Metal-directed supra­molecular architectures based on the bifunctional ligand 2,5-bis­(1H-1,2,4-triazol-1-yl)terephthalic acid

By the solvothermal reactions of 2,5-bis­(1H-1,2,4-triazol-1-yl)terephthalic acid (H₂L) with transition-metal ions, two novel polymeric com­plexes, namely, poly[di­aqua­[μ₄-2,5-bis­(1H-1,2,4-triazol-1-yl)terephthalato]cobalt(II)], [Co(C₁₂H₆N₆O₄)(H₂O)₂]_n, (1), and poly[[di­aqua­[μ₄-2,5-bis­(1H-1,2,4-triazol-1-yl)tere­ph­thalato]nickel(II)] dihydrate], {[Ni(C₁₂H₆N₆O₄)(H₂O)₂]·2H₂O}_n, (2), were isolated. Both polymers have been characterized by FT–IR spectroscopy, elemental analysis and single-crystal X-ray diffraction analysis. The com­plexes have similar two-dimensional layered structures and coordination modes. Furthermore, the two-dimensional layered structures bear distinct inter­molecular hydrogen-bonding inter­actions and π–π stacking inter­actions to form two different three-dimensional supra­molecular networks based on 4⁴-subnets. The structural variation depends on the nature of the metal cations. The results of variable-temperature magnetization measurements (χ_MT−T and χ_M⁻¹−T) show that com­plexes (1) and (2) display anti­ferromagnetic behaviour.

Computing details

For both structures, data collection: SMART (Bruker, 2010); cell refinement: SMART (Bruker, 2010); data reduction: SAINT (Bruker, 2010); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: SHELXTL (Bruker, 2010); software used to prepare material for publication: SHELXTL (Bruker, 2010).

Poly[diaqua[µ₄-2,5-bis(1H-1,2,4-triazol-1-yl)terephthalato]cobalt(II)] (160717a)

Crystal data

[Co(C12H6N6O4)(H2O)2]	F(000) = 398
Mr = 393.19	Dx = 1.829 Mg m^−3
Monoclinic, P21/c	Mo Kα radiation, λ = 0.71073 Å
a = 7.134 (3) Å	Cell parameters from 959 reflections
b = 9.882 (3) Å	θ = 2.9–26.8°
c = 10.300 (4) Å	µ = 1.25 mm^−1
β = 100.484 (4)°	T = 296 K
V = 714.0 (4) Å^3	Block, pale-red
Z = 2	0.22 × 0.16 × 0.10 mm

Data collection

Bruker SMART CCD area detector diffractometer	1408 independent reflections
Radiation source: fine-focus sealed tube	1007 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.041
phi and ω scans	θmax = 26.0°, θmin = 2.9°
Absorption correction: multi-scan (SADABS; Bruker, 2010)	h = −8→8
Tmin = 0.770, Tmax = 0.885	k = −6→12
3624 measured reflections	l = −12→11

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)] = 0.048	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.122	H atoms treated by a mixture of independent and constrained refinement
S = 1.05	w = 1/[σ^2(Fo^2) + (0.0598P)^2 + 0.5618P] where P = (Fo^2 + 2Fc^2)/3
1408 reflections	(Δ/σ)max < 0.001
115 parameters	Δρmax = 0.69 e Å^−3
0 restraints	Δρmin = −0.81 e Å^−3

Special details

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds 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 > 2sigma(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
Co1	0.5000	0.5000	0.5000	0.0219 (3)
O1	0.6815 (4)	0.5172 (3)	0.6808 (2)	0.0282 (7)
O2	0.6061 (5)	0.3321 (4)	0.7816 (3)	0.0513 (10)
O3	0.4703 (5)	0.7116 (3)	0.5001 (3)	0.0456 (9)
H3A	0.3750	0.7287	0.4398	0.068*
H3B	0.4753	0.7530	0.5730	0.068*
N1	1.2599 (5)	0.4664 (4)	0.5935 (3)	0.0291 (9)
N2	1.0440 (5)	0.3469 (4)	0.6781 (3)	0.0380 (10)
N3	1.1054 (5)	0.4546 (4)	0.7554 (3)	0.0260 (8)
C1	1.2321 (6)	0.5256 (5)	0.7032 (4)	0.0296 (11)
H1	1.2916	0.6045	0.7385	0.035*
C2	1.1425 (6)	0.3592 (5)	0.5829 (4)	0.0369 (12)
H2	1.1317	0.2982	0.5131	0.044*
C3	1.0474 (5)	0.4786 (4)	0.8789 (3)	0.0236 (10)
C4	0.8584 (5)	0.4673 (4)	0.8902 (3)	0.0216 (9)
C5	1.1870 (5)	0.5100 (4)	0.9858 (3)	0.0253 (9)
H5	1.3133	0.5161	0.9749	0.030*
C6	0.7012 (5)	0.4366 (5)	0.7745 (4)	0.0253 (9)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Co1	0.0228 (4)	0.0382 (5)	0.0058 (3)	−0.0002 (4)	0.0057 (2)	−0.0003 (3)
O1	0.0292 (15)	0.045 (2)	0.0094 (13)	−0.0060 (13)	0.0008 (10)	0.0060 (12)
O2	0.049 (2)	0.069 (3)	0.0283 (18)	−0.0297 (19)	−0.0123 (15)	0.0211 (17)
O3	0.071 (3)	0.043 (2)	0.0216 (15)	0.0079 (18)	0.0061 (16)	−0.0032 (15)
N1	0.0268 (18)	0.050 (3)	0.0116 (16)	0.0008 (16)	0.0066 (13)	−0.0005 (15)
N2	0.043 (2)	0.053 (3)	0.0219 (19)	−0.010 (2)	0.0163 (17)	−0.0117 (17)
N3	0.0257 (18)	0.045 (2)	0.0096 (15)	−0.0043 (15)	0.0078 (13)	−0.0036 (14)
C1	0.029 (2)	0.047 (3)	0.0145 (19)	−0.0033 (19)	0.0091 (16)	−0.0024 (18)
C2	0.034 (3)	0.061 (3)	0.019 (2)	−0.010 (2)	0.0135 (19)	−0.016 (2)
C3	0.0244 (19)	0.039 (3)	0.0094 (16)	−0.0034 (17)	0.0080 (14)	0.0011 (16)
C4	0.0190 (19)	0.036 (3)	0.0101 (17)	−0.0015 (16)	0.0030 (14)	0.0036 (15)
C5	0.0197 (17)	0.045 (3)	0.0123 (17)	−0.0057 (19)	0.0055 (14)	0.0037 (18)
C6	0.020 (2)	0.043 (3)	0.0124 (19)	−0.0039 (19)	0.0009 (15)	0.0000 (18)

Geometric parameters (Å, º)

Co1—O1	2.073 (2)	N2—C2	1.312 (5)
Co1—O1^i	2.073 (2)	N2—N3	1.353 (5)
Co1—O3	2.102 (3)	N3—C1	1.332 (5)
Co1—O3^i	2.102 (3)	N3—C3	1.427 (4)
Co1—N1^ii	2.138 (3)	C1—H1	0.9300
Co1—N1^iii	2.138 (3)	C2—H2	0.9300
O1—C6	1.239 (5)	C3—C5	1.379 (5)
O2—C6	1.245 (5)	C3—C4	1.380 (5)
O3—H3A	0.8501	C4—C5^v	1.391 (5)
O3—H3B	0.8501	C4—C6	1.511 (5)
N1—C1	1.319 (5)	C5—C4^v	1.391 (5)
N1—C2	1.342 (5)	C5—H5	0.9300
N1—Co1^iv	2.138 (3)
O1—Co1—O1^i	179.999 (1)	C2—N2—N3	102.0 (4)
O1—Co1—O3	87.97 (12)	C1—N3—N2	110.3 (3)
O1^i—Co1—O3	92.03 (12)	C1—N3—C3	126.9 (4)
O1—Co1—O3^i	92.03 (12)	N2—N3—C3	122.7 (3)
O1^i—Co1—O3^i	87.97 (12)	N1—C1—N3	109.3 (4)
O3—Co1—O3^i	180.0	N1—C1—H1	125.3
O1—Co1—N1^ii	88.34 (11)	N3—C1—H1	125.3
O1^i—Co1—N1^ii	91.66 (11)	N2—C2—N1	115.0 (4)
O3—Co1—N1^ii	86.24 (13)	N2—C2—H2	122.5
O3^i—Co1—N1^ii	93.76 (13)	N1—C2—H2	122.5
O1—Co1—N1^iii	91.66 (11)	C5—C3—C4	121.7 (3)
O1^i—Co1—N1^iii	88.34 (11)	C5—C3—N3	117.7 (3)
O3—Co1—N1^iii	93.76 (13)	C4—C3—N3	120.6 (3)
O3^i—Co1—N1^iii	86.24 (13)	C3—C4—C5^v	117.3 (3)
N1^ii—Co1—N1^iii	180.0	C3—C4—C6	123.1 (3)
C6—O1—Co1	127.6 (3)	C5^v—C4—C6	119.6 (3)
Co1—O3—H3A	105.4	C3—C5—C4^v	121.0 (3)
Co1—O3—H3B	119.4	C3—C5—H5	119.5
H3A—O3—H3B	116.6	C4^v—C5—H5	119.5
C1—N1—C2	103.4 (3)	O1—C6—O2	126.4 (4)
C1—N1—Co1^iv	125.0 (3)	O1—C6—C4	116.6 (4)
C2—N1—Co1^iv	128.7 (3)	O2—C6—C4	117.0 (4)
O1^i—Co1—O1—C6	−160 (12)	N2—N3—C3—C5	132.7 (4)
O3—Co1—O1—C6	146.2 (3)	C1—N3—C3—C4	138.7 (4)
O3^i—Co1—O1—C6	−33.8 (3)	N2—N3—C3—C4	−45.8 (6)
N1^ii—Co1—O1—C6	−127.5 (3)	C5—C3—C4—C5^v	0.6 (7)
N1^iii—Co1—O1—C6	52.5 (3)	N3—C3—C4—C5^v	178.9 (4)
C2—N2—N3—C1	0.9 (5)	C5—C3—C4—C6	178.4 (4)
C2—N2—N3—C3	−175.2 (4)	N3—C3—C4—C6	−3.2 (6)
C2—N1—C1—N3	0.9 (5)	C4—C3—C5—C4^v	−0.6 (7)
Co1^iv—N1—C1—N3	−161.2 (3)	N3—C3—C5—C4^v	−179.0 (4)
N2—N3—C1—N1	−1.2 (5)	Co1—O1—C6—O2	−4.8 (6)
C3—N3—C1—N1	174.8 (4)	Co1—O1—C6—C4	174.1 (2)
N3—N2—C2—N1	−0.4 (5)	C3—C4—C6—O1	−58.7 (6)
C1—N1—C2—N2	−0.3 (5)	C5^v—C4—C6—O1	119.0 (4)
Co1^iv—N1—C2—N2	160.9 (3)	C3—C4—C6—O2	120.3 (5)
C1—N3—C3—C5	−42.8 (6)	C5^v—C4—C6—O2	−62.0 (6)

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

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
O3—H3A···O2^i	0.85	2.39	2.886 (4)	118
O3—H3B···O2^vi	0.85	1.87	2.686 (5)	160

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

Poly[[diaqua[µ₄-2,5-bis(1H-1,2,4-triazol-1-yl)terephthalato]nickel(II)] dihydrate] (160716b)

Crystal data

[Ni(C12H6N6O4)(H2O)2]·2H2O	Z = 1
Mr = 429.00	F(000) = 220
Triclinic, P1	Dx = 1.772 Mg m^−3
a = 7.3533 (11) Å	Mo Kα radiation, λ = 0.71073 Å
b = 7.9541 (12) Å	Cell parameters from 1018 reflections
c = 8.1396 (12) Å	θ = 2.8–28.9°
α = 65.416 (2)°	µ = 1.27 mm^−1
β = 88.890 (2)°	T = 296 K
γ = 69.788 (2)°	Block, pale-blue
V = 401.92 (10) Å^3	0.24 × 0.16 × 0.10 mm

Data collection

Bruker SMART CCD area detector diffractometer	1554 independent reflections
Radiation source: fine-focus sealed tube	1404 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.015
phi and ω scans	θmax = 26.0°, θmin = 2.8°
Absorption correction: multi-scan (SADABS; Bruker, 2010)	h = −9→9
Tmin = 0.751, Tmax = 0.884	k = −9→8
2275 measured reflections	l = −9→10

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)] = 0.033	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.074	H atoms treated by a mixture of independent and constrained refinement
S = 1.06	w = 1/[σ^2(Fo^2) + (0.029P)^2 + 0.3943P] where P = (Fo^2 + 2Fc^2)/3
1554 reflections	(Δ/σ)max < 0.001
124 parameters	Δρmax = 0.32 e Å^−3
0 restraints	Δρmin = −0.35 e Å^−3

Special details

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds 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 > 2sigma(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	0.0000	0.0000	0.0000	0.01590 (14)
O1	0.1783 (3)	0.0696 (3)	0.1277 (2)	0.0238 (4)
O2	−0.0129 (3)	0.2317 (3)	0.2726 (3)	0.0421 (6)
O3	−0.2351 (3)	0.0953 (3)	0.1293 (2)	0.0275 (4)
H3A	−0.3420	0.1628	0.0559	0.041*
H3B	−0.1994	0.1532	0.1825	0.041*
O4	0.6086 (3)	0.6950 (4)	0.0815 (3)	0.0509 (6)
H4A	0.6568	0.6149	0.1923	0.076*
H4B	0.6589	0.7818	0.0281	0.076*
N1	0.0940 (3)	0.7165 (3)	0.2172 (3)	0.0201 (5)
N2	0.2247 (3)	0.4829 (3)	0.5033 (3)	0.0223 (5)
N3	0.2541 (3)	0.4001 (3)	0.3823 (3)	0.0187 (4)
C1	0.1760 (4)	0.5418 (4)	0.2138 (3)	0.0214 (5)
H1	0.1789	0.5208	0.1094	0.026*
C2	0.1266 (4)	0.6718 (4)	0.3973 (3)	0.0220 (5)
H2	0.0829	0.7674	0.4414	0.026*
C3	0.3723 (4)	0.1949 (3)	0.4425 (3)	0.0175 (5)
C4	0.3308 (4)	0.0778 (4)	0.3714 (3)	0.0187 (5)
C5	0.4630 (4)	−0.1177 (4)	0.4310 (3)	0.0197 (5)
H5	0.4397	−0.1983	0.3844	0.024*
C6	0.1472 (4)	0.1363 (4)	0.2464 (3)	0.0203 (5)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Ni1	0.0176 (2)	0.0123 (2)	0.0159 (2)	−0.00286 (18)	−0.00463 (17)	−0.00636 (19)
O1	0.0235 (9)	0.0297 (10)	0.0231 (9)	−0.0099 (8)	−0.0032 (7)	−0.0158 (8)
O2	0.0230 (11)	0.0562 (14)	0.0578 (15)	−0.0008 (10)	−0.0065 (10)	−0.0453 (13)
O3	0.0226 (10)	0.0313 (10)	0.0300 (10)	−0.0069 (8)	0.0001 (8)	−0.0170 (9)
O4	0.0391 (13)	0.0659 (16)	0.0376 (13)	−0.0241 (12)	−0.0034 (10)	−0.0093 (12)
N1	0.0226 (11)	0.0144 (10)	0.0194 (11)	−0.0041 (9)	−0.0049 (9)	−0.0056 (9)
N2	0.0281 (12)	0.0174 (11)	0.0206 (11)	−0.0062 (9)	−0.0021 (9)	−0.0092 (9)
N3	0.0226 (11)	0.0146 (10)	0.0171 (10)	−0.0045 (9)	−0.0047 (8)	−0.0068 (9)
C1	0.0272 (14)	0.0147 (12)	0.0177 (13)	−0.0036 (10)	−0.0063 (10)	−0.0058 (10)
C2	0.0253 (13)	0.0177 (12)	0.0239 (13)	−0.0062 (11)	0.0003 (11)	−0.0112 (11)
C3	0.0210 (12)	0.0123 (11)	0.0164 (12)	−0.0047 (10)	−0.0033 (10)	−0.0045 (10)
C4	0.0209 (13)	0.0164 (12)	0.0167 (12)	−0.0051 (10)	−0.0047 (10)	−0.0065 (10)
C5	0.0238 (13)	0.0184 (12)	0.0200 (13)	−0.0080 (10)	−0.0041 (10)	−0.0107 (10)
C6	0.0239 (13)	0.0160 (12)	0.0197 (13)	−0.0067 (10)	−0.0057 (10)	−0.0066 (10)

Geometric parameters (Å, º)

Ni1—O1	2.0347 (16)	N1—Ni1^iv	2.084 (2)
Ni1—O1^i	2.0347 (16)	N2—C2	1.311 (3)
Ni1—N1^ii	2.084 (2)	N2—N3	1.372 (3)
Ni1—N1^iii	2.084 (2)	N3—C1	1.333 (3)
Ni1—O3^i	2.0878 (18)	N3—C3	1.425 (3)
Ni1—O3	2.0879 (18)	C1—H1	0.9300
O1—C6	1.265 (3)	C2—H2	0.9300
O2—C6	1.231 (3)	C3—C5^v	1.384 (3)
O3—H3A	0.8500	C3—C4	1.395 (3)
O3—H3B	0.8500	C4—C5	1.394 (3)
O4—H4A	0.8500	C4—C6	1.517 (3)
O4—H4B	0.8501	C5—C3^v	1.384 (3)
N1—C1	1.322 (3)	C5—H5	0.9300
N1—C2	1.361 (3)
O1—Ni1—O1^i	180.0	C2—N2—N3	102.5 (2)
O1—Ni1—N1^ii	91.32 (8)	C1—N3—N2	109.73 (19)
O1^i—Ni1—N1^ii	88.68 (8)	C1—N3—C3	129.8 (2)
O1—Ni1—N1^iii	88.68 (8)	N2—N3—C3	120.12 (19)
O1^i—Ni1—N1^iii	91.32 (8)	N1—C1—N3	109.9 (2)
N1^ii—Ni1—N1^iii	180.0	N1—C1—H1	125.0
O1—Ni1—O3^i	87.59 (7)	N3—C1—H1	125.0
O1^i—Ni1—O3^i	92.41 (7)	N2—C2—N1	114.5 (2)
N1^ii—Ni1—O3^i	89.88 (8)	N2—C2—H2	122.8
N1^iii—Ni1—O3^i	90.12 (8)	N1—C2—H2	122.8
O1—Ni1—O3	92.41 (7)	C5^v—C3—C4	120.8 (2)
O1^i—Ni1—O3	87.59 (7)	C5^v—C3—N3	117.1 (2)
N1^ii—Ni1—O3	90.12 (8)	C4—C3—N3	122.0 (2)
N1^iii—Ni1—O3	89.88 (8)	C5—C4—C3	117.4 (2)
O3^i—Ni1—O3	180.0	C5—C4—C6	116.0 (2)
C6—O1—Ni1	130.31 (17)	C3—C4—C6	126.4 (2)
Ni1—O3—H3A	111.9	C3^v—C5—C4	121.8 (2)
Ni1—O3—H3B	105.5	C3^v—C5—H5	119.1
H3A—O3—H3B	116.5	C4—C5—H5	119.1
H4A—O4—H4B	116.3	O2—C6—O1	126.8 (2)
C1—N1—C2	103.4 (2)	O2—C6—C4	119.5 (2)
C1—N1—Ni1^iv	128.76 (17)	O1—C6—C4	113.6 (2)
C2—N1—Ni1^iv	126.28 (16)
O1^i—Ni1—O1—C6	−53 (100)	N2—N3—C3—C5^v	−38.0 (3)
N1^ii—Ni1—O1—C6	93.1 (2)	C1—N3—C3—C4	−42.7 (4)
N1^iii—Ni1—O1—C6	−86.9 (2)	N2—N3—C3—C4	145.1 (2)
O3^i—Ni1—O1—C6	−177.1 (2)	C5^v—C3—C4—C5	−0.9 (4)
O3—Ni1—O1—C6	2.9 (2)	N3—C3—C4—C5	175.9 (2)
C2—N2—N3—C1	0.9 (3)	C5^v—C3—C4—C6	173.8 (2)
C2—N2—N3—C3	174.6 (2)	N3—C3—C4—C6	−9.4 (4)
C2—N1—C1—N3	−0.2 (3)	C3—C4—C5—C3^v	0.9 (4)
Ni1^iv—N1—C1—N3	166.03 (16)	C6—C4—C5—C3^v	−174.4 (2)
N2—N3—C1—N1	−0.5 (3)	Ni1—O1—C6—O2	−24.4 (4)
C3—N3—C1—N1	−173.3 (2)	Ni1—O1—C6—C4	152.05 (17)
N3—N2—C2—N1	−1.1 (3)	C5—C4—C6—O2	135.4 (3)
C1—N1—C2—N2	0.8 (3)	C3—C4—C6—O2	−39.5 (4)
Ni1^iv—N1—C2—N2	−165.83 (17)	C5—C4—C6—O1	−41.4 (3)
C1—N3—C3—C5^v	134.2 (3)	C3—C4—C6—O1	143.8 (3)

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

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
O3—H3A···O4^vi	0.85	1.97	2.812 (3)	171
O3—H3B···O2	0.85	1.95	2.749 (3)	155
O4—H4A···N2^vii	0.85	2.35	3.148 (3)	157
O4—H4B···O1^viii	0.85	2.03	2.852 (3)	163

Symmetry codes: (vi) −x, −y+1, −z+1; (vii) −x+1, −y+1, −z+1; (viii) −x+1, −y+1, −z.