Li₄Ru₂OCl₁₀·10H₂O: crystal structure, magnetic properties and bonding interactions in ruthenium-oxo complexes

The results of the structural determination, magnetic characterization, and theoretical calculations of a new ruthenium-oxo complex, Li₄[Ru₂OCl₁₀]·10H₂O, are presented. Single crystals were grown using solvent methods and the crystal structure was characterized by single crystal X-ray diffraction. Li₄[Ru₂OCl₁₀]·10H₂O crystallizes into a low-symmetry triclinic structure (P1) due to the much smaller Li⁺ cation compared to K⁺ cation in the tetragonal complex K₄[Ru₂OCl₁₀]·H₂O. The X-ray photoelectron spectra confirm only the single valent Ru⁴⁺ in Li₄[Ru₂OCl₁₀]·10H₂O even though two distinct Ru sites exist in the crystal structure. Magnetic measurements reveal the diamagnetic property of Li₄[Ru₂OCl₁₀]·10H₂O with unpaired electrons existing on Ru⁴⁺. Furthermore, the molecular orbital analysis matches well with the observed UV and magnetic measurements.

Computing details

Data collection: Bruker APEX2; cell refinement: Bruker SAINT; data reduction: Bruker SAINT; program(s) used to solve structure: SHELXS97 (Sheldrick 2008); program(s) used to refine structure: SHELXL2013 (Sheldrick, 2013); molecular graphics: Bruker SHELXTL; software used to prepare material for publication: Bruker SHELXTL.

(I)

Crystal data

H20Cl10Li4O11Ru2	Z = 2
Mr = 780.56	F(000) = 756
Triclinic, P1	Dx = 2.345 Mg m^−3
a = 8.0548 (6) Å	Mo Kα radiation, λ = 0.71073 Å
b = 10.7806 (8) Å	Cell parameters from 63759 reflections
c = 13.7195 (11) Å	θ = 1.5–35.3°
α = 78.551 (2)°	µ = 2.61 mm^−1
β = 80.825 (2)°	T = 90 K
γ = 72.1629 (19)°	Regular, brown
V = 1105.28 (15) Å^3	0.03 × 0.02 × 0.01 mm

Data collection

Bruker APEX-II CCD diffractometer	8673 reflections with I > 2σ(I)
φ and ω scans	Rint = 0.024
Absorption correction: multi-scan xprep	θmax = 35.3°, θmin = 1.5°
Tmin = 0.346, Tmax = 0.746	h = −13→13
63759 measured reflections	k = −17→17
9864 independent reflections	l = −22→22

Refinement

Refinement on F^2	4 restraints
Least-squares matrix: full	Hydrogen site location: difference Fourier map
R[F^2 > 2σ(F^2)] = 0.021	Only H-atom coordinates refined
wR(F^2) = 0.046	w = 1/[σ^2(Fo^2) + (0.0157P)^2 + 1.3455P] where P = (Fo^2 + 2Fc^2)/3
S = 1.06	(Δ/σ)max = 0.031
9864 reflections	Δρmax = 1.82 e Å^−3
302 parameters	Δρmin = −1.06 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.

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

	x	y	z	Uiso*/Ueq
Ru1	0.37849 (2)	0.10506 (2)	0.58791 (2)	0.00420 (2)
Ru2	0.37460 (2)	0.58996 (2)	0.09581 (2)	0.00434 (2)
Cl3	0.63765 (4)	0.14070 (3)	0.62171 (2)	0.00779 (5)
Cl4	0.33885 (4)	0.29470 (3)	0.46345 (2)	0.00806 (5)
Cl8	0.63405 (4)	0.61891 (3)	0.13761 (2)	0.00800 (5)
Cl7	0.41255 (4)	0.40057 (3)	0.21788 (2)	0.00838 (5)
Cl10	0.10674 (4)	0.56616 (3)	0.06751 (2)	0.00904 (5)
Cl5	0.10693 (4)	0.08117 (3)	0.56365 (2)	0.00821 (5)
Cl6	0.22222 (4)	0.71033 (3)	0.22261 (2)	0.00878 (5)
Cl2	0.60008 (4)	0.06858 (3)	0.27565 (2)	0.00810 (5)
Cl1	0.22556 (4)	0.24841 (3)	0.70076 (2)	0.00879 (5)
Cl9	0.67977 (4)	0.20764 (3)	0.01955 (2)	0.00779 (5)
O1	0.5000	0.0000	0.5000	0.0054 (2)
O2	0.5000	0.5000	0.0000	0.0058 (2)
O3	0.49687 (15)	0.05887 (11)	0.89613 (8)	0.01091 (18)
H31	0.513 (3)	0.865 (2)	0.1204 (16)	0.013*
H32	0.477 (3)	0.031 (2)	0.8477 (17)	0.013*
O4	0.17532 (14)	0.16390 (11)	0.31416 (8)	0.01025 (18)
H41	0.269 (3)	0.174 (2)	0.3095 (16)	0.012*
H42	0.194 (3)	0.088 (2)	0.3398 (16)	0.012*
O5	0.84836 (15)	0.30161 (11)	0.18461 (9)	0.01250 (19)
H51	0.785 (3)	0.278 (2)	0.1599 (16)	0.015*
H52	0.830 (3)	0.3791 (15)	0.1725 (17)	0.015*
O6	0.79177 (15)	0.24740 (11)	0.39721 (8)	0.01058 (18)
H61	0.787 (3)	0.177 (2)	0.4155 (17)	0.013*
H62	0.778 (3)	0.267 (2)	0.3404 (17)	0.013*
O7	0.26147 (15)	0.22695 (11)	0.07677 (9)	0.01236 (19)
H71	0.234 (3)	0.284 (2)	0.0348 (17)	0.015*
H72	0.344 (3)	0.240 (2)	0.0884 (17)	0.015*
O8	0.54025 (15)	0.53274 (11)	0.39269 (8)	0.01099 (18)
H81	0.415 (3)	0.417 (2)	0.6305 (17)	0.013*
H82	0.498 (3)	0.517 (2)	0.3492 (17)	0.013*
O9	0.09657 (15)	0.01971 (11)	0.14488 (9)	0.01292 (19)
H91	0.019 (2)	0.000 (2)	0.1298 (17)	0.016*
H92	0.119 (3)	0.0811 (17)	0.1336 (17)	0.016*
O10	0.10879 (15)	0.11581 (11)	0.92565 (9)	0.01243 (19)
H101	0.148 (3)	0.144 (2)	0.8696 (18)	0.015*
H102	0.028 (3)	0.170 (2)	0.9346 (17)	0.015*
O11	0.13884 (14)	0.61075 (11)	0.46687 (8)	0.01122 (18)*
H111	0.114 (3)	0.651 (2)	0.4207 (17)	0.013*
H112	0.145 (3)	0.533 (2)	0.4670 (16)	0.013*
O12	0.01300 (18)	0.44829 (12)	0.31954 (11)	0.0197 (3)
H121	0.024 (4)	0.496 (3)	0.690 (2)	0.024*
H122	0.098 (3)	0.450 (2)	0.2809 (19)	0.024*
Li1	0.2961 (4)	0.0471 (3)	0.0216 (2)	0.0143 (5)
Li2	0.0140 (4)	0.2890 (3)	0.4167 (2)	0.0150 (5)
Li3	0.0835 (4)	0.1886 (3)	0.1848 (2)	0.0134 (5)
Li4	0.6351 (4)	0.3666 (3)	0.4937 (2)	0.0154 (5)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Ru1	0.00449 (4)	0.00398 (4)	0.00420 (4)	−0.00086 (3)	−0.00043 (3)	−0.00135 (3)
Ru2	0.00449 (4)	0.00439 (4)	0.00417 (4)	−0.00112 (3)	−0.00036 (3)	−0.00109 (3)
Cl3	0.00706 (12)	0.00806 (12)	0.00958 (12)	−0.00320 (10)	−0.00223 (10)	−0.00175 (10)
Cl4	0.01010 (13)	0.00619 (12)	0.00716 (12)	−0.00197 (10)	−0.00166 (10)	0.00067 (9)
Cl8	0.00770 (12)	0.00881 (12)	0.00881 (12)	−0.00364 (10)	−0.00242 (10)	−0.00114 (10)
Cl7	0.00854 (12)	0.00731 (12)	0.00797 (12)	−0.00227 (10)	−0.00064 (10)	0.00165 (9)
Cl10	0.00627 (12)	0.01080 (13)	0.01114 (13)	−0.00316 (10)	−0.00164 (10)	−0.00252 (10)
Cl5	0.00621 (12)	0.00848 (12)	0.01061 (13)	−0.00243 (10)	−0.00201 (10)	−0.00161 (10)
Cl6	0.01015 (13)	0.00884 (12)	0.00671 (12)	−0.00130 (10)	0.00055 (10)	−0.00313 (10)
Cl2	0.01008 (13)	0.00671 (12)	0.00717 (12)	−0.00269 (10)	−0.00162 (10)	0.00076 (9)
Cl1	0.01001 (13)	0.00772 (12)	0.00797 (12)	−0.00099 (10)	0.00115 (10)	−0.00376 (10)
Cl9	0.00972 (12)	0.00598 (11)	0.00713 (12)	−0.00184 (10)	−0.00161 (10)	0.00016 (9)
O1	0.0055 (5)	0.0049 (5)	0.0063 (5)	−0.0021 (4)	−0.0010 (4)	−0.0009 (4)
O2	0.0054 (5)	0.0081 (5)	0.0047 (5)	−0.0036 (4)	−0.0001 (4)	−0.0006 (4)
O3	0.0144 (5)	0.0089 (4)	0.0105 (4)	−0.0039 (4)	−0.0030 (4)	−0.0018 (4)
O4	0.0086 (4)	0.0104 (4)	0.0107 (4)	−0.0004 (3)	−0.0027 (3)	−0.0015 (3)
O5	0.0125 (5)	0.0115 (4)	0.0137 (5)	−0.0012 (4)	−0.0056 (4)	−0.0024 (4)
O6	0.0131 (5)	0.0098 (4)	0.0087 (4)	−0.0038 (4)	0.0002 (4)	−0.0016 (3)
O7	0.0127 (5)	0.0116 (5)	0.0128 (5)	−0.0048 (4)	−0.0012 (4)	0.0001 (4)
O8	0.0133 (5)	0.0113 (4)	0.0088 (4)	−0.0049 (4)	−0.0011 (4)	−0.0005 (4)
O9	0.0146 (5)	0.0103 (4)	0.0159 (5)	−0.0064 (4)	−0.0022 (4)	−0.0019 (4)
O10	0.0096 (4)	0.0113 (4)	0.0125 (5)	0.0007 (4)	−0.0004 (4)	0.0007 (4)
O12	0.0148 (5)	0.0121 (5)	0.0241 (6)	−0.0005 (4)	0.0065 (5)	0.0038 (5)
Li1	0.0123 (12)	0.0171 (13)	0.0129 (12)	−0.0030 (10)	−0.0023 (9)	−0.0020 (10)
Li2	0.0146 (12)	0.0122 (12)	0.0176 (13)	−0.0026 (10)	−0.0001 (10)	−0.0041 (10)
Li3	0.0129 (12)	0.0145 (12)	0.0120 (11)	−0.0027 (10)	−0.0007 (9)	−0.0028 (9)
Li4	0.0149 (12)	0.0166 (13)	0.0136 (12)	−0.0032 (10)	0.0000 (10)	−0.0028 (10)

Geometric parameters (Å, º)

Ru1—O1	1.7839 (1)	O7—Li1	2.143 (3)
Ru1—Cl1	2.3405 (3)	O7—H71	0.76 (2)
Ru1—Cl2^i	2.3562 (3)	O7—H72	0.77 (2)
Ru1—Cl5	2.3607 (4)	O8—Li4	2.056 (3)
Ru1—Cl4	2.3633 (3)	O8—Li4^vi	2.153 (3)
Ru1—Cl3	2.3645 (4)	O8—H82	0.80 (2)
Ru2—O2	1.7808 (1)	O9—Li3	1.972 (3)
Ru2—Cl6	2.3358 (3)	O9—Li1	2.181 (3)
Ru2—Cl7	2.3431 (3)	O9—H91	0.783 (15)
Ru2—Cl10	2.3489 (4)	O9—H92	0.721 (15)
Ru2—Cl8	2.3762 (4)	O10—Li1^iv	2.030 (3)
Ru2—Cl9^ii	2.3922 (3)	O10—H101	0.82 (2)
Cl3—Li4	2.701 (3)	O10—H102	0.74 (2)
Cl4—Li2	2.813 (3)	O11—Li4^vi	2.076 (3)
Cl4—Li4	2.836 (3)	O11—Li2^vii	2.117 (3)
Cl5—Li2	2.702 (3)	O11—H111	0.71 (2)
Cl2—Ru1^i	2.3563 (3)	O11—H112	0.83 (2)
Cl9—Ru2^ii	2.3922 (3)	O12—Li2	1.953 (3)
Cl9—Li1^iii	2.855 (3)	O12—H122	0.80 (3)
O1—Ru1^i	1.7839 (1)	Li1—O10^viii	2.030 (3)
O2—Ru2^ii	1.7808 (1)	Li1—O3^i	2.069 (3)
O3—Li1^i	2.069 (3)	Li1—O3^viii	2.179 (3)
O3—Li1^iv	2.179 (3)	Li1—Cl9^iii	2.855 (3)
O3—H32	0.84 (2)	Li1—Li3	2.968 (4)
O4—Li3	1.969 (3)	Li1—Li1^iii	3.121 (6)
O4—Li2	2.140 (3)	Li1—H92	1.93 (2)
O4—H41	0.78 (2)	Li2—O6^ix	2.042 (3)
O4—H42	0.80 (2)	Li2—O11^vii	2.116 (3)
O5—Li3^v	1.913 (3)	Li2—Li4^ix	2.982 (4)
O5—H51	0.776 (15)	Li2—Li3	3.468 (4)
O5—H52	0.790 (15)	Li3—O5^ix	1.913 (3)
O6—Li4	2.035 (3)	Li3—H92	1.41 (2)
O6—Li2^v	2.042 (3)	Li4—O11^vi	2.076 (3)
O6—H61	0.76 (2)	Li4—O8^vi	2.153 (3)
O6—H62	0.78 (2)	Li4—Li2^v	2.982 (4)
O7—Li3	1.966 (3)	Li4—Li4^vi	3.047 (6)
O1—Ru1—Cl1	177.816 (10)	O3^i—Li1—Li3	95.35 (12)
O1—Ru1—Cl2^i	94.139 (11)	O7—Li1—Li3	41.45 (8)
Cl1—Ru1—Cl2^i	87.354 (13)	O3^viii—Li1—Li3	142.95 (14)
O1—Ru1—Cl5	93.552 (10)	O9—Li1—Li3	41.63 (8)
Cl1—Ru1—Cl5	88.061 (13)	Cl9^iii—Li1—Li3	129.80 (12)
Cl2^i—Ru1—Cl5	89.093 (12)	O10^viii—Li1—Li1^iii	130.08 (17)
O1—Ru1—Cl4	92.116 (11)	O3^i—Li1—Li1^iii	44.08 (8)
Cl1—Ru1—Cl4	86.397 (13)	O7—Li1—Li1^iii	100.83 (15)
Cl2^i—Ru1—Cl4	173.744 (12)	O3^viii—Li1—Li1^iii	41.34 (8)
Cl5—Ru1—Cl4	90.407 (12)	O9—Li1—Li1^iii	137.87 (17)
O1—Ru1—Cl3	91.164 (10)	Cl9^iii—Li1—Li1^iii	83.15 (12)
Cl1—Ru1—Cl3	87.223 (13)	Li3—Li1—Li1^iii	128.23 (16)
Cl2^i—Ru1—Cl3	90.628 (12)	O10^viii—Li1—H92	90.5 (7)
Cl5—Ru1—Cl3	175.283 (12)	O3^i—Li1—H92	97.1 (7)
Cl4—Ru1—Cl3	89.357 (12)	O7—Li1—H92	64.2 (5)
O2—Ru2—Cl6	177.250 (10)	O3^viii—Li1—H92	165.6 (5)
O2—Ru2—Cl7	93.010 (11)	O9—Li1—H92	19.0 (5)
Cl6—Ru2—Cl7	87.268 (13)	Cl9^iii—Li1—H92	107.2 (5)
O2—Ru2—Cl10	94.485 (10)	Li3—Li1—H92	22.9 (5)
Cl6—Ru2—Cl10	88.251 (13)	Li1^iii—Li1—H92	139.2 (7)
Cl7—Ru2—Cl10	89.830 (12)	O12—Li2—O6^ix	103.50 (15)
O2—Ru2—Cl8	90.100 (10)	O12—Li2—O11^vii	95.47 (13)
Cl6—Ru2—Cl8	87.171 (12)	O6^ix—Li2—O11^vii	86.47 (12)
Cl7—Ru2—Cl8	88.545 (12)	O12—Li2—O4	92.21 (13)
Cl10—Ru2—Cl8	175.210 (12)	O6^ix—Li2—O4	93.17 (12)
O2—Ru2—Cl9^ii	91.354 (10)	O11^vii—Li2—O4	172.17 (16)
Cl6—Ru2—Cl9^ii	88.402 (13)	O12—Li2—Cl5	162.16 (15)
Cl7—Ru2—Cl9^ii	175.594 (12)	O6^ix—Li2—Cl5	94.34 (11)
Cl10—Ru2—Cl9^ii	89.225 (12)	O11^vii—Li2—Cl5	85.12 (10)
Cl8—Ru2—Cl9^ii	92.055 (12)	O4—Li2—Cl5	87.11 (10)
Ru1—Cl3—Li4	100.25 (6)	O12—Li2—Cl4	87.37 (11)
Ru1—Cl4—Li2	95.85 (6)	O6^ix—Li2—Cl4	168.69 (14)
Ru1—Cl4—Li4	96.58 (6)	O11^vii—Li2—Cl4	95.68 (11)
Li2—Cl4—Li4	166.15 (8)	O4—Li2—Cl4	83.18 (10)
Ru1—Cl5—Li2	98.90 (6)	Cl5—Li2—Cl4	74.84 (8)
Ru2^ii—Cl9—Li1^iii	126.64 (6)	O12—Li2—Li4^ix	97.57 (13)
Ru1—O1—Ru1^i	180.0	O6^ix—Li2—Li4^ix	42.93 (9)
Ru2^ii—O2—Ru2	180.0	O11^vii—Li2—Li4^ix	44.12 (8)
Li1^i—O3—Li1^iv	94.57 (11)	O4—Li2—Li4^ix	136.10 (14)
Li1^i—O3—H32	121.0 (15)	Cl5—Li2—Li4^ix	95.22 (10)
Li1^iv—O3—H32	110.6 (15)	Cl4—Li2—Li4^ix	139.71 (12)
Li3—O4—Li2	115.08 (12)	O12—Li2—Li3	74.92 (11)
Li3—O4—H41	113.9 (16)	O6^ix—Li2—Li3	72.97 (10)
Li2—O4—H41	106.0 (16)	O11^vii—Li2—Li3	154.13 (14)
Li3—O4—H42	109.8 (16)	O4—Li2—Li3	30.93 (7)
Li2—O4—H42	109.9 (16)	Cl5—Li2—Li3	111.41 (10)
H41—O4—H42	101 (2)	Cl4—Li2—Li3	107.57 (10)
Li3^v—O5—H51	115.1 (17)	Li4^ix—Li2—Li3	112.34 (12)
Li3^v—O5—H52	120.3 (17)	O5^ix—Li3—O7	119.00 (15)
H51—O5—H52	112 (2)	O5^ix—Li3—O4	111.94 (14)
Li4—O6—Li2^v	93.98 (13)	O7—Li3—O4	108.84 (14)
Li4—O6—H61	109.2 (17)	O5^ix—Li3—O9	109.92 (14)
Li2^v—O6—H61	117.2 (17)	O7—Li3—O9	93.47 (13)
Li4—O6—H62	118.3 (16)	O4—Li3—O9	112.39 (14)
Li2^v—O6—H62	108.4 (16)	O5^ix—Li3—Li1	127.25 (14)
H61—O6—H62	109 (2)	O7—Li3—Li1	46.18 (9)
Li3—O7—Li1	92.37 (12)	O4—Li3—Li1	120.72 (13)
Li3—O7—H71	118.9 (17)	O9—Li3—Li1	47.29 (9)
Li1—O7—H71	108.3 (17)	O5^ix—Li3—Li2	78.35 (10)
Li3—O7—H72	120.8 (17)	O7—Li3—Li2	124.54 (13)
Li1—O7—H72	117.1 (17)	O4—Li3—Li2	33.98 (8)
H71—O7—H72	100 (2)	O9—Li3—Li2	132.16 (13)
Li4—O8—Li4^vi	92.76 (12)	Li1—Li3—Li2	154.35 (12)
Li4—O8—H82	111.7 (16)	O5^ix—Li3—H92	114.9 (9)
Li4^vi—O8—H82	117.9 (16)	O7—Li3—H92	78.4 (8)
Li3—O9—Li1	91.08 (12)	O4—Li3—H92	120.1 (9)
Li3—O9—H91	126.4 (17)	O9—Li3—H92	15.4 (8)
Li1—O9—H91	114.2 (17)	Li1—Li3—H92	32.3 (8)
Li3—O9—H92	31.2 (18)	Li2—Li3—H92	145.8 (8)
Li1—O9—H92	60.5 (18)	O6—Li4—O8	99.27 (13)
H91—O9—H92	131 (2)	O6—Li4—O11^vi	87.73 (12)
Li1^iv—O10—H101	112.1 (16)	O8—Li4—O11^vi	104.99 (14)
Li1^iv—O10—H102	122.2 (18)	O6—Li4—O8^vi	171.67 (17)
H101—O10—H102	101 (2)	O8—Li4—O8^vi	87.25 (12)
Li4^vi—O11—Li2^vii	90.66 (12)	O11^vi—Li4—O8^vi	95.64 (13)
Li4^vi—O11—H111	112.1 (18)	O6—Li4—Cl3	85.61 (10)
Li2^vii—O11—H111	107.1 (18)	O8—Li4—Cl3	158.15 (15)
Li4^vi—O11—H112	112.9 (15)	O11^vi—Li4—Cl3	96.43 (11)
Li2^vii—O11—H112	120.5 (15)	O8^vi—Li4—Cl3	86.44 (10)
H111—O11—H112	112 (2)	O6—Li4—Cl4	90.98 (11)
Li2—O12—H122	118.9 (18)	O8—Li4—Cl4	84.84 (10)
O10^viii—Li1—O3^i	167.40 (17)	O11^vi—Li4—Cl4	170.17 (14)
O10^viii—Li1—O7	98.16 (13)	O8^vi—Li4—Cl4	84.41 (10)
O3^i—Li1—O7	94.22 (12)	Cl3—Li4—Cl4	73.75 (7)
O10^viii—Li1—O3^viii	89.76 (12)	O6—Li4—Li2^v	43.09 (9)
O3^i—Li1—O3^viii	85.43 (11)	O8—Li4—Li2^v	101.17 (12)
O7—Li1—O3^viii	101.50 (13)	O11^vi—Li4—Li2^v	45.22 (9)
O10^viii—Li1—O9	89.89 (12)	O8^vi—Li4—Li2^v	140.86 (15)
O3^i—Li1—O9	93.97 (12)	Cl3—Li4—Li2^v	96.89 (10)
O7—Li1—O9	83.08 (11)	Cl4—Li4—Li2^v	134.05 (12)
O3^viii—Li1—O9	175.41 (16)	O6—Li4—Li4^vi	143.87 (18)
O10^viii—Li1—Cl9^iii	85.44 (10)	O8—Li4—Li4^vi	44.88 (8)
O3^i—Li1—Cl9^iii	82.69 (10)	O11^vi—Li4—Li4^vi	104.12 (16)
O7—Li1—Cl9^iii	170.61 (14)	O8^vi—Li4—Li4^vi	42.36 (8)
O3^viii—Li1—Cl9^iii	87.13 (10)	Cl3—Li4—Li4^vi	125.61 (15)
O9—Li1—Cl9^iii	88.28 (10)	Cl4—Li4—Li4^vi	82.56 (12)
O10^viii—Li1—Li3	95.50 (12)	Li2^v—Li4—Li4^vi	132.74 (17)

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