	O5—S1—O7	112.1 (3)
C3—C4—H4A	120.0	O6—S1—O7	113.9 (3)
C4—C5—C6	120.3 (5)	O5—S1—C5	105.9 (2)
C4—C5—S1	120.3 (4)	O6—S1—C5	107.3 (2)
C6—C5—S1	119.4 (4)	O7—S1—C5	106.5 (2)
C1—C6—C5	119.6 (5)
O8—Cu1—O1—C7	−94.2 (4)	C2—C1—C7—O2	−178.6 (5)
O3^i—Cu1—O1—C7	172.0 (8)	C6—C1—C7—O1	179.1 (5)
N1—Cu1—O1—C7	69.6 (4)	C2—C1—C7—O1	−0.7 (7)
O1—Cu1—N1—C9	−131.3 (3)	Cu1^ii—O3—C8—O4	5.8 (6)
O8—Cu1—N1—C9	−30.3 (8)	Cu1^ii—O3—C8—C3	−173.3 (4)
O3^i—Cu1—N1—C9	58.7 (3)	C4—C3—C8—O4	8.6 (7)
O1—Cu1—N1—C12	104.6 (4)	C2—C3—C8—O4	−170.9 (5)
O8—Cu1—N1—C12	−154.4 (6)	C4—C3—C8—O3	−172.2 (5)
O3^i—Cu1—N1—C12	−65.4 (4)	C2—C3—C8—O3	8.3 (7)
C6—C1—C2—C3	0.5 (8)	C12—N1—C9—C10	−53.4 (6)
C7—C1—C2—C3	−179.8 (5)	Cu1—N1—C9—C10	−179.6 (4)
C1—C2—C3—C4	−0.4 (8)	C11—N2—C10—C9	−59.5 (6)
C1—C2—C3—C8	179.1 (5)	N1—C9—C10—N2	56.3 (6)
C2—C3—C4—C5	−0.6 (8)	C10—N2—C11—C12	59.4 (6)
C8—C3—C4—C5	179.9 (5)	C9—N1—C12—C11	54.0 (6)
C3—C4—C5—C6	1.5 (8)	Cu1—N1—C12—C11	177.5 (4)
C3—C4—C5—S1	−174.6 (4)	N2—C11—C12—N1	−56.5 (6)
C2—C1—C6—C5	0.4 (8)	C4—C5—S1—O5	113.8 (4)
C7—C1—C6—C5	−179.3 (5)	C6—C5—S1—O5	−62.3 (5)
C4—C5—C6—C1	−1.4 (8)	C4—C5—S1—O6	−128.0 (4)
S1—C5—C6—C1	174.7 (4)	C6—C5—S1—O6	55.9 (5)
Cu1—O1—C7—O2	−2.4 (7)	C4—C5—S1—O7	−5.7 (5)
Cu1—O1—C7—C1	179.7 (3)	C6—C5—S1—O7	178.2 (4)
C6—C1—C7—O2	1.1 (8)

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

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
O8—H8A···O7^iii	0.90 (4)	1.82 (5)	2.700 (6)	165 (6)
O8—H8B···O2^iv	0.90 (5)	1.82 (3)	2.687 (6)	162 (8)
N1—H1A···O4^v	0.93 (4)	2.07 (5)	2.958 (6)	159 (5)
N2—H2B···O5^vi	0.93 (5)	1.88 (3)	2.735 (6)	153 (6)
N2—H2C···O6^vii	0.93 (6)	1.94 (7)	2.842 (7)	165 (8)

Symmetry codes: (iii) −x+2, −y+1, −z+1; (iv) x, −y+1/2, z−1/2; (v) −x+1, −y+1, −z+1; (vi) −x+1, y−1/2, −z+3/2; (vii) x−1, −y+1/2, z−1/2.