Ammonium hydrogen tartronate at 240 and 20 K

The low-temperature structure determination of the title compound, alternatively called ammonium hydrogen hydroxy­propane­dioate, NH₄⁺·C₃H₃O₅⁻, has revealed that the H atom involved in a very short asymmetric O—H...O hydrogen bond [O...O = 2.448 (2) Å at 240 K and 2.4393 (10) Å at 20 K] is disordered.

S1. Comment

A previous study of the title compound, (I), at 295 K showed that a very short asymmetric O—H···O hydrogen bond [O···O 2.443 (2) Å] is formed between the carboxylate and carboxyl groups of neighboring moieties related by a twofold screw axis (Taka et al., 1998). A least-squares refinement of (I) resulted in a long O—H bond length of 1.18 (3) Å in the short hydrogen bond, as reported earlier for several short hydrogen bonds (Olovsson & Jönsson, 1976; Misaki et al., 1986). The structure was not refined by assuming a disorder of the H atom involved in the short hydrogen bond, because no peak was observed in a difference Fourier map around the region expected for a disordered H atom and the residual electron density in the region was as low as 0.12 e Å^-3 (Taka et al., 1998). Valuable temperature experiments have been undertaken recently for several compounds by neutron or X-ray diffraction method in order to obtain information on the bonding behaviour of the H atom involved in short hydrogen bonds (Olovsson et al., 2001; Wilson, 2001; Kashino et al., 1998, 2001).

The present work is part of a study on the behaviour of the H atom in the very short hydrogen bond in (I) formed between O5 and O3ⁱ [symmetry code: (i) -x, y - 1/2, 3/2 - z] (Figs. 1–3, and Tables 2 and 4). It has been confirmed that at 240 and 20 K, the H atom involved in the short hydrogen bond is disordered between two positions with non-equivalent occupancy factors. This indicates that the potential function is of an asymmetric double-minimum type for this short hydrogen bond. The apparent H3A···H3B distance is 0.89 (7) Å at 240 K and 0.78 (5) Å at 20 K. It is noted that the O5—C3 bond is significantly longer than the O3—C2 bond (Tables 1 and 3), in accordance with the fact that the occupancy factor of H3A (0.60 at 240 K and 0.55 at 20 K) is larger than that of H3B (0.40 at 240 K and 0.45 at 20 K). The O5···O3ⁱ distances at 240 and 20 K [2.448 (2) and 2.4393 (10) Å, respectively] are essentially the same as that observed at 295 K [2.443 (2) Å]. The longest two N···O distances of the N—H···O hydrogen bonds are shortened by 0.024 (6)–0.017 (5) Å at 245 K and by 0.053 (4)–0.056 (4) Å at 20 K.

A recent study of short asymmetric O—H···O hydrogen bond [O···O 2.400 (5) Å at 150 K, 2.405 (6) Å at 250 K, 2.420 (7) Å at 295 K and 2.430 (9) Å at 335 K] in urea–phosphoric acid (1/1) by neutron diffraction showed that the position of the H atom involved in the hydrogen bond becomes effectively centred at the highest temperatures studied (Wilson, 2001). The disorder and occupancy of the H atom were examined (Wilson, 2001). The short hydrogen bond in (I) should further be studied in order to clarify the temperature dependence of the occupancy of the H atom and the nature of the disorder by combining neutron and X-ray diffraction methods at low temperatures.

Related literature

For related literature, see: Kashino et al. (1998, 2001); Misaki et al. (1986); Olovsson & Jönsson (1976); Olovsson et al. (2001); Taka et al. (1998); Wilson (2001).

S2. Experimental

Crystals of (I) were grown by slow evaporation from an aqueous solution containing equimolar amounts of tartronic acid and ammonia. A crystal was mounted on a sapphire stick and a thermocouple was fixed to the stick adjacent to the crystal. The temperature was regulated to within ±0.2 K of the setting temperature using a closed-cycle He refrigerator (Cryogenics HC-2) equipped with a temperature control (Chino, KP1000). Cell constants were checked over the temperature range 20–298 K. Intensity data for structure analysis were collected at 20 and 240 K. The χ range was limited within 173 and 473 K for the low-temperature measurements. Reflections which suffered from scattering by the sapphire stick were removed from the diffraction data.

S3. Refinement

All H atoms were located from difference Fourier maps and refined isotropically. The sum of the occupancy factors of the H3A and H3B atoms was constrained to 1.0. A t the final stage of the least-squares refinement, the occupancy factor of H3A was fixed to 0.6 for the 240 K determination and at 0.55 for the 20 K determination. The O5—H3A and O3—H3B distances became essentially the same [0.78 (4) and 0.78 (5) Å, respectively, at 240 K, and 0.86 (3) and 0.82 (4) Å, respectively, at 20 K]. The U_iso values of H3A were close to those of H3B, and the U_eq values of O5 [0.0284 (3) Ų at 240 K and 0.00650 (13) Ų at 2 0 K] were close to those of O3 [0.0277 (3) Ų at 240 K and 0.00608 (13) Ų at 20 K]. Thus, the refinements were terminated. U_iso values at 240 K: 0.029 (5) Ų for H1, 0.026 (9) Ų for H3A and 0.030 (15) Ų for H3B; the range of U_iso values for the H atoms of the ammonium ion was 0.061 (8)–0.082 (10) Ų. U_iso values at 20 K: 0.009 (3) Ų for H1, 0.018 (7) Ų for H3A and 0.020 (9) Ų for H3B; the range of U_iso values for the H atoms of the ammonium ion was 0.022 (4)–0.027 (4) Ų. Refined distances: C—H = 0.99 (2) Å, N—H = 0.86 (3)–1.00 (3) Å and O—H = 0.78 (5)–0.79 (3) Å at 240 K; C—H = 1.008 (14) Å, N—H = 0.870 (16)–0.906 (17) Å and O—H = 0.790 (19)–0.86 (3) Å at 20 K.

Computing details

For both compounds, data collection: MSC/AFC Diffractometer Control Software (Molecular Structure Corporation, 1990); cell refinement: MSC/AFC Diffractometer Control Software; data reduction: TEXSAN for Windows (Molecular Structure Corpotation, 1999); program(s) used to solve structure: SIR92 (Altomare et al., 1993); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997); molecular graphics: ORTEP-3 (Farrugia, 1997); software used to prepare material for publication: TEXSAN for Windows.

Figures

	Fig. 1. ORTEP-3 (Farrugia, 1997) drawing of the formula unit of (I) at 240 K with the atomic numbering scheme. Displacement ellipsoids are drawn at the 50% probability level. The occupancy factors for H3A and H3B are 0.6 and 0.4, respectively. Hydrogen bonds are indicated by dashed lines.
	Fig. 2. ORTEP-3 (Farrugia, 1997) drawing of the formula unit of (I) at 20 K with the atomic numbering scheme. Displacement ellipsoids are drawn at the 50% probability level. The occupancy factors for H3A and H3B are 0.55 and 0.45, respectively. Hydrogen bonds are indicated by dashed lines.
	Fig. 3. The molecular arrangement based on the atomic parameters of (I) at 240 K. Hydrogen bonds are indicated by dashed lines. The H3B atom with an occupancy factor of 0.4 and the formula units (ii) and (iii) have been omitted for clarity. The symmetry codes are as given in Table 2.

(240K) Ammonium hydrogen hydroxypropanedioate

Crystal data

NH4^+·C3H3O5^−	F(000) = 288
Mr = 137.10	Dx = 1.640 Mg m^−3
Monoclinic, P21/c	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybc	Cell parameters from 25 reflections
a = 4.299 (3) Å	θ = 9.5–11°
b = 9.0650 (16) Å	µ = 0.16 mm^−1
c = 14.262 (6) Å	T = 240 K
β = 92.67 (5)°	Prismatic, colorless
V = 555.2 (5) Å^3	0.40 × 0.30 × 0.30 mm
Z = 4

Data collection

Huber off-center four-circle diffractometer	Rint = 0.032
Radiation source: Rigaku rotating anode	θmax = 30.0°, θmin = 2.7°
Graphite monochromator	h = 0→6
ω–2θ scans	k = 0→12
1811 measured reflections	l = −20→20
1584 independent reflections	3 standard reflections every 97 reflections
1059 reflections with I > 2σ(I)	intensity decay: 2.3%

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)] = 0.046	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.125	All H-atom parameters refined
S = 0.98	w = 1/[σ^2(Fo^2) + (0.1P)^2 + 0.1104P] where P = (Fo^2 + 2Fc^2)/3
1584 reflections	(Δ/σ)max = 0.001
114 parameters	Δρmax = 0.25 e Å^−3
0 restraints	Δρmin = −0.25 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.

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
O1	0.6356 (3)	0.29310 (15)	0.55964 (9)	0.0329 (3)
O2	0.2672 (3)	0.52135 (14)	0.59125 (8)	0.0314 (3)
O3	0.1668 (3)	0.44548 (14)	0.73593 (8)	0.0277 (3)
O4	0.2693 (3)	0.06613 (14)	0.59727 (7)	0.0314 (3)
O5	0.2171 (3)	0.14138 (14)	0.74516 (8)	0.0284 (3)
N1	0.1078 (5)	0.19683 (19)	0.41576 (11)	0.0308 (4)
C1	0.4986 (4)	0.29647 (18)	0.64725 (10)	0.0220 (3)
C2	0.2919 (4)	0.43368 (17)	0.65716 (10)	0.0204 (3)
C3	0.3137 (4)	0.15489 (18)	0.66120 (10)	0.0210 (3)
H1	0.660 (5)	0.298 (2)	0.6987 (14)	0.029 (5)*
H2	0.631 (6)	0.370 (3)	0.5344 (18)	0.056 (8)*
H3A	0.090 (8)	0.081 (4)	0.751 (2)	0.026 (9)*	0.60
H3B	0.049 (12)	0.510 (6)	0.741 (4)	0.030 (15)*	0.40
H4	0.238 (7)	0.195 (3)	0.464 (2)	0.073 (9)*
H5	−0.026 (8)	0.286 (3)	0.425 (2)	0.079 (9)*
H6	0.208 (6)	0.209 (3)	0.360 (2)	0.061 (8)*
H7	−0.033 (8)	0.126 (4)	0.415 (2)	0.082 (10)*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
O1	0.0435 (8)	0.0249 (6)	0.0321 (7)	0.0038 (6)	0.0230 (6)	0.0046 (5)
O2	0.0462 (8)	0.0269 (6)	0.0219 (6)	0.0064 (6)	0.0101 (5)	0.0066 (5)
O3	0.0412 (8)	0.0223 (6)	0.0206 (6)	0.0024 (6)	0.0129 (5)	−0.0005 (5)
O4	0.0478 (8)	0.0264 (6)	0.0203 (6)	−0.0072 (6)	0.0048 (5)	−0.0021 (5)
O5	0.0444 (8)	0.0208 (6)	0.0209 (6)	−0.0050 (6)	0.0112 (5)	0.0012 (5)
N1	0.0418 (9)	0.0327 (9)	0.0181 (7)	−0.0009 (8)	0.0044 (6)	−0.0020 (6)
C1	0.0249 (7)	0.0237 (8)	0.0180 (7)	−0.0008 (7)	0.0062 (6)	0.0008 (6)
C2	0.0253 (8)	0.0190 (7)	0.0172 (7)	−0.0037 (6)	0.0042 (6)	−0.0015 (5)
C3	0.0235 (8)	0.0210 (7)	0.0187 (7)	0.0021 (6)	0.0017 (6)	0.0027 (6)

Geometric parameters (Å, º)

O1—C1	1.406 (2)	C1—C3	1.528 (2)
O1—H2	0.79 (3)	C1—C2	1.539 (2)
O2—C2	1.2318 (19)	C1—H1	0.99 (2)
O3—C2	1.2724 (19)	N1—H4	0.86 (3)
O3—H3B	0.78 (5)	N1—H5	1.00 (3)
O4—C3	1.2243 (19)	N1—H6	0.93 (3)
O5—C3	1.2914 (19)	N1—H7	0.88 (3)
O5—H3A	0.78 (4)
C1—O1—H2	112.7 (19)	O3—C2—C1	114.75 (13)
C2—O3—H3B	116 (4)	O4—C3—O5	125.63 (16)
C3—O5—H3A	115 (2)	O4—C3—C1	121.28 (14)
O1—C1—C3	109.85 (13)	O5—C3—C1	113.09 (14)
O1—C1—C2	111.51 (13)	H4—N1—H5	105 (3)
C3—C1—C2	111.15 (13)	H4—N1—H6	112 (3)
O1—C1—H1	110.6 (12)	H5—N1—H6	108 (2)
C3—C1—H1	105.5 (11)	H4—N1—H7	114 (3)
C2—C1—H1	108.1 (11)	H5—N1—H7	101 (3)
O2—C2—O3	126.47 (16)	H6—N1—H7	115 (2)
O2—C2—C1	118.76 (13)
O1—C1—C2—O2	0.3 (2)	O1—C1—C3—O4	−8.5 (2)
C3—C1—C2—O2	−122.63 (16)	C2—C1—C3—O4	115.37 (17)
O1—C1—C2—O3	−178.31 (14)	O1—C1—C3—O5	171.06 (14)
C3—C1—C2—O3	58.75 (18)	C2—C1—C3—O5	−65.04 (17)

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
O5—H3A···O3^i	0.78 (4)	1.67 (4)	2.448 (2)	177 (4)
O3—H3B···O5^ii	0.78 (5)	1.67 (5)	2.448 (2)	177 (6)
O1—H2···O2	0.79 (3)	2.26 (3)	2.657 (2)	112 (2)
O1—H2···O2^iii	0.79 (3)	2.11 (3)	2.7781 (19)	143 (3)
N1—H4···O4	0.86 (3)	2.23 (3)	2.902 (2)	134 (3)
N1—H4···O1	0.86 (3)	2.32 (3)	3.112 (3)	153 (3)
N1—H5···O2^iv	1.00 (3)	2.04 (3)	3.020 (2)	164 (3)
N1—H6···O5^v	0.93 (3)	2.13 (3)	2.898 (2)	139 (2)
N1—H6···O3^v	0.93 (3)	2.25 (3)	2.893 (2)	125 (2)
N1—H7···O4^vi	0.88 (3)	2.02 (3)	2.884 (2)	166 (3)

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

(20K) Ammonium hydrogen hydroxypropanedioate

Crystal data

NH4^+·C3H3O5^−	F(000) = 288
Mr = 137.10	Dx = 1.678 Mg m^−3
Monoclinic, P21/c	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ybc	Cell parameters from 25 reflections
a = 4.2418 (7) Å	θ = 9.5–11°
b = 9.0258 (12) Å	µ = 0.16 mm^−1
c = 14.217 (3) Å	T = 20 K
β = 94.335 (15)°	Prismatic, colorless
V = 542.76 (16) Å^3	0.40 × 0.30 × 0.30 mm
Z = 4

Data collection

Huber off-center four-circle diffractometer	Rint = 0.034
Radiation source: Rigaku rotating anode	θmax = 34.8°, θmin = 2.7°
Graphite monochromator	h = 0→6
ω–2θ scans	k = 0→14
2464 measured reflections	l = −22→22
2216 independent reflections	3 standard reflections every 97 reflections
1738 reflections with I > 2σ(I)	intensity decay: 7.0%

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)] = 0.037	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.096	All H-atom parameters refined
S = 1.02	w = 1/[σ^2(Fo^2) + (0.0526P)^2] where P = (Fo^2 + 2Fc^2)/3
2216 reflections	(Δ/σ)max = 0.003
114 parameters	Δρmax = 0.45 e Å^−3
0 restraints	Δρmin = −0.26 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.

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

	x	y	z	Uiso*/Ueq	Occ. (<1)
O1	0.64234 (16)	0.29303 (8)	0.56170 (5)	0.00692 (13)
O2	0.26543 (15)	0.52342 (7)	0.58982 (4)	0.00690 (13)
O3	0.17058 (16)	0.44755 (7)	0.73593 (4)	0.00608 (13)
O4	0.27449 (16)	0.06407 (7)	0.59768 (4)	0.00702 (13)
O5	0.21690 (16)	0.14304 (7)	0.74550 (4)	0.00650 (13)
N1	0.1103 (2)	0.19626 (9)	0.41561 (6)	0.00724 (14)
C1	0.5041 (2)	0.29761 (9)	0.64857 (6)	0.00478 (14)
C2	0.2940 (2)	0.43537 (9)	0.65696 (6)	0.00481 (15)
C3	0.31649 (19)	0.15494 (9)	0.66179 (6)	0.00459 (14)
H1	0.663 (3)	0.2995 (15)	0.7050 (10)	0.009 (3)*
H2	0.626 (4)	0.371 (2)	0.5364 (12)	0.031 (5)*
H3A	0.078 (7)	0.074 (4)	0.7507 (19)	0.018 (7)*	0.55
H3B	0.035 (9)	0.512 (5)	0.735 (2)	0.020 (9)*	0.45
H4	0.250 (4)	0.1867 (18)	0.4634 (11)	0.023 (4)*
H5	0.000 (4)	0.2813 (18)	0.4210 (11)	0.023 (4)*
H6	0.194 (4)	0.2000 (18)	0.3615 (13)	0.027 (4)*
H7	−0.020 (4)	0.1213 (19)	0.4130 (11)	0.022 (4)*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
O1	0.0100 (3)	0.0058 (3)	0.0055 (3)	0.0008 (2)	0.0044 (2)	0.0010 (2)
O2	0.0097 (3)	0.0060 (3)	0.0051 (3)	0.0010 (2)	0.0015 (2)	0.0015 (2)
O3	0.0086 (3)	0.0054 (3)	0.0045 (3)	0.0009 (2)	0.0026 (2)	−0.0002 (2)
O4	0.0102 (3)	0.0060 (3)	0.0048 (3)	−0.0010 (2)	0.0005 (2)	−0.0010 (2)
O5	0.0091 (3)	0.0063 (3)	0.0044 (3)	−0.0015 (2)	0.0026 (2)	0.0001 (2)
N1	0.0094 (3)	0.0071 (3)	0.0052 (3)	−0.0003 (3)	0.0008 (3)	0.0001 (2)
C1	0.0058 (3)	0.0051 (3)	0.0038 (3)	−0.0001 (3)	0.0020 (2)	0.0005 (3)
C2	0.0054 (3)	0.0047 (3)	0.0043 (3)	−0.0007 (3)	0.0004 (3)	−0.0010 (3)
C3	0.0045 (3)	0.0043 (3)	0.0048 (3)	0.0008 (3)	0.0000 (2)	0.0009 (3)

Geometric parameters (Å, º)

O1—C1	1.4072 (11)	C1—C3	1.5329 (12)
O1—H2	0.790 (19)	C1—C2	1.5399 (12)
O2—C2	1.2410 (10)	C1—H1	1.008 (14)
O3—C2	1.2788 (11)	N1—H4	0.870 (16)
O3—H3B	0.82 (4)	N1—H5	0.906 (17)
O4—C3	1.2291 (10)	N1—H6	0.872 (19)
O5—C3	1.2971 (11)	N1—H7	0.874 (17)
O5—H3A	0.86 (3)
C1—O1—H2	110.2 (13)	O2—C2—C1	118.75 (7)
C2—O3—H3B	112 (2)	O3—C2—C1	114.53 (7)
C3—O5—H3A	114.3 (18)	O4—C3—O5	125.89 (8)
H4—N1—H5	110.1 (14)	O4—C3—C1	121.14 (8)
O1—C1—C3	109.73 (7)	O5—C3—C1	112.97 (7)
O1—C1—C2	111.95 (7)	H4—N1—H6	113.3 (15)
C3—C1—C2	111.09 (7)	H5—N1—H6	106.8 (14)
O1—C1—H1	113.7 (9)	H4—N1—H7	110.3 (15)
C3—C1—H1	103.8 (8)	H5—N1—H7	109.2 (15)
C2—C1—H1	106.3 (8)	H6—N1—H7	107.0 (14)
O2—C2—O3	126.70 (8)
O1—C1—C2—O2	0.69 (11)	O1—C1—C3—O4	−7.69 (11)
C3—C1—C2—O2	−122.36 (8)	C2—C1—C3—O4	116.63 (9)
O1—C1—C2—O3	−177.83 (7)	O1—C1—C3—O5	171.58 (7)
C3—C1—C2—O3	59.11 (9)	C2—C1—C3—O5	−64.10 (9)

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
O5—H3A···O3^i	0.86 (3)	1.58 (3)	2.4393 (10)	178 (3)
O3—H3B···O5^ii	0.82 (4)	1.63 (4)	2.4393 (10)	169 (4)
O1—H2···O2	0.790 (19)	2.234 (19)	2.6715 (10)	115.6 (16)
O1—H2···O2^iii	0.790 (19)	2.114 (18)	2.7686 (10)	140.4 (18)
N1—H4···O4	0.870 (16)	2.202 (16)	2.8871 (11)	135.4 (15)
N1—H4···O1	0.870 (16)	2.301 (16)	3.0756 (11)	148.2 (14)
N1—H5···O2^iv	0.906 (17)	2.090 (17)	2.9882 (11)	170.9 (14)
N1—H6···O5^v	0.872 (19)	2.182 (18)	2.8849 (11)	137.5 (15)
N1—H6···O3^v	0.872 (19)	2.224 (18)	2.8940 (11)	133.5 (14)
N1—H7···O4^vi	0.874 (17)	1.990 (17)	2.8592 (11)	173.0 (16)

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