D-Mannosamine hydrochloride (2-amino-2-deoxy-
D-mannose hydrochloride), C
6H
14NO
5+·Cl
−, (I), crystallized from a methanol/ethyl acetate/
n-hexane solvent mixture at room temperature in a
4C1 chair conformation that is slightly distorted towards the
C3,O5B form. A comparison of the structural parameters of (I) with the corresponding parameters in α-
D-glucosamine hydrochloride, (II), and β-
D-galactosamine hydrochloride, (III)/(III′), was undertaken to evaluate the effects of ionic hydrogen bonding on structural properties. Three types of ionic hydrogen bonds are present in the crystals of (I)–(III)/(III′),
i.e. N
+—H
O, N
+—H
Cl
−, and O—H
Cl
−. The exocyclic structural parameters in (I), (II), and (III)/(III′) appear to be most influenced by this bonding, especially the exocyclic hydroxy groups, which adopt eclipsed conformations enabled by ionic hydrogen bonding to the chloride anion. Anomeric disorder was observed in crystals of (I), with an α:β ratio of 37:63. However, anomeric configuration appears to exert minimal structural effects; that is, bond lengths, bond angles, and torsion angles are essentially identical in both anomers. The observed disorder at the anomeric C atom of (I) appears to be caused by the presence of the chloride anion and atom O3 or O4 in proximal voids, which provide opportunities for hydrogen bonding to atom O1 in both axial and equatorial orientations.
Supporting information
CCDC reference: 2154074
Data collection: APEX3 (Bruker, 2018); cell refinement: SAINT (Bruker, 2018); data reduction: SAINT (Bruker, 2018); program(s) used to solve structure: SHELXT2014 (Sheldrick, 2015a); program(s) used to refine structure: SHELXL2018 (Sheldrick, 2015b); molecular graphics: Mercury (Macrae et al., 2020); software used to prepare material for publication: SHELXL2018 (Sheldrick, 2015b).
2-Amino-2-deoxy-
D-mannose hydrochloride
top
Crystal data top
C6H14NO5+·Cl− | Dx = 1.540 Mg m−3 |
Mr = 215.63 | Cu Kα radiation, λ = 1.54178 Å |
Orthorhombic, P212121 | Cell parameters from 9972 reflections |
a = 8.3512 (4) Å | θ = 5.9–70.4° |
b = 10.0750 (5) Å | µ = 3.65 mm−1 |
c = 11.0555 (6) Å | T = 120 K |
V = 930.19 (8) Å3 | Block, colorless |
Z = 4 | 0.28 × 0.20 × 0.18 mm |
F(000) = 456 | |
Data collection top
Bruker PHOTON II diffractometer | 1773 independent reflections |
Radiation source: Ius micro-focus | 1770 reflections with I > 2σ(I) |
Detector resolution: 7.41 pixels mm-1 | Rint = 0.068 |
ω and φ–scans | θmax = 70.5°, θmin = 5.9° |
Absorption correction: numerical (SADABS; Krause et al., 2015) | h = −10→10 |
Tmin = 0.462, Tmax = 0.589 | k = −12→12 |
19393 measured reflections | l = −13→13 |
Refinement top
Refinement on F2 | Secondary atom site location: difference Fourier map |
Least-squares matrix: full | Hydrogen site location: mixed |
R[F2 > 2σ(F2)] = 0.037 | H atoms treated by a mixture of independent and constrained refinement |
wR(F2) = 0.089 | w = 1/[σ2(Fo2) + 1.3591P] where P = (Fo2 + 2Fc2)/3 |
S = 1.22 | (Δ/σ)max < 0.001 |
1773 reflections | Δρmax = 0.22 e Å−3 |
160 parameters | Δρmin = −0.25 e Å−3 |
0 restraints | Absolute structure: Flack x determined using 708 quotients [(I+)-(I-)]/[(I+)+(I-)]
(Parsons et al., 2013) |
Primary atom site location: dual | Absolute structure parameter: 0.060 (10) |
Special details top
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 (Å2) top | x | y | z | Uiso*/Ueq | Occ. (<1) |
Cl1 | 0.62782 (14) | 0.46942 (10) | 0.83365 (9) | 0.0272 (3) | |
O1 | 0.4210 (7) | 0.4273 (5) | 0.1434 (4) | 0.0275 (16) | 0.628 (11) |
H1O | 0.418 (19) | 0.515 (17) | 0.114 (14) | 0.12 (6)* | 0.628 (11) |
O1A | 0.2130 (11) | 0.3879 (10) | 0.2779 (9) | 0.036 (3) | 0.372 (11) |
H1OA | 0.16 (2) | 0.452 (16) | 0.241 (14) | 0.05 (4)* | 0.372 (11) |
O3 | 0.5395 (4) | 0.1644 (3) | 0.4828 (3) | 0.0235 (7) | |
H3O | 0.494 (9) | 0.122 (7) | 0.531 (7) | 0.06 (2)* | |
O4 | 0.3977 (4) | 0.3747 (3) | 0.6342 (3) | 0.0227 (7) | |
H4O | 0.460 (7) | 0.391 (6) | 0.686 (5) | 0.034 (16)* | |
O5 | 0.4320 (3) | 0.5213 (3) | 0.3310 (2) | 0.0192 (6) | |
O6 | 0.3460 (4) | 0.7488 (3) | 0.4689 (3) | 0.0272 (7) | |
H6O | 0.360 (8) | 0.810 (6) | 0.514 (5) | 0.034 (14)* | |
N2 | 0.6321 (5) | 0.2938 (3) | 0.2770 (3) | 0.0162 (7) | |
H2NC | 0.690 (6) | 0.232 (5) | 0.316 (5) | 0.032 (14)* | |
H2NB | 0.631 (5) | 0.285 (4) | 0.201 (4) | 0.006 (9)* | |
H2NA | 0.673 (6) | 0.383 (6) | 0.295 (4) | 0.030 (14)* | |
C1 | 0.3767 (6) | 0.4101 (4) | 0.2640 (4) | 0.0224 (8) | |
H1 | 0.257820 | 0.401209 | 0.271205 | 0.027* | 0.628 (11) |
H1A | 0.401375 | 0.423626 | 0.176312 | 0.027* | 0.372 (11) |
C2 | 0.4593 (5) | 0.2851 (4) | 0.3081 (3) | 0.0172 (8) | |
H2 | 0.411714 | 0.206877 | 0.265613 | 0.021* | |
C3 | 0.4359 (5) | 0.2680 (4) | 0.4440 (3) | 0.0167 (8) | |
H3 | 0.323017 | 0.238937 | 0.458421 | 0.020* | |
C4 | 0.4651 (5) | 0.3944 (4) | 0.5169 (3) | 0.0157 (8) | |
H4 | 0.582936 | 0.409874 | 0.524592 | 0.019* | |
C5 | 0.3873 (5) | 0.5150 (4) | 0.4575 (3) | 0.0167 (8) | |
H5 | 0.268152 | 0.508716 | 0.464874 | 0.020* | |
C6 | 0.4445 (6) | 0.6424 (4) | 0.5127 (4) | 0.0217 (9) | |
H6A | 0.437407 | 0.637221 | 0.601987 | 0.026* | |
H6B | 0.557658 | 0.658379 | 0.490346 | 0.026* | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
Cl1 | 0.0383 (5) | 0.0222 (5) | 0.0212 (4) | −0.0120 (5) | 0.0001 (5) | 0.0021 (4) |
O1 | 0.046 (3) | 0.021 (3) | 0.016 (3) | 0.007 (2) | −0.004 (2) | 0.0019 (18) |
O1A | 0.033 (5) | 0.030 (5) | 0.044 (6) | 0.015 (4) | −0.026 (4) | −0.010 (4) |
O3 | 0.0240 (15) | 0.0191 (14) | 0.0273 (16) | 0.0069 (13) | 0.0075 (13) | 0.0079 (13) |
O4 | 0.0245 (16) | 0.0300 (16) | 0.0135 (13) | −0.0005 (13) | 0.0023 (12) | 0.0053 (12) |
O5 | 0.0286 (14) | 0.0154 (13) | 0.0135 (12) | 0.0019 (11) | 0.0022 (11) | 0.0002 (12) |
O6 | 0.045 (2) | 0.0156 (14) | 0.0209 (14) | 0.0106 (15) | −0.0004 (14) | −0.0013 (12) |
N2 | 0.0181 (17) | 0.0140 (16) | 0.0165 (16) | 0.0013 (14) | −0.0003 (15) | 0.0025 (12) |
C1 | 0.029 (2) | 0.0185 (19) | 0.0197 (18) | 0.0055 (18) | −0.0071 (19) | −0.0032 (15) |
C2 | 0.0171 (19) | 0.0154 (18) | 0.0192 (19) | 0.0011 (15) | −0.0020 (15) | −0.0023 (15) |
C3 | 0.0150 (18) | 0.0124 (18) | 0.023 (2) | 0.0033 (15) | 0.0035 (15) | 0.0030 (16) |
C4 | 0.0150 (18) | 0.0172 (18) | 0.0150 (18) | −0.0012 (15) | 0.0019 (15) | 0.0015 (15) |
C5 | 0.0210 (19) | 0.0152 (17) | 0.0140 (16) | 0.0021 (16) | 0.0025 (15) | 0.0011 (14) |
C6 | 0.029 (2) | 0.017 (2) | 0.020 (2) | 0.0000 (17) | 0.0002 (17) | 0.0011 (16) |
Geometric parameters (Å, º) top
O1—C1 | 1.394 (6) | N2—H2NA | 0.98 (6) |
O1—H1O | 0.94 (17) | C1—C2 | 1.516 (5) |
O1A—C1 | 1.394 (11) | C1—H1 | 1.0000 |
O1A—H1OA | 0.90 (18) | C1—H1A | 1.0000 |
O3—C3 | 1.422 (5) | C2—C3 | 1.525 (5) |
O3—H3O | 0.78 (8) | C2—H2 | 1.0000 |
O4—C4 | 1.428 (5) | C3—C4 | 1.527 (5) |
O4—H4O | 0.79 (6) | C3—H3 | 1.0000 |
O5—C1 | 1.420 (5) | C4—C5 | 1.526 (5) |
O5—C5 | 1.448 (4) | C4—H4 | 1.0000 |
O6—C6 | 1.435 (5) | C5—C6 | 1.500 (5) |
O6—H6O | 0.81 (6) | C5—H5 | 1.0000 |
N2—C2 | 1.487 (5) | C6—H6A | 0.9900 |
N2—H2NC | 0.90 (6) | C6—H6B | 0.9900 |
N2—H2NB | 0.84 (4) | | |
| | | |
C1—O1—H1O | 116 (9) | C1—C2—H2 | 108.8 |
C1—O1A—H1OA | 111 (10) | C3—C2—H2 | 108.8 |
C3—O3—H3O | 108 (5) | O3—C3—C2 | 107.6 (3) |
C4—O4—H4O | 112 (4) | O3—C3—C4 | 110.9 (3) |
C1—O5—C5 | 112.7 (3) | C2—C3—C4 | 113.9 (3) |
C6—O6—H6O | 106 (4) | O3—C3—H3 | 108.1 |
C2—N2—H2NC | 112 (3) | C2—C3—H3 | 108.1 |
C2—N2—H2NB | 102 (3) | C4—C3—H3 | 108.1 |
H2NC—N2—H2NB | 114 (4) | O4—C4—C5 | 109.5 (3) |
C2—N2—H2NA | 110 (3) | O4—C4—C3 | 107.5 (3) |
H2NC—N2—H2NA | 110 (4) | C5—C4—C3 | 111.7 (3) |
H2NB—N2—H2NA | 108 (4) | O4—C4—H4 | 109.4 |
O1—C1—O5 | 108.3 (4) | C5—C4—H4 | 109.4 |
O1A—C1—O5 | 112.8 (5) | C3—C4—H4 | 109.4 |
O1—C1—C2 | 106.9 (4) | O5—C5—C6 | 105.9 (3) |
O1A—C1—C2 | 106.1 (5) | O5—C5—C4 | 109.9 (3) |
O5—C1—C2 | 109.8 (3) | C6—C5—C4 | 111.7 (3) |
O1—C1—H1 | 110.6 | O5—C5—H5 | 109.8 |
O5—C1—H1 | 110.6 | C6—C5—H5 | 109.8 |
C2—C1—H1 | 110.6 | C4—C5—H5 | 109.8 |
O1A—C1—H1A | 109.4 | O6—C6—C5 | 108.6 (3) |
O5—C1—H1A | 109.4 | O6—C6—H6A | 110.0 |
C2—C1—H1A | 109.4 | C5—C6—H6A | 110.0 |
N2—C2—C1 | 108.6 (3) | O6—C6—H6B | 110.0 |
N2—C2—C3 | 111.0 (3) | C5—C6—H6B | 110.0 |
C1—C2—C3 | 110.7 (3) | H6A—C6—H6B | 108.3 |
N2—C2—H2 | 108.8 | | |
| | | |
C5—O5—C1—O1 | 178.1 (4) | O3—C3—C4—O4 | −74.2 (4) |
C5—O5—C1—O1A | 52.6 (6) | C2—C3—C4—O4 | 164.3 (3) |
C5—O5—C1—C2 | −65.5 (4) | O3—C3—C4—C5 | 165.7 (3) |
O1—C1—C2—N2 | 50.3 (5) | C2—C3—C4—C5 | 44.2 (5) |
O1A—C1—C2—N2 | 170.9 (5) | C1—O5—C5—C6 | −176.3 (3) |
O5—C1—C2—N2 | −67.0 (4) | C1—O5—C5—C4 | 62.9 (4) |
O1—C1—C2—C3 | 172.4 (4) | O4—C4—C5—O5 | −169.3 (3) |
O1A—C1—C2—C3 | −67.0 (5) | C3—C4—C5—O5 | −50.4 (4) |
O5—C1—C2—C3 | 55.1 (5) | O4—C4—C5—C6 | 73.5 (4) |
N2—C2—C3—O3 | −48.8 (4) | C3—C4—C5—C6 | −167.6 (3) |
C1—C2—C3—O3 | −169.4 (3) | O5—C5—C6—O6 | 70.9 (4) |
N2—C2—C3—C4 | 74.6 (4) | C4—C5—C6—O6 | −169.5 (3) |
C1—C2—C3—C4 | −46.1 (5) | | |
Hydrogen-bond geometry (Å, º) top
D—H···A | D—H | H···A | D···A | D—H···A |
O3—H3O···Cl1i | 0.78 (8) | 2.38 (8) | 3.151 (3) | 173 (7) |
O6—H6O···Cl1ii | 0.81 (6) | 2.33 (6) | 3.123 (3) | 171 (6) |
O1a—H1Oa···O3iii | 0.94 (17) | 1.88 (17) | 2.785 (6) | 160 (14) |
O1A—H1OA···Cl1iv | 0.90 (18) | 2.70 (18) | 3.247 (9) | 120 (12) |
O1A—H1OA···O4iv | 0.90 (18) | 2.15 (17) | 3.017 (10) | 160 (15) |
O4—H4O···Cl1 | 0.79 (6) | 2.29 (6) | 3.077 (3) | 172 (6) |
N2—H2NB···O6v | 0.84 (4) | 1.92 (4) | 2.762 (4) | 172 (4) |
N2—H2NC···O4vi | 0.90 (6) | 2.11 (6) | 2.960 (5) | 157 (5) |
N2—H2NA···Cl1vii | 0.98 (6) | 2.27 (6) | 3.179 (4) | 153 (4) |
Symmetry codes: (i) −x+1, y−1/2, −z+3/2; (ii) −x+1, y+1/2, −z+3/2; (iii) −x+1, y+1/2, −z+1/2; (iv) −x+1/2, −y+1, z−1/2; (v) −x+1, y−1/2, −z+1/2; (vi) x+1/2, −y+1/2, −z+1; (vii) −x+3/2, −y+1, z−1/2. |
Cremer–Pople structural parameters for compounds (I)–(III)/(III') topCompound | φ (°) | θ (°) | Q (Å) | q2 | q3 |
(I) | 13.777 | 11.033 | 0.545 | 0.104 | 0.535 |
(II) | 302.622 | 4.781 | 0.570 | 0.048 | 0.568 |
(III) | 295.270 | 4.440 | 0.594 | 0.046 | 0.592 |
(III') | 301.512 | 5.955 | 0.594 | 0.062 | 0.591 |
Selected structural parameters (Å, °) in (I)–(III)/(III'). topParameter | Compound | | | |
| (I) | (II) | (III)a | (III')a |
C—C bond lengths (Å) | | | | |
C1—C2 | 1.516 (6) | 1.527 (2) | 1.523 | 1.532 |
C2—C3 | 1.525 (5) | 1.525 (2) | 1.515 | 1.519 |
C3—C4 | 1.527 (5) | 1.528 (2) | 1.528 | 1.522 |
C4—C5 | 1.526 (6) | 1.528 (2) | 1.517 | 1.522 |
C5—C6 | 1.499 (6) | 1.515 (2) | 1.525 | 1.496 |
| | | | |
C—O bond lengths (Å) | | | | |
C1—O1 | 1.394 (6)/1.394 (11) | 1.390 (2) | 1.390 | 1.398 |
C1—O5 | 1.420 (5) | 1.422 (2) | 1.435 | 1.429 |
C3—O3 | 1.422 (5) | 1.431 (2) | 1.426 | 1.428 |
C4—O4 | 1.428 (5) | 1.424 (2) | 1.447 | 1.453 |
C5—O5 | 1.449 (4) | 1.442 (2) | 1.446 | 1.441 |
C6—O6 | 1.435 (5) | 1.415 (2) | 1.427 | 1.447 |
| | | | |
C—N bond lengths (Å) | | | | |
C2—N2 | 1.486 (6) | 1.490 (2) | 1.504 | 1.503 |
| | | | |
Bond angles (°) | | | | |
C1—C2—C3 | 110.6 (3) | 112.6 (1) | 111.2 | 111.7 |
C2—C3—C4 | 113.9 (3) | 109.7 (1) | 109.2 | 109.7 |
C3—C4—C5 | 111.6 (3) | 109.6 (1) | 108.1 | 108.0 |
C4—C5—O5 | 109.9 (3) | 109.0 (1) | 110.5 | 109.4 |
C5—O5—C1 | 112.6 (3) | 114.1 (1) | 110.1 | 111.1 |
O5—C1—C2 | 109.8 (3) | 109.2 (1) | 110.7 | 109.7 |
C4—C5—C6 | 111.8 (3) | 113.5 (1) | 110.2 | 112.0 |
| | | | |
Torsion angles (°)b | | | | |
C1—C2—C3—C4 | -46.1 (4) | -51.6 (2) | -53.8 | -52.6 |
C2—C3—C4—C5 | 44.2 (4) | 53.6 (2) | 56.0 | 55.6 |
C3—C4—C5—O5 | -50.4 (4) | -58.5 (2) | –61.8 | -62.1 |
C4—C5—O5—C1 | 62.8 (4) | 63.5 (2) | 64.3 | 66.0 |
C5—O5—C1—C2 | -65.5 (4) | –59.8 (2) | -60.1 | -60.8 |
O5—C1—C2—C3 | 55.2 (4) | 53.1 (2) | 55.8 | 54.3 |
C2—C1—O1—H | -154 (11)/-170 (13) | -163.2 | 143.9 | 149.3 |
C1—C2—N2—HA | 43 (3) | 52.9 | 53.3 | 53.6 |
C1—C2—N2—HB | -71 (3) | -67.1 | -53.3 | -59.2 |
C1—C2—N2—HC | 167 (4) | 173 | -177.0 | 174.5 |
C2—C3—O3—H | -141 (6) | 55.7 | -125.4 | -135.7 |
C3—C4—O4—H | 128 (5) | -64.1 | -107.3 | -98.2 |
C3—C4—C5—C6 | -167.6 (3) | -177.0 (1) | 178.5 | 176.7 |
C4—C5—C6—O6 | -169.4 (3) (gt) | 62.1 (2) (gg) | –170.6 (gt) | -167.3 (gt) |
C5—C6—O6—H | 163 (5) | -78.5 | 178.3 | -98.8 |
Notes: (a) s.u. values were not reported in the original article.
(b) Definitions of the gg (gauche–gauche),
gt (gauche–trans), and tg
(trans–gauche) conformers of the exocyclic hydroxymethyl groups
in (I)–(III)/(III') are as follows: gg, H5 anti to O6;
gt, C4 anti to O6; tg, O5 anti to O6. |
Hydrogen-bond geometry in (I)–(III') (Å, °) topD—H···A | D—H | H···A | D···A | D—H···A |
Compound (I) | | | | |
O1—H1O···O3iii | 0.94 (17) | 1.88 (17) | 2.785 (6) | 160 (14) |
O1A—H1OA···Cl1iv | 0.90 (18) | 2.70 (18) | 3.247 (9) | 120 (12) |
O1A—H1OA···O4iv | 0.90 (18) | 2.15 (17) | 3.017 (10) | 160 (15) |
N2—H2NB···O6v | 0.84 (4) | 1.92 (4) | 2.762 (4) | 172 (4) |
N2—H2NC···O4vi | 0.90 (6) | 2.11 (6) | 2.960 (5) | 157 (5) |
N2—H2NA···Cl1vii | 0.98 (6) | 2.27 (6) | 3.179 (4) | 153 (4) |
O3—H3O···Cl1i | 0.78 (8) | 2.38 (8) | 3.151 (3) | 173 (7) |
O4—H4O···Cl1 | 0.79 (6) | 2.29 (6) | 3.077 (3) | 172 (6) |
O6—H6O···Cl1ii | 0.81 (6) | 2.33 (6) | 3.123 (3) | 171 (6) |
O1—H1O···Cl1iv | 0.82 | 2.35 | 3.145 (12) | 162 |
| | | | |
Compound (II) | | | | |
N2—H2A···O6i | 0.89 | 1.89 | 2.777 (17) | 172 |
N2—H2B···O4ii | 0.89 | 2.15 | 2.893 (19) | 141 |
N2—H2C···Cl1iii | 0.89 | 2.38 | 3.174 (13) | 149 |
O3—H3O···Cl1v | 0.84 | 2.35 | 3.191 (14) | 173 |
O4—H4O···Cl1 | 0.82 | 2.35 | 3.167 (14) | 175 |
O6—H6O···O3v | 0.81 | 1.95 | 2.737 (17) | 163 |
| | | | |
Compound (III) | | | | |
O1—H1O···Cl2i | 0.98 | 2.11 | 3.045 | 159 |
N2—H2A···O3ii | 0.92 | 2.08 | 2.980 | 166 |
N2—H2B···Cl1ii | 0.95 | 2.23 | 3.176 | 170 |
N2—H2C···Cl2iii | 0.91 | 2.34 | 3.218 | 162 |
O3—H3O···O4?iii | 0.87 | 2.02 | 2.865 | 164 |
O4—H4O···O6iv | 0.88 | 2.02 | 2.896 | 176 |
O6—H6O···Cl1v | 0.89 | 2.31 | 3.168 | 161 |
O1'—H1?O···Cl2 | 0.88 | 2.18 | 3.058 | 175 |
N2'—H2?A···O3?vi | 0.90 | 2.13 | 2.937 | 150 |
N2'—H2?B···Cl1 | 0.95 | 2.32 | 3.260 | 170 |
N2'—H2?C···Cl2vi | 0.90 | 2.45 | 3.270 | 151 |
O3'—H3?O···O4iii | 0.77 | 2.32 | 3.073 | 165 |
O4'—H4?O···O6?vii | 0.96 | 1.78 | 2.733 | 175 |
O6'—H6?O···O6i | 0.89 | 2.01 | 2.847 | 156 |
Symmetry codes for (I):
(i) -x+1, y-1/2, -z+3/2;
(ii) -x+1, y+1/2, -z+3/2;
(iii) -x+1, y+1/2, -z+1/2;
(iv) -x+1/2, -y+1, z-1/2;
(v) -x+1, y-1/2, -z+1/2;
(vi) x+1/2, -y+1/2, -z+1;
(vii) -x+3/2, -y+1, z-1/2.
Symmetry codes for (II):
(i) -x, y+1/2, -z+2;
(ii) -x+1, y+1/2, -z+2;
(iii) x, y, z+1;
(iv) x-1, y, z;
(v) -x+1, y-1/2, -z+2.
Symmetry codes for (III):
(i) x-1/2, -y-1/2, -z-1;
(ii) -x-1/2, -y, z-1/2;
(iii) -x, y+1/2, -z-1/2;
(iv) -x+1/2, -y, z+1/2;
(v) x+1, y, z;
(vi) x-1/2, -y-1/2, -z;
(vii) x+1/2, -y-1/2, -z-1. |