Hexakis­(imidazole-N³)­nickel(II) bis(4-methoxy­benzoate)

In the title complex of [Ni(C₃H₄N₂)₆](C₈H₇O₃)₂, the Ni atom is in an octahedral environment formed by the tertiary N atom of the imidazole moieties. The methoxy­benzoate moieties act as a bridge connecting two hexakis­(imidazole)nickel(II) mol­ecules through N-H...O hydrogen bonds.

S1. Comment

Imidazole is of considerable interest as a potential binding site for metal ions in many biological systems. It is an unidentate ligand and forms complexes with metal ions through its tertiary nitrogen atom. Some complexes of imidazole and its derivatives with transition-metal ions have been reported (Brooks & Davidson, 1960; Davis & Smith, 1971). Complexes of copper(II) and cobalt(II) atoms with carboxylate and imidazole ligands have been studied as models for metalloproteins since both contain functionalities in the side chain (Sigel, 1980; Bernarducci et al., 1983; Abuhijleh & Woods, 1992). In addition, some of these copper(II) complexes were found to have a variety of pharmacological activities (Tamura et al., 1987) and superoxide dismutase activities (Bhirud & Srivastava, 1990). Complexes of [Co(RCOO)₂(Im)₂] (Im = –CH₃, –C₂H₅) and model complexes for cobalt(II)-substituted zinc metalloenzymes, have been reported by Horrocks group (Horrocks et al., 1982).

The asymmetric unit of the title complex is formed by one-half of the hexakis(imidazole)nickel(II) molecule; the other half is related by inversion symmetry through the Ni atom located at the inversion centre and one methoxybenzoate molecule. The Ni atom is in an octahedral environment formed by the tertiary N atom of the imidazole moieties. The bond distances involved in this octahedral geometry, (Ni1—N1, Ni1—N2, Ni1—N3) ranging from 2.127 (2) Å to 2.141 (2) Å are comparable with those found in the complex of hexakis(imidazole)nickel(II) disalicylate (Jian et al., 1999). All the imidazole rings are planar. The methoxybenzoate moiety is planar with the maximum deviation of 0.111 (2) Å for the O3 atom.

In the solid state, intermolecular C—H.·O(x, y − 1, z) interactions connect the methoxybenzoate with the NiN₆ chromophore. The complex forms a three-dimensional network through N—H···O intermolecular hydrogen bonds with the methoxybenzoate moieties acting as a bridge connecting the two hexakis(imidazole) nickel(II) molecules (figure 2). In addition, intra- and intermolecular C—H···π interactions contribute to the molecular packing (C3—H3A = 0.93; H3A···Cg1 = 2.77; C3···Cg1 = 3.481 Å; C3—H3···Cg1 = 134°, C16—H16C = 0.93; H16A···Cg2(1 − x, 1 − y, 1 − z) = 2.836; C16···Cg2 = 3.664 Å; C16—H16A···Cg2 = 145°, Cg1 = centroid of imidazole ring N5, N6, C7—C9 and Cg2 = centroid of C10—C15 ring).

S2. Experimental

The title complex was prepared by the reaction of imidazole (1 g, 15 mmol) with Ni(p—CH₃O·C₆H₄COO)₂·4H₂O (Ephraim & Pfister, 1925) (1 g, 2.5 mmol) in warm ethanol (50 ml). Single crystals suitable for X-ray analysis were obtained by recrystallization from ethanol.

S3. Refinement

After checking their presence in the difference map, all H-atoms were geometrically fixed and allowed to ride on their attached atoms.

Computing details

Data collection: SMART (Siemens, 1996); cell refinement: SAINT (Siemens, 1996); data reduction: SAINT; program(s) used to solve structure: SHELXTL (Sheldrick, 1997); program(s) used to refine structure: SHELXTL; molecular graphics: SHELXTL; software used to prepare material for publication: SHELXTL and PLATON (Spek, 1990).

Figures

	Fig. 1. The molecule structure of the title complex. Displacement ellipsoids are shown at the 30% probability level. H atoms are omitted for clarity.
	Fig. 2. Part of the three-dimensional network.

Hexakis(imidazole)nickel(II) di(p-methoxybenzoate), [Ni(Im)₆(mB)₂] (Im = imidazole, mB = p-methoxybenzoate)

Crystal data

[Ni(C3H4N2)6](C8H7O3)2	Z = 1
Mr = 769.48	F(000) = 402
Triclinic, P1	Dx = 1.377 Mg m^−3
a = 9.8185 (4) Å	Mo Kα radiation, λ = 0.71073 Å
b = 10.5191 (4) Å	Cell parameters from 5621 reflections
c = 10.8556 (5) Å	θ = 2.7–61.1°
α = 61.474 (1)°	µ = 0.58 mm^−1
β = 76.764 (1)°	T = 293 K
γ = 70.973 (1)°	Parallelopiped, pale blue
V = 927.62 (7) Å^3	0.50 × 0.24 × 0.14 mm

Data collection

Siemens SMART CCD area detector diffractometer	4138 independent reflections
Radiation source: fine-focus sealed tube	3474 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.039
Detector resolution: 8.33 pixels mm^-1	θmax = 27.5°, θmin = 2.7°
ω scans	h = −12→12
Absorption correction: empirical (using intensity measurements) (SADABS; Sheldrick, 1996)	k = −13→12
Tmin = 0.759, Tmax = 0.923	l = −14→14
6314 measured reflections

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)] = 0.052	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.140	H-atom parameters constrained
S = 1.00	w = 1/[σ^2(Fo^2) + (0.0853P)^2] where P = (Fo^2 + 2Fc^2)/3
4140 reflections	(Δ/σ)max < 0.001
241 parameters	Δρmax = 0.74 e Å^−3
0 restraints	Δρmin = −1.05 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. 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 30 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 97% complete. Crystal decay was monitored by repeating thirty initial frames at the end of data collection and analysing the duplicate reflections, and was found to be negligible.

The structure was solved by direct methods and refined by full-matrix.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Ni1	1.0000	0.0000	0.0000	0.02620 (14)
O1	0.7181 (2)	0.5723 (2)	0.37875 (19)	0.0516 (5)
O2	0.4355 (2)	0.11618 (19)	0.35781 (16)	0.0413 (4)
O3	0.4316 (2)	0.28877 (18)	0.13656 (16)	0.0407 (4)
N1	0.9064 (2)	−0.0372 (2)	0.20911 (19)	0.0340 (4)
N2	0.7497 (3)	−0.0666 (3)	0.3960 (2)	0.0458 (5)
H2A	0.6694	−0.0718	0.4482	0.055*
N3	1.0730 (2)	−0.2351 (2)	0.0674 (2)	0.0330 (4)
N4	1.1969 (3)	−0.4591 (2)	0.0882 (3)	0.0473 (5)
H4A	1.2612	−0.5281	0.0706	0.057*
N5	1.1954 (2)	0.0005 (2)	0.0552 (2)	0.0321 (4)
N6	1.3347 (2)	0.0761 (3)	0.1252 (2)	0.0448 (5)
H6A	1.3674	0.1341	0.1395	0.054*
C1	0.7706 (3)	−0.0366 (3)	0.2592 (3)	0.0386 (5)
H1A	0.6981	−0.0177	0.2061	0.046*
C2	0.8783 (4)	−0.0871 (4)	0.4362 (3)	0.0572 (8)
H2B	0.8971	−0.1089	0.5253	0.069*
C3	0.9751 (3)	−0.0696 (4)	0.3213 (3)	0.0503 (7)
H3A	1.0729	−0.0784	0.3190	0.060*
C4	1.0162 (3)	−0.3416 (3)	0.1835 (3)	0.0499 (7)
H4B	0.9371	−0.3219	0.2437	0.060*
C5	1.0932 (4)	−0.4790 (3)	0.1967 (3)	0.0583 (8)
H5A	1.0776	−0.5700	0.2669	0.070*
C6	1.1806 (3)	−0.3111 (3)	0.0132 (3)	0.0422 (6)
H6B	1.2389	−0.2675	−0.0677	0.051*
C7	1.3081 (3)	−0.1177 (3)	0.1186 (3)	0.0411 (6)
H7A	1.3236	−0.2143	0.1289	0.049*
C8	1.3936 (3)	−0.0727 (3)	0.1639 (3)	0.0459 (6)
H8A	1.4759	−0.1317	0.2117	0.055*
C9	1.2171 (3)	0.1152 (3)	0.0613 (3)	0.0418 (6)
H9A	1.1572	0.2115	0.0251	0.050*
C10	0.6589 (3)	0.4831 (3)	0.3587 (2)	0.0363 (5)
C11	0.6449 (3)	0.3424 (3)	0.4589 (2)	0.0410 (6)
H11A	0.6789	0.3009	0.5467	0.049*
C12	0.5797 (3)	0.2634 (3)	0.4272 (2)	0.0363 (5)
H12A	0.5693	0.1695	0.4950	0.044*
C13	0.5302 (2)	0.3222 (2)	0.2964 (2)	0.0288 (4)
C14	0.5459 (3)	0.4635 (3)	0.1963 (2)	0.0342 (5)
H14A	0.5136	0.5042	0.1079	0.041*
C15	0.6089 (3)	0.5440 (3)	0.2267 (2)	0.0378 (5)
H15A	0.6179	0.6385	0.1593	0.045*
C16	0.7736 (4)	0.5142 (4)	0.5106 (3)	0.0561 (8)
H16A	0.8117	0.5870	0.5115	0.084*
H16B	0.8491	0.4245	0.5245	0.084*
H16C	0.6974	0.4919	0.5848	0.084*
C17	0.4598 (2)	0.2357 (2)	0.2626 (2)	0.0289 (4)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Ni1	0.0262 (2)	0.0272 (2)	0.0262 (2)	−0.00938 (15)	0.00287 (14)	−0.01298 (17)
O1	0.0689 (13)	0.0571 (12)	0.0438 (10)	−0.0382 (11)	0.0003 (9)	−0.0219 (10)
O2	0.0572 (11)	0.0391 (9)	0.0270 (8)	−0.0267 (9)	0.0030 (7)	−0.0078 (7)
O3	0.0645 (12)	0.0300 (8)	0.0280 (8)	−0.0128 (8)	−0.0131 (8)	−0.0086 (7)
N1	0.0356 (10)	0.0363 (10)	0.0303 (9)	−0.0111 (8)	0.0053 (8)	−0.0169 (9)
N2	0.0548 (13)	0.0442 (12)	0.0365 (11)	−0.0219 (11)	0.0201 (10)	−0.0207 (10)
N3	0.0337 (10)	0.0293 (10)	0.0340 (9)	−0.0098 (8)	−0.0004 (8)	−0.0123 (8)
N4	0.0540 (13)	0.0300 (11)	0.0570 (13)	−0.0038 (10)	−0.0074 (11)	−0.0217 (10)
N5	0.0309 (9)	0.0344 (10)	0.0338 (9)	−0.0110 (8)	−0.0004 (8)	−0.0165 (8)
N6	0.0446 (12)	0.0496 (13)	0.0551 (13)	−0.0161 (10)	−0.0082 (10)	−0.0302 (11)
C1	0.0414 (13)	0.0406 (13)	0.0354 (12)	−0.0160 (11)	0.0079 (10)	−0.0192 (11)
C2	0.0655 (19)	0.072 (2)	0.0300 (13)	−0.0177 (16)	0.0024 (13)	−0.0220 (14)
C3	0.0421 (14)	0.072 (2)	0.0339 (12)	−0.0145 (14)	−0.0006 (11)	−0.0220 (13)
C4	0.0624 (17)	0.0394 (14)	0.0413 (14)	−0.0205 (13)	0.0099 (13)	−0.0139 (12)
C5	0.085 (2)	0.0319 (13)	0.0465 (15)	−0.0187 (14)	−0.0014 (15)	−0.0077 (13)
C6	0.0424 (13)	0.0336 (12)	0.0484 (14)	−0.0104 (10)	0.0037 (11)	−0.0189 (11)
C7	0.0393 (13)	0.0368 (13)	0.0503 (14)	−0.0081 (10)	−0.0105 (11)	−0.0196 (12)
C8	0.0400 (13)	0.0494 (15)	0.0515 (15)	−0.0092 (11)	−0.0149 (12)	−0.0213 (13)
C9	0.0397 (13)	0.0366 (13)	0.0534 (15)	−0.0090 (11)	−0.0095 (11)	−0.0211 (12)
C10	0.0418 (12)	0.0392 (12)	0.0344 (12)	−0.0185 (10)	0.0054 (10)	−0.0197 (11)
C11	0.0545 (15)	0.0434 (14)	0.0258 (11)	−0.0190 (12)	−0.0067 (10)	−0.0102 (10)
C12	0.0504 (14)	0.0305 (11)	0.0262 (10)	−0.0157 (10)	−0.0014 (10)	−0.0084 (9)
C13	0.0343 (11)	0.0274 (10)	0.0236 (9)	−0.0098 (9)	0.0024 (8)	−0.0112 (9)
C14	0.0449 (13)	0.0310 (11)	0.0243 (10)	−0.0121 (10)	0.0004 (9)	−0.0101 (9)
C15	0.0501 (14)	0.0319 (12)	0.0293 (11)	−0.0184 (11)	0.0034 (10)	−0.0098 (10)
C16	0.0664 (19)	0.068 (2)	0.0525 (16)	−0.0307 (16)	−0.0052 (15)	−0.0326 (16)
C17	0.0320 (10)	0.0276 (10)	0.0251 (10)	−0.0070 (8)	0.0028 (8)	−0.0126 (9)

Geometric parameters (Å, º)

Ni1—N3	2.127 (2)	C2—C3	1.358 (4)
Ni1—N3^i	2.1266 (19)	C2—H2B	0.9300
Ni1—N5	2.140 (2)	C3—H3A	0.9300
Ni1—N5^i	2.1395 (18)	C4—C5	1.350 (4)
Ni1—N1	2.141 (2)	C4—H4B	0.9300
Ni1—N1^i	2.1406 (17)	C5—H5A	0.9300
O1—C10	1.370 (3)	C6—H6B	0.9300
O1—C16	1.422 (3)	C7—C8	1.357 (3)
O2—C17	1.244 (2)	C7—H7A	0.9300
O3—C17	1.266 (3)	C8—H8A	0.9300
N1—C1	1.319 (3)	C9—H9A	0.9300
N1—C3	1.372 (3)	C10—C11	1.385 (3)
N2—C1	1.342 (3)	C10—C15	1.394 (3)
N2—C2	1.347 (4)	C11—C12	1.392 (3)
N2—H2A	0.8600	C11—H11A	0.9300
N3—C6	1.310 (3)	C12—C13	1.384 (3)
N3—C4	1.375 (3)	C12—H12A	0.9300
N4—C6	1.342 (3)	C13—C14	1.395 (3)
N4—C5	1.350 (4)	C13—C17	1.511 (3)
N4—H4A	0.8600	C14—C15	1.382 (3)
N5—C9	1.324 (3)	C14—H14A	0.9300
N5—C7	1.370 (3)	C15—H15A	0.9300
N6—C9	1.330 (3)	C16—H16A	0.9600
N6—C8	1.362 (4)	C16—H16B	0.9600
N6—H6A	0.8600	C16—H16C	0.9600
C1—H1A	0.9300
N3—Ni1—N3^i	180.00 (17)	C5—C4—H4B	125.3
N3—Ni1—N5	90.40 (7)	N3—C4—H4B	125.3
N3^i—Ni1—N5	89.60 (7)	C4—C5—N4	106.8 (2)
N3—Ni1—N5^i	89.60 (7)	C4—C5—H5A	126.6
N3^i—Ni1—N5^i	90.40 (7)	N4—C5—H5A	126.6
N5—Ni1—N5^i	180.00 (11)	N3—C6—N4	112.0 (2)
N3—Ni1—N1	89.75 (7)	N3—C6—H6B	124.0
N3^i—Ni1—N1	90.25 (7)	N4—C6—H6B	124.0
N5—Ni1—N1	88.57 (7)	C8—C7—N5	109.8 (2)
N5^i—Ni1—N1	91.43 (7)	C8—C7—H7A	125.1
N3—Ni1—N1^i	90.25 (7)	N5—C7—H7A	125.1
N3^i—Ni1—N1^i	89.75 (7)	C7—C8—N6	106.2 (2)
N5—Ni1—N1^i	91.43 (7)	C7—C8—H8A	126.9
N5^i—Ni1—N1^i	88.57 (7)	N6—C8—H8A	126.9
N1—Ni1—N1^i	180.00 (12)	N5—C9—N6	112.0 (2)
C10—O1—C16	117.6 (2)	N5—C9—H9A	124.0
C1—N1—C3	104.8 (2)	N6—C9—H9A	124.0
C1—N1—Ni1	127.63 (17)	O1—C10—C11	124.8 (2)
C3—N1—Ni1	127.57 (16)	O1—C10—C15	115.3 (2)
C1—N2—C2	107.1 (2)	C11—C10—C15	119.9 (2)
C1—N2—H2A	126.5	C10—C11—C12	119.6 (2)
C2—N2—H2A	126.5	C10—C11—H11A	120.2
C6—N3—C4	104.9 (2)	C12—C11—H11A	120.2
C6—N3—Ni1	129.20 (16)	C13—C12—C11	121.1 (2)
C4—N3—Ni1	125.88 (17)	C13—C12—H12A	119.4
C6—N4—C5	106.8 (2)	C11—C12—H12A	119.4
C6—N4—H4A	126.6	C12—C13—C14	118.6 (2)
C5—N4—H4A	126.6	C12—C13—C17	120.85 (19)
C9—N5—C7	104.8 (2)	C14—C13—C17	120.58 (19)
C9—N5—Ni1	124.70 (17)	C15—C14—C13	121.0 (2)
C7—N5—Ni1	129.16 (15)	C15—C14—H14A	119.5
C9—N6—C8	107.2 (2)	C13—C14—H14A	119.5
C9—N6—H6A	126.4	C14—C15—C10	119.8 (2)
C8—N6—H6A	126.4	C14—C15—H15A	120.1
N1—C1—N2	111.9 (2)	C10—C15—H15A	120.1
N1—C1—H1A	124.1	O1—C16—H16A	109.5
N2—C1—H1A	124.1	O1—C16—H16B	109.5
N2—C2—C3	106.7 (2)	H16A—C16—H16B	109.5
N2—C2—H2B	126.6	O1—C16—H16C	109.5
C3—C2—H2B	126.6	H16A—C16—H16C	109.5
C2—C3—N1	109.5 (2)	H16B—C16—H16C	109.5
C2—C3—H3A	125.2	O2—C17—O3	123.7 (2)
N1—C3—H3A	125.2	O2—C17—C13	119.01 (19)
C5—C4—N3	109.5 (3)	O3—C17—C13	117.28 (18)
N3—Ni1—N1—C1	−97.2 (2)	N2—C2—C3—N1	0.4 (4)
N3^i—Ni1—N1—C1	82.8 (2)	C1—N1—C3—C2	−0.3 (3)
N5—Ni1—N1—C1	172.4 (2)	Ni1—N1—C3—C2	−179.2 (2)
N5^i—Ni1—N1—C1	−7.6 (2)	C6—N3—C4—C5	0.8 (3)
N1^i—Ni1—N1—C1	−40 (100)	Ni1—N3—C4—C5	−177.6 (2)
N3—Ni1—N1—C3	81.5 (2)	N3—C4—C5—N4	−0.7 (4)
N3^i—Ni1—N1—C3	−98.5 (2)	C6—N4—C5—C4	0.3 (3)
N5—Ni1—N1—C3	−8.9 (2)	C4—N3—C6—N4	−0.6 (3)
N5^i—Ni1—N1—C3	171.1 (2)	Ni1—N3—C6—N4	177.74 (16)
N1^i—Ni1—N1—C3	139 (100)	C5—N4—C6—N3	0.2 (3)
N3^i—Ni1—N3—C6	−111 (100)	C9—N5—C7—C8	1.2 (3)
N5—Ni1—N3—C6	−67.2 (2)	Ni1—N5—C7—C8	−165.85 (18)
N5^i—Ni1—N3—C6	112.8 (2)	N5—C7—C8—N6	−1.3 (3)
N1—Ni1—N3—C6	−155.8 (2)	C9—N6—C8—C7	0.9 (3)
N1^i—Ni1—N3—C6	24.2 (2)	C7—N5—C9—N6	−0.6 (3)
N3^i—Ni1—N3—C4	67 (100)	Ni1—N5—C9—N6	167.19 (16)
N5—Ni1—N3—C4	110.9 (2)	C8—N6—C9—N5	−0.2 (3)
N5^i—Ni1—N3—C4	−69.1 (2)	C16—O1—C10—C11	2.1 (4)
N1—Ni1—N3—C4	22.3 (2)	C16—O1—C10—C15	−178.7 (2)
N1^i—Ni1—N3—C4	−157.7 (2)	O1—C10—C11—C12	178.6 (2)
N3—Ni1—N5—C9	−168.7 (2)	C15—C10—C11—C12	−0.6 (4)
N3^i—Ni1—N5—C9	11.3 (2)	C10—C11—C12—C13	0.8 (4)
N5^i—Ni1—N5—C9	96 (93)	C11—C12—C13—C14	−0.4 (4)
N1—Ni1—N5—C9	−79.0 (2)	C11—C12—C13—C17	179.9 (2)
N1^i—Ni1—N5—C9	101.0 (2)	C12—C13—C14—C15	−0.3 (4)
N3—Ni1—N5—C7	−4.0 (2)	C17—C13—C14—C15	179.4 (2)
N3^i—Ni1—N5—C7	176.0 (2)	C13—C14—C15—C10	0.6 (4)
N5^i—Ni1—N5—C7	−100 (93)	O1—C10—C15—C14	−179.4 (2)
N1—Ni1—N5—C7	85.7 (2)	C11—C10—C15—C14	−0.1 (4)
N1^i—Ni1—N5—C7	−94.3 (2)	C12—C13—C17—O2	6.2 (3)
C3—N1—C1—N2	0.0 (3)	C14—C13—C17—O2	−173.6 (2)
Ni1—N1—C1—N2	178.94 (16)	C12—C13—C17—O3	−172.4 (2)
C2—N2—C1—N1	0.2 (3)	C14—C13—C17—O3	7.8 (3)
C1—N2—C2—C3	−0.4 (3)

Symmetry code: (i) −x+2, −y, −z.

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
N2—H2A···O2	0.86	2.52	3.057 (4)	121
N2—H2A···O2^ii	0.86	2.02	2.790 (3)	147
N4—H4A···O3^iii	0.86	2.04	2.803 (3)	147
N6—H6A···O2^iv	0.86	2.51	3.157 (3)	132
N6—H6A···O3^iv	0.86	1.91	2.762 (3)	169
C4—H4B···O1^v	0.93	2.56	3.307 (4)	137
C9—H9A···N3^i	0.93	2.57	3.075 (4)	114
C14—H14A···O3^vi	0.93	2.58	3.303 (3)	134

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