Supporting information
Crystallographic Information File (CIF) https://doi.org/10.1107/S0108270113016491/sk3495sup1.cif | |
Rietveld powder data file (CIF format) https://doi.org/10.1107/S0108270113016491/sk3495Isup2.rtv |
CCDC reference: 958840
Bisphosphonates are biologically relevant therapeutics for bone disorders and cancer (Abdou & Shaddy, 2009; Fleisch, 1998). These compounds exhibit specific affinity towards bone, which makes them an excellent therapeutic for bone-resorption diseases (especially osteoporosis, Paget's disease, tumour-induced osteolysis and hypercalcaemia originating from malignancy) by inhibition of farnesyl diphosphate synthase, and for bone tumours caused by metastatic breast tumours (Body et al., 1998; Russell & Rogers, 1999; Szabo et al., 2002; Fleisch, 2003). The biological activity of bisphosphonates is dependent on the structure (Graham & Russell, 2007), lipophilicity (Zhang et al., 2009) and bone-binding affinity (Nancollas et al., 2006) of the compounds. Organic diphosphonic acids are potentially very powerful chelating agents used in metal extraction and have been tested by the pharmaceutical industry for use as efficient drugs to prevent calcification and to inhibit bone resorption (Tromelin et al., 1986; Matczak-Jon & Videnova-Adrabinska, 2005). However, it is still not clearly understood why small structural modifications of bisphosphonates may lead to extensive alterations in their physicochemical, biological and toxicological characteristics (Matczak-Jon & Videnova-Adrabinska, 2005). As a consequence, determination of the structure of bisphosphonates is very important to understand the influence of structural modifications on their complex-forming abilities and physiological activities, and for deriving structure–property relationships in general.
The crystal structures of tetrahydrofuranyl-2,2-bisphosphonic acid disodium salt (Maltezou et al., 2010) and (1-ammonioethane-1,1-diyl)bis(hydrogenphosphonato)]diaquachloridodisodium (Rukiah & Assaad, 2012) have been reported previously. In this paper, the crystal structure of the title compound, (I), is reported. This compound crystallizes as a fine white powder and, since no single crystal of sufficient size and quality was obtained, structure determination by powder X-ray diffraction was undertaken. We used laboratory powder X-ray diffraction data to solve and refine the crystal structure of (I).
This is a 12-atom (non-H) problem which requires careful measurement and interpretation of the in-house data in order to optimize the quality of the results. Compound (I) crystallizes in the triclinic system in a centrosymmetric space group (P1), with one molecule in the asymmetric unit. The asymmetric unit (Fig. 1) contains one (1-azaniumylethane-1,1-diyl)bis(hydrogen phosphonate) anion and one Na+ cation. The bisphosphonate anion in (I) has an overall -1 charge, and thus two of the phosphonate O atoms are deprotonated (P—O-), two have H atoms attached (P—OH) and two form double bonds with P (P═O). Two of the four phosphonate H atoms are used in the protonation, one for the amino group and the other for the Na+ cation. [Remove?]
The bond lengths and angles in (I) are in the normal ranges (Allen et al., 1987). Both P atoms exhibit a distorted tetrahedral geometry, with average P—OH(protonated) = 1.592 Å, average P—O(deprotonated) = 1.554 Å and average P═O = 1.490 Å (Table 1). Although the P—OH bonds can be located very easily by inspection of the bond lengths, the P—O- and P═O bonds cannot be distinguished because of the charge delocalization over the O-—P═O groups. The Na—O bond lengths range from 2.202 (9) to 2.680 (13) Å (average 2.462 Å), with Na1—O2(x, y + 1, z) being the longest. The central Na+ cations exhibit a distorted octahedral coordination geometry involving six bisphosphonate O atoms (two oxide, two hydroxy and two double-bond O atoms). Two neighbouring sodium-centred octahedra form an edge-sharing environment, connected to each other by the bisphosphonate anions to form two-dimensional infinite sheets parallel to the (001) plane. The two-dimensional networks are stacked on top of each other along the c axis. The layers are further connected by extensive N—H···O and O—H···O hydrogen bonding (Table 2); the strongest is O1—H1···O3(-x, -y, -z + 1), which links the layers into a three-dimensional network (Fig. 2 and Table 2).
For the syntheses of (I), a mixture of acetonitrile (150 ml) and phosphorus [phosphoric?] acid (16.8 g, 0.2 mol) in acetic acid (10 g, 0.167 mol) was heated at 328–338 K and phosphoryl trichloride (POCl3; 51.7 g, 0.334 mol) was added slowly with stirring. After completion of the addition, the reaction mixture was kept for 24 h at 343–348 K. The reaction mixture was then cooled to 333–338 K and water (150 ml) was added slowly at the same temperature. The temperature was increased to 363–373 K and maintained for 4–6 h. The solid product which formed was separated by filtration and washed with water and finally with methanol. Sodium hydroxide (1 N, 15 ml) was added to this solid (200 mg) and the pH was adjusted to 6–7. The mixture was stirred at room temperature for 18 h. The resulting solution was freeze-dried and the crude product was further purified by recrystallization from absolute ethanol and water [Solvent ratio?] at 273 K to produce the product, (I) (white powder; m.p. 549–553 K) in 77% yield. Spectroscopic analysis: 1H NMR (D2O, δ, p.p.m.): 1.60 (t, 3H, CH3, J = 12.00 Hz); 13C{1H} NMR (D2O, δ, p.p.m.): 20.45 (1C, CH3), 56.84 (1C, C—CH3); 31P{1H} NMR (D2O, δ, p.p.m.): 13.73 (2P, P—OH); IR (KBr, ν, cm-1): 3450 (NH2), 3419 (OH), 1634.7 (O═P—O—H). Analytical data for (I), found: C 11.51, H 4.32, N 6.30%; calculated: C 10.58, H 3.55, N 6.17%.
For pattern indexing, the extraction of the peak positions was carried out with the program WinPLOTR (Roisnel & Rodriguez-Carvajal, 2001). Pattern indexing was performed with the program DICVOL4.0 (Boultif & Louër, 2004). The first 20 lines of the powder pattern were completely indexed on the basis of a primitive triclinic cell. The absolute error on each observed line was fixed at 0.02° (2θ). The figures of merit are sufficiently good to support the obtained indexing results [M(20) = 41.0 and F(20) = 72.8(0.0081, 34)]. The whole powder diffraction pattern from 8 to 85° (2θ) was subsequently refined with cell and resolution constraints (Le Bail et al., 1988) in space group P1, using the `profile matching' option of the program FULLPROF (Rodriguez-Carvajal, 2001). The number of molecules per unit cell was estimated to be Z = 2. The structure was solved by direct methods using the program EXPO2009 (Altomare et al., 2009). The model found by this program was introduced into the program GSAS (Larson & Von Dreele, 2004) implemented in EXPGUI (Toby, 2001) for Rietveld refinement. The background was refined using a shifted Chebyshev polynomial with 20 coefficients. During the Rietveld refinement, the effect of asymmetry of low-order peaks was corrected using a pseudo-Voigt description of the peak shape (Thompson et al., 1987), which allows for angle-dependent asymmetry with axial divergence (Finger et al., 1994) and microstrain broadening as described by Stephens (1999). The two asymmetry parameters of this function, S/L and D/L, were both fixed at 0.0235 during the Rietveld refinement. Intensities were corrected from absorption effects with a µ.d value of 0.1. The preferred orientation was modelled with six coefficients using a spherical harmonics correction (Von Dreele, 1997) of intensities in the final refinement. The use of the preferred orientation correction leads to better molecular geometry with better agreement factors. Two C—CH3 and C—NH3 bond lengths were restrained to their normal values. The Rietveld refinement cycles were performed using a global isotropic atomic displacement parameter for each type of non-H atom. The hydroxy H atoms were located in a difference map. The methyl and amino H atoms were positioned in their idealized geometries using a riding model, with C—H = 0.97 Å and N—H = 0.87 Å. The coordinates of these H atoms were not refined. The isotropic displacement parameters were set at 0.02 Å2. The final Rietveld agreement factors are Rp = 0.023, Rwp = 0.030, Rexp = 0.022, χ2 = 1.904 and RF2 = 0.01638. The final Rietveld plot of the X-ray diffraction pattern is given in Fig. 3.
For related literature, see: Abdou & Shaddy (2009); Allen et al. (1987); Altomare et al. (2009); Body et al. (1998); Boultif & Louër (2004); Finger et al. (1994); Fleisch (1998, 2003); Graham & Russell (2007); Larson & Von Dreele (2004); Le Bail, Duroy & Fourquet (1988); Maltezou et al. (2010); Matczak-Jon & Videnova-Adrabinska (2005); Nancollas et al. (2006); Rodriguez-Carvajal (2001); Roisnel & Rodriguez-Carvajal (2001); Rukiah & Assaad (2012); Russell & Rogers (1999); Stephens (1999); Szabo et al. (2002); Thompson et al. (1987); Toby (2001); Tromelin et al. (1986); Von Dreele (1997); Zhang (2009).
Data collection: WinXPOW (Stoe & Cie, 1999); cell refinement: GSAS (Larson & Von Dreele, 2004); data reduction: WinXPOW (Stoe & Cie, 1999); program(s) used to solve structure: EXPO2009 (Altomare et al., 2009); program(s) used to refine structure: GSAS (Larson & Von Dreele, 2004); molecular graphics: ORTEP-3 (Farrugia, 2012); software used to prepare material for publication: publCIF (Westrip, 2010).
[Na(C2H8NO6P2)] | Z = 2 |
Mr = 227.02 | F(000) = 232 |
Triclinic, P1 | Dx = 1.96 Mg m−3 |
Hall symbol: -P 1 | Melting point: 551 K |
a = 5.53651 (11) Å | Cu Kα1 radiation, λ = 1.5406 Å |
b = 8.1058 (3) Å | µ = 5.76 mm−1 |
c = 9.1108 (2) Å | T = 298 K |
α = 98.086 (2)° | Particle morphology: Fine powder |
β = 102.0847 (16)° | white |
γ = 101.4987 (19)° | flat sheet, 8 × 8 mm |
V = 384.62 (2) Å3 | Specimen preparation: Prepared at 298 K and 101.3 kPa |
Stoe STADI P diffractometer | Scan method: step |
Radiation source: sealed X-ray tube | Absorption correction: for a cylinder mounted on the ϕ axis [GSAS absorption/surface roughness correction (Larson & Von Dreele, 2004): function number 4; flat plate in transmission mode, absorbtion correction term (µ.d) not refined] |
Ge (111) monochromator | Tmin = 0.811, Tmax = 0.857 |
Specimen mounting: powder loaded between two Mylar foils | 2θmin = 5.0°, 2θmax = 85°, 2θstep = 0.02° |
Data collection mode: transmission |
Least-squares matrix: full | Profile function: pseudo-Voigt function of the peak shape for angle-dependent asymmetry with axial divergence |
Rp = 0.023 | 112 parameters |
Rwp = 0.030 | 2 restraints |
Rexp = 0.022 | 3 constraints |
R(F2) = 0.01639 | H-atom parameters not refined |
χ2 = 1.904 | (Δ/σ)max = 0.02 |
4000 data points | Background function: shifted Chebyshev polynomial with 20 coefficients. |
Excluded region(s): none | Preferred orientation correction: spherical harmonics correction |
[Na(C2H8NO6P2)] | γ = 101.4987 (19)° |
Mr = 227.02 | V = 384.62 (2) Å3 |
Triclinic, P1 | Z = 2 |
a = 5.53651 (11) Å | Cu Kα1 radiation, λ = 1.5406 Å |
b = 8.1058 (3) Å | µ = 5.76 mm−1 |
c = 9.1108 (2) Å | T = 298 K |
α = 98.086 (2)° | flat sheet, 8 × 8 mm |
β = 102.0847 (16)° |
Stoe STADI P diffractometer | Absorption correction: for a cylinder mounted on the ϕ axis [GSAS absorption/surface roughness correction (Larson & Von Dreele, 2004): function number 4; flat plate in transmission mode, absorbtion correction term (µ.d) not refined] |
Specimen mounting: powder loaded between two Mylar foils | Tmin = 0.811, Tmax = 0.857 |
Data collection mode: transmission | 2θmin = 5.0°, 2θmax = 85°, 2θstep = 0.02° |
Scan method: step |
Rp = 0.023 | 4000 data points |
Rwp = 0.030 | 112 parameters |
Rexp = 0.022 | 2 restraints |
R(F2) = 0.01639 | H-atom parameters not refined |
χ2 = 1.904 |
Experimental. The sample was ground lightly in a mortar, loaded between two Mylar foils and fixed in the sample holder with a mask of 8.0 mm internal diameter |
Geometry. Bond distances, angles etc. have been calculated using the rounded fractional coordinates. All su's are estimated from the variances of the (full) variance-covariance matrix. The cell e.s.d.'s are taken into account in the estimation of distances, angles and torsion angles |
x | y | z | Uiso*/Ueq | ||
P1 | −0.1518 (7) | −0.0122 (8) | 0.2627 (4) | 0.0059 (8)* | |
P2 | −0.0944 (7) | 0.3408 (7) | 0.1498 (4) | 0.0059 (8)* | |
Na1 | −0.5425 (7) | 0.6265 (9) | 0.1563 (5) | 0.0181 (15)* | |
O1 | −0.3231 (14) | −0.1030 (15) | 0.3623 (9) | 0.0068 (11)* | |
O2 | −0.2119 (14) | −0.1095 (15) | 0.1071 (8) | 0.0068 (11)* | |
O3 | 0.1283 (10) | 0.0369 (14) | 0.3564 (8) | 0.0068 (11)* | |
O4 | −0.2585 (12) | 0.4757 (12) | 0.1325 (7) | 0.0068 (11)* | |
O5 | −0.1653 (11) | 0.2225 (14) | −0.0154 (7) | 0.0068 (11)* | |
O6 | 0.1881 (11) | 0.4144 (13) | 0.2207 (8) | 0.0068 (11)* | |
N1 | −0.5181 (12) | 0.1551 (15) | 0.2009 (7) | 0.0043 (17)* | |
C1 | −0.2364 (14) | 0.2020 (19) | 0.2643 (10) | 0.0043 (17)* | |
C2 | −0.2002 (16) | 0.3029 (17) | 0.4283 (10) | 0.0043 (17)* | |
H1 | −0.23908 | −0.08842 | 0.45130 | 0.0200* | |
H1N1 | −0.59564 | 0.12310 | 0.26919 | 0.0200* | |
H2N1 | −0.56649 | 0.24522 | 0.17275 | 0.0200* | |
H2A | −0.02088 | 0.35841 | 0.47085 | 0.0200* | |
H2B | −0.29612 | 0.39237 | 0.42348 | 0.0200* | |
H2C | −0.25879 | 0.22830 | 0.49336 | 0.0200* | |
H3N1 | −0.55873 | 0.07302 | 0.12121 | 0.0200* | |
H5 | −0.03482 | 0.20831 | −0.03895 | 0.0200* |
P1—O1 | 1.592 (11) | Na1—O4iii | 2.575 (8) |
P1—O2 | 1.458 (9) | Na1—O5iii | 2.475 (10) |
P1—O3 | 1.548 (8) | O1—H1 | 0.83 |
P1—C1 | 1.885 (15) | O5—H5 | 0.82 |
P2—O4 | 1.559 (10) | N1—C1 | 1.495 (11) |
P2—O5 | 1.592 (8) | N1—H1N1 | 0.87 |
P2—O6 | 1.522 (8) | N1—H2N1 | 0.88 |
P2—C1 | 1.815 (13) | N1—H3N1 | 0.87 |
Na1—O4 | 2.202 (9) | C1—C2 | 1.552 (14) |
Na1—O6i | 2.270 (11) | C2—H2A | 0.97 |
Na1—O1ii | 2.575 (12) | C2—H2B | 0.98 |
Na1—O2ii | 2.680 (13) | C2—H2C | 0.96 |
O1—P1—O2 | 111.6 (6) | P1—O2—Na1iv | 95.0 (5) |
O1—P1—O3 | 108.3 (5) | P2—O4—Na1 | 162.8 (5) |
O1—P1—C1 | 103.3 (5) | P2—O4—Na1iii | 96.4 (4) |
O2—P1—O3 | 118.5 (5) | Na1—O4—Na1iii | 91.8 (3) |
O2—P1—C1 | 111.0 (6) | P2—O5—Na1iii | 99.5 (5) |
O3—P1—C1 | 102.9 (6) | P2—O6—Na1v | 128.5 (5) |
O4—P2—O5 | 104.7 (4) | Na1iv—O1—H1 | 133 |
O4—P2—O6 | 115.3 (6) | P1—O1—H1 | 110 |
O4—P2—C1 | 104.9 (5) | P2—O5—H5 | 109 |
O5—P2—O6 | 115.8 (4) | Na1iii—O5—H5 | 123 |
O5—P2—C1 | 103.6 (6) | C1—N1—H2N1 | 109 |
O6—P2—C1 | 111.3 (5) | C1—N1—H3N1 | 110 |
O4—Na1—O6i | 94.7 (4) | C1—N1—H1N1 | 111 |
O1ii—Na1—O4 | 107.9 (3) | H2N1—N1—H3N1 | 109 |
O2ii—Na1—O4 | 86.6 (3) | H1N1—N1—H3N1 | 110 |
O4—Na1—O4iii | 88.2 (3) | H1N1—N1—H2N1 | 109 |
O4—Na1—O5iii | 143.4 (3) | P1—C1—C2 | 112.8 (8) |
O1ii—Na1—O6i | 117.5 (3) | P1—C1—N1 | 103.5 (9) |
O2ii—Na1—O6i | 174.8 (3) | N1—C1—C2 | 103.9 (7) |
O4iii—Na1—O6i | 93.5 (3) | P2—C1—N1 | 108.3 (7) |
O5iii—Na1—O6i | 102.8 (3) | P2—C1—C2 | 110.2 (10) |
O1ii—Na1—O2ii | 57.3 (3) | P1—C1—P2 | 117.0 (5) |
O1ii—Na1—O4iii | 142.7 (4) | C1—C2—H2A | 109 |
O1ii—Na1—O5iii | 92.2 (4) | C1—C2—H2B | 109 |
O2ii—Na1—O4iii | 91.6 (3) | C1—C2—H2C | 111 |
O2ii—Na1—O5iii | 78.9 (3) | H2A—C2—H2B | 108 |
O4iii—Na1—O5iii | 59.2 (3) | H2A—C2—H2C | 110 |
P1—O1—Na1iv | 95.8 (4) | H2B—C2—H2C | 109 |
O2—P1—O1—Na1iv | 5.2 (6) | O5—P2—O6—Na1v | 72.8 (9) |
O3—P1—O1—Na1iv | −127.0 (5) | C1—P2—O6—Na1v | −169.2 (6) |
C1—P1—O1—Na1iv | 124.4 (4) | O4—P2—C1—P1 | 162.9 (5) |
O1—P1—O2—Na1iv | −5.0 (6) | O4—P2—C1—N1 | 46.6 (9) |
O3—P1—O2—Na1iv | 121.8 (6) | O4—P2—C1—C2 | −66.5 (7) |
C1—P1—O2—Na1iv | −119.5 (5) | O5—P2—C1—P1 | 53.3 (6) |
O1—P1—C1—P2 | −175.8 (5) | O5—P2—C1—N1 | −63.0 (9) |
O1—P1—C1—N1 | −56.9 (6) | O5—P2—C1—C2 | −176.1 (6) |
O1—P1—C1—C2 | 54.8 (8) | O6—P2—C1—P1 | −71.8 (7) |
O2—P1—C1—P2 | −56.2 (7) | O6—P2—C1—N1 | 171.9 (7) |
O2—P1—C1—N1 | 62.8 (7) | O6—P2—C1—C2 | 58.8 (8) |
O2—P1—C1—C2 | 174.5 (7) | O6i—Na1—O4—Na1iii | 93.3 (3) |
O3—P1—C1—P2 | 71.6 (6) | O1ii—Na1—O4—Na1iii | −145.7 (4) |
O3—P1—C1—N1 | −169.5 (5) | O2ii—Na1—O4—Na1iii | −91.7 (3) |
O3—P1—C1—C2 | −57.8 (8) | O4iii—Na1—O4—Na1iii | 0.0 (3) |
O5—P2—O4—Na1iii | −3.2 (5) | O5iii—Na1—O4—Na1iii | −25.6 (8) |
O6—P2—O4—Na1iii | 125.3 (4) | O4—Na1—O6i—P2i | −127.8 (6) |
C1—P2—O4—Na1iii | −111.9 (4) | O4—Na1—O1ii—P1ii | 70.5 (5) |
O4—P2—O5—Na1iii | 3.3 (5) | O4—Na1—O2ii—P1ii | −110.4 (5) |
O6—P2—O5—Na1iii | −124.9 (5) | O4—Na1—O4iii—P2iii | −164.8 (5) |
C1—P2—O5—Na1iii | 113.0 (5) | O4—Na1—O4iii—Na1iii | 0.0 (4) |
O4—P2—O6—Na1v | −49.9 (7) | O4—Na1—O5iii—P2iii | 32.5 (9) |
Symmetry codes: (i) x−1, y, z; (ii) x, y+1, z; (iii) −x−1, −y+1, −z; (iv) x, y−1, z; (v) x+1, y, z. |
D—H···A | D—H | H···A | D···A | D—H···A |
O1—H1···O3vi | 0.83 | 1.69 | 2.499 (11) | 165 |
N1—H1N1···O3i | 0.87 | 1.92 | 2.748 (11) | 159 |
N1—H2N1···O6i | 0.88 | 2.18 | 2.911 (13) | 141 |
N1—H3N1···O5vii | 0.87 | 2.51 | 3.235 (14) | 142 |
O5—H5···O2viii | 0.82 | 1.88 | 2.679 (11) | 163 |
Symmetry codes: (i) x−1, y, z; (vi) −x, −y, −z+1; (vii) −x−1, −y, −z; (viii) −x, −y, −z. |
Experimental details
Crystal data | |
Chemical formula | [Na(C2H8NO6P2)] |
Mr | 227.02 |
Crystal system, space group | Triclinic, P1 |
Temperature (K) | 298 |
a, b, c (Å) | 5.53651 (11), 8.1058 (3), 9.1108 (2) |
α, β, γ (°) | 98.086 (2), 102.0847 (16), 101.4987 (19) |
V (Å3) | 384.62 (2) |
Z | 2 |
Radiation type | Cu Kα1, λ = 1.5406 Å |
µ (mm−1) | 5.76 |
Specimen shape, size (mm) | Flat sheet, 8 × 8 |
Data collection | |
Diffractometer | Stoe STADI P diffractometer |
Specimen mounting | Powder loaded between two Mylar foils |
Data collection mode | Transmission |
Scan method | Step |
Absorption correction | For a cylinder mounted on the ϕ axis [GSAS absorption/surface roughness correction (Larson & Von Dreele, 2004): function number 4; flat plate in transmission mode, absorbtion correction term (µ.d) not refined] |
Tmin, Tmax | 0.811, 0.857 |
2θ values (°) | 2θmin = 5.0 2θmax = 85 2θstep = 0.02 |
Refinement | |
R factors and goodness of fit | Rp = 0.023, Rwp = 0.030, Rexp = 0.022, R(F2) = 0.01639, χ2 = 1.904 |
No. of data points | 4000 |
No. of parameters | 112 |
No. of restraints | 2 |
H-atom treatment | H-atom parameters not refined |
Computer programs: WinXPOW (Stoe & Cie, 1999), GSAS (Larson & Von Dreele, 2004), EXPO2009 (Altomare et al., 2009), ORTEP-3 (Farrugia, 2012), publCIF (Westrip, 2010).
P1—O1 | 1.592 (11) | Na1—O4 | 2.202 (9) |
P1—O2 | 1.458 (9) | Na1—O6i | 2.270 (11) |
P1—O3 | 1.548 (8) | Na1—O1ii | 2.575 (12) |
P2—O4 | 1.559 (10) | Na1—O2ii | 2.680 (13) |
P2—O5 | 1.592 (8) | Na1—O4iii | 2.575 (8) |
P2—O6 | 1.522 (8) | Na1—O5iii | 2.475 (10) |
Symmetry codes: (i) x−1, y, z; (ii) x, y+1, z; (iii) −x−1, −y+1, −z. |
D—H···A | D—H | H···A | D···A | D—H···A |
O1—H1···O3iv | 0.83 | 1.69 | 2.499 (11) | 165 |
N1—H1N1···O3i | 0.87 | 1.92 | 2.748 (11) | 159 |
N1—H2N1···O6i | 0.88 | 2.18 | 2.911 (13) | 141 |
N1—H3N1···O5v | 0.87 | 2.51 | 3.235 (14) | 142 |
O5—H5···O2vi | 0.82 | 1.88 | 2.679 (11) | 163 |
Symmetry codes: (i) x−1, y, z; (iv) −x, −y, −z+1; (v) −x−1, −y, −z; (vi) −x, −y, −z. |
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