A 16-connected three-dimensional hydrogen-bonding network in tetra­kis­(4-amino­pyridinium) perylene-3,4,9,10-tetra­carboxyl­ate octa­hydrate

The novel title organic salt, 4C₅H₇N₂⁺·C₂₄H₈O₈⁴⁻·8H₂O, was obtained from the reaction of perylene-3,4,9,10-tetra­carb­oxy­lic acid (H₄ptca) with 4-amino­pyridine (4-ap). The asymmetric unit contains half a perylene-3,4,9,10-tetra­carboxyl­ate (ptca⁴⁻) anion with twofold symmetry, two 4-amino­pyridinium (4-Hap⁺) cations and four water mol­ecules. Strong N—H...O hydrogen bonds connect each ptca⁴⁻ anion with four 4-Hap⁺ cations to form a one-dimensional linear chain along the [010] direction, decorated by additional 4-Hap⁺ cations attached by weak N—H...O hydrogen bonds to the ptca⁴⁻ anions. Inter­molecular O—H...O inter­actions of water mol­ecules with ptca⁴⁻ and 4-Hap⁺ ions complete the three-dimensional hydrogen-bonding network. From the viewpoint of topology, each ptca⁴⁻ anion acts as a 16-connected node by hydrogen bonding to six 4-Hap⁺ cations and ten water mol­ecules to yield a highly connected hydrogen-bonding framework. π–π inter­actions between 4-Hap⁺ cations, and between 4-Hap⁺ cations and ptca⁴⁻ anions, further stabilize the three-dimensional hydrogen-bonding network.

Computing details

Data collection: APEX2 (Bruker, 2007); cell refinement: APEX2 and SAINT (Bruker, 2007); data reduction: SAINT (Bruker, 2007); program(s) used to solve structure: SHELXTL (Sheldrick, 2008); program(s) used to refine structure: SHELXTL (Sheldrick, 2008); molecular graphics: SHELXTL (Sheldrick, 2008) and DIAMOND (Brandenburg, 2005); software used to prepare material for publication: SHELXTL (Sheldrick, 2008).

Tetrakis(4-aminopyridinium) perylene-3,4,9,10-tetracarboxylate octahydrate

Crystal data

4C5H7N2^+·C24H8O8^4^−·8H2O	F(000) = 2000
Mr = 948.94	Dx = 1.388 Mg m^−3
Monoclinic, C2/c	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: -C 2yc	Cell parameters from 1793 reflections
a = 17.771 (3) Å	θ = 2.5–25.9°
b = 19.703 (3) Å	µ = 0.11 mm^−1
c = 14.210 (3) Å	T = 296 K
β = 114.097 (6)°	Block, red
V = 4541.8 (14) Å^3	0.25 × 0.20 × 0.18 mm
Z = 4

Data collection

Bruker APEXII CCD area-detector diffractometer	4696 independent reflections
Radiation source: fine-focus sealed tube	2381 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.096
φ and ω scans	θmax = 26.5°, θmin = 1.6°
Absorption correction: multi-scan (SADABS; Sheldrick, 2003)	h = −22→18
Tmin = 0.612, Tmax = 0.7459	k = −24→24
14116 measured reflections	l = −17→17

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)] = 0.061	Hydrogen site location: inferred from neighbouring sites
wR(F^2) = 0.137	H-atom parameters constrained
S = 1.04	w = 1/[σ^2(Fo^2) + (0.0407P)^2 + 0.001P] where P = (Fo^2 + 2Fc^2)/3
4696 reflections	(Δ/σ)max < 0.001
309 parameters	Δρmax = 0.33 e Å^−3
0 restraints	Δρmin = −0.28 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
C1	0.9710 (2)	0.32278 (13)	0.6368 (2)	0.0290 (7)
C2	0.96368 (16)	0.39818 (12)	0.6530 (2)	0.0232 (6)
C3	1.0000	0.43261 (17)	0.7500	0.0215 (8)
C4	1.0000	0.50553 (16)	0.7500	0.0196 (8)
C5	0.96691 (15)	0.54217 (12)	0.65490 (19)	0.0206 (6)
C6	0.92824 (16)	0.50641 (12)	0.5652 (2)	0.0286 (7)
H6	0.9025	0.5297	0.5034	0.034*
C7	0.92681 (17)	0.43610 (12)	0.5651 (2)	0.0289 (7)
H7	0.8997	0.4137	0.5027	0.035*
C8	0.97470 (15)	0.61650 (12)	0.65513 (19)	0.0219 (6)
C9	1.0000	0.65332 (17)	0.7500	0.0202 (8)
C10	1.0000	0.72572 (17)	0.7500	0.0212 (9)
C11	0.97895 (16)	0.75952 (12)	0.65388 (19)	0.0234 (6)
C12	0.96146 (17)	0.72232 (12)	0.5663 (2)	0.0301 (7)
H12	0.9516	0.7448	0.5048	0.036*
C13	0.95803 (17)	0.65224 (12)	0.5662 (2)	0.0286 (7)
H13	0.9441	0.6289	0.5045	0.034*
C14	0.96652 (19)	0.83524 (13)	0.6386 (2)	0.0293 (7)
C15	0.1790 (2)	0.55143 (19)	0.7441 (3)	0.0503 (9)
H15	0.1430	0.5465	0.6753	0.060*
C16	0.19905 (19)	0.61441 (18)	0.7813 (3)	0.0504 (9)
H16	0.1781	0.6523	0.7397	0.061*
C17	0.2536 (2)	0.62123 (17)	0.8870 (3)	0.0531 (9)
C18	0.2828 (2)	0.56314 (17)	0.9438 (3)	0.0565 (10)
H18	0.3188	0.5664	1.0129	0.068*
C19	0.2593 (2)	0.5000 (2)	0.8992 (3)	0.0680 (11)
H19	0.2797	0.4612	0.9386	0.082*
C20	0.0109 (2)	0.05439 (16)	0.6115 (3)	0.0508 (9)
H20	−0.0345	0.0513	0.5487	0.061*
C21	0.0386 (2)	0.11590 (15)	0.6509 (2)	0.0435 (8)
H21	0.0124	0.1547	0.6153	0.052*
C22	0.10709 (19)	0.12177 (14)	0.7457 (2)	0.0356 (7)
C23	0.1449 (2)	0.06081 (14)	0.7944 (2)	0.0419 (8)
H23	0.1911	0.0622	0.8567	0.050*
C24	0.1139 (2)	0.00069 (16)	0.7508 (3)	0.0503 (9)
H24	0.1392	−0.0392	0.7836	0.060*
N1	0.20815 (18)	0.49415 (15)	0.8013 (3)	0.0610 (8)
H1	0.1980	0.4516	0.7763	0.092*
N2	0.27533 (19)	0.68099 (14)	0.9274 (2)	0.0746 (10)
H2A	0.3087	0.6849	0.9911	0.089*
H2B	0.2563	0.7167	0.8904	0.089*
N3	0.04715 (19)	−0.00305 (13)	0.6604 (2)	0.0522 (8)
H3	0.0282	−0.0448	0.6370	0.078*
N4	0.13400 (16)	0.18156 (11)	0.7888 (2)	0.0492 (7)
H4A	0.1094	0.2181	0.7587	0.059*
H4B	0.1761	0.1839	0.8470	0.059*
O1	0.90681 (12)	0.29259 (9)	0.57955 (15)	0.0388 (5)
O2	1.04202 (12)	0.29812 (9)	0.67590 (14)	0.0344 (5)
O3	1.00914 (13)	0.86639 (9)	0.60088 (15)	0.0409 (5)
O4	0.91029 (13)	0.86101 (9)	0.65815 (15)	0.0400 (5)
O5	0.79773 (13)	0.79637 (11)	0.71060 (18)	0.0617 (7)
H5A	0.8121	0.7884	0.7722	0.093*
H5B	0.8304	0.8111	0.6882	0.093*
O6	0.74939 (13)	0.34649 (10)	0.50149 (18)	0.0602 (7)
H6A	0.7164	0.3314	0.5231	0.090*
H6B	0.7966	0.3318	0.5224	0.090*
O7	0.86557 (14)	0.20037 (11)	0.42567 (18)	0.0667 (7)
H7A	0.9034	0.1786	0.4215	0.100*
H7B	0.8745	0.2329	0.4648	0.100*
O8	0.16710 (13)	0.38081 (10)	0.69289 (17)	0.0580 (7)
H8A	0.1287	0.3583	0.6943	0.087*
H8B	0.2031	0.3634	0.6788	0.087*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
C1	0.046 (2)	0.0192 (15)	0.0270 (16)	−0.0003 (15)	0.0207 (15)	0.0008 (13)
C2	0.0262 (17)	0.0153 (14)	0.0288 (16)	−0.0006 (12)	0.0119 (13)	−0.0040 (12)
C3	0.024 (2)	0.018 (2)	0.024 (2)	0.000	0.0112 (17)	0.000
C4	0.019 (2)	0.0154 (19)	0.026 (2)	0.000	0.0111 (17)	0.000
C5	0.0233 (16)	0.0158 (13)	0.0231 (15)	0.0024 (11)	0.0099 (12)	−0.0005 (11)
C6	0.0386 (18)	0.0210 (15)	0.0221 (15)	0.0008 (13)	0.0081 (13)	0.0026 (12)
C7	0.0395 (18)	0.0205 (15)	0.0222 (15)	−0.0006 (13)	0.0080 (13)	−0.0033 (12)
C8	0.0248 (16)	0.0194 (14)	0.0216 (15)	0.0006 (12)	0.0095 (12)	0.0009 (11)
C9	0.022 (2)	0.020 (2)	0.019 (2)	0.000	0.0079 (17)	0.000
C10	0.025 (2)	0.018 (2)	0.022 (2)	0.000	0.0117 (18)	0.000
C11	0.0282 (17)	0.0157 (14)	0.0247 (15)	−0.0017 (12)	0.0092 (13)	0.0012 (12)
C12	0.045 (2)	0.0219 (15)	0.0230 (15)	0.0009 (13)	0.0141 (14)	0.0043 (12)
C13	0.0449 (19)	0.0201 (15)	0.0203 (15)	−0.0013 (13)	0.0129 (13)	−0.0024 (12)
C14	0.046 (2)	0.0183 (15)	0.0173 (15)	−0.0007 (14)	0.0059 (14)	−0.0015 (12)
C15	0.043 (2)	0.063 (2)	0.043 (2)	0.0052 (19)	0.0156 (17)	0.0030 (19)
C16	0.037 (2)	0.059 (2)	0.050 (2)	−0.0034 (17)	0.0137 (18)	−0.0041 (18)
C17	0.045 (2)	0.049 (2)	0.063 (3)	−0.0059 (18)	0.0190 (19)	0.000 (2)
C18	0.061 (3)	0.049 (2)	0.043 (2)	−0.0147 (19)	0.0050 (18)	−0.0026 (18)
C19	0.059 (3)	0.079 (3)	0.059 (3)	0.001 (2)	0.018 (2)	0.013 (2)
C20	0.061 (3)	0.044 (2)	0.049 (2)	−0.0063 (19)	0.0248 (18)	−0.0028 (18)
C21	0.055 (2)	0.0307 (18)	0.045 (2)	−0.0007 (16)	0.0208 (18)	0.0023 (15)
C22	0.047 (2)	0.0264 (17)	0.042 (2)	−0.0002 (14)	0.0275 (17)	0.0002 (14)
C23	0.055 (2)	0.0241 (17)	0.051 (2)	0.0057 (15)	0.0261 (17)	0.0061 (15)
C24	0.071 (3)	0.033 (2)	0.060 (3)	0.0059 (18)	0.040 (2)	0.0086 (17)
N1	0.054 (2)	0.062 (2)	0.075 (2)	−0.0158 (16)	0.0331 (18)	−0.0170 (18)
N2	0.087 (3)	0.050 (2)	0.070 (2)	−0.0023 (18)	0.0148 (19)	−0.0086 (17)
N3	0.079 (2)	0.0310 (16)	0.064 (2)	−0.0195 (16)	0.0467 (18)	−0.0145 (15)
N4	0.063 (2)	0.0238 (14)	0.0520 (18)	0.0036 (13)	0.0143 (14)	−0.0024 (13)
O1	0.0424 (13)	0.0225 (11)	0.0444 (13)	−0.0087 (10)	0.0105 (10)	−0.0101 (10)
O2	0.0394 (13)	0.0221 (11)	0.0401 (12)	0.0072 (9)	0.0146 (10)	0.0002 (9)
O3	0.0702 (16)	0.0190 (10)	0.0424 (13)	−0.0064 (10)	0.0321 (12)	0.0021 (9)
O4	0.0482 (14)	0.0295 (12)	0.0433 (13)	0.0111 (10)	0.0196 (11)	0.0029 (10)
O5	0.0542 (16)	0.0649 (16)	0.0703 (17)	−0.0008 (12)	0.0297 (13)	0.0008 (13)
O6	0.0433 (15)	0.0573 (15)	0.0714 (17)	−0.0034 (12)	0.0145 (13)	0.0078 (13)
O7	0.0576 (16)	0.0742 (17)	0.0740 (18)	−0.0203 (13)	0.0328 (13)	−0.0405 (14)
O8	0.0482 (15)	0.0538 (15)	0.0726 (17)	−0.0064 (12)	0.0253 (12)	−0.0186 (13)

Geometric parameters (Å, º)

C1—O1	1.248 (3)	C17—N2	1.298 (4)
C1—O2	1.251 (3)	C17—C18	1.374 (4)
C1—C2	1.517 (3)	C18—C19	1.381 (5)
C2—C7	1.371 (3)	C18—H18	0.9300
C2—C3	1.432 (3)	C19—N1	1.320 (4)
C3—C2^i	1.432 (3)	C19—H19	0.9300
C3—C4	1.437 (4)	C20—C21	1.341 (4)
C4—C5^i	1.430 (3)	C20—N3	1.347 (4)
C4—C5	1.430 (3)	C20—H20	0.9300
C5—C6	1.370 (3)	C21—C22	1.404 (4)
C5—C8	1.471 (3)	C21—H21	0.9300
C6—C7	1.386 (3)	C22—N4	1.324 (3)
C6—H6	0.9300	C22—C23	1.412 (4)
C7—H7	0.9300	C23—C24	1.346 (4)
C8—C13	1.369 (3)	C23—H23	0.9300
C8—C9	1.432 (3)	C24—N3	1.348 (4)
C9—C10	1.427 (5)	C24—H24	0.9300
C9—C8^i	1.432 (3)	N1—H1	0.9001
C10—C11^i	1.425 (3)	N2—H2A	0.8600
C10—C11	1.425 (3)	N2—H2B	0.8600
C11—C12	1.366 (3)	N3—H3	0.9001
C11—C14	1.511 (3)	N4—H4A	0.8600
C12—C13	1.382 (3)	N4—H4B	0.8600
C12—H12	0.9300	O5—H5A	0.8204
C13—H13	0.9300	O5—H5B	0.8202
C14—O4	1.249 (3)	O6—H6A	0.8203
C14—O3	1.253 (3)	O6—H6B	0.8202
C15—C16	1.339 (4)	O7—H7A	0.8202
C15—N1	1.364 (4)	O7—H7B	0.8202
C15—H15	0.9300	O8—H8A	0.8204
C16—C17	1.423 (5)	O8—H8B	0.8200
C16—H16	0.9300
O1—C1—O2	126.3 (2)	N1—C15—H15	118.1
O1—C1—C2	117.0 (3)	C15—C16—C17	117.5 (3)
O2—C1—C2	116.5 (3)	C15—C16—H16	121.3
C7—C2—C3	118.6 (2)	C17—C16—H16	121.3
C7—C2—C1	115.7 (2)	N2—C17—C18	121.6 (4)
C3—C2—C1	125.3 (2)	N2—C17—C16	120.3 (3)
C2—C3—C2^i	123.4 (3)	C18—C17—C16	118.2 (3)
C2—C3—C4	118.28 (16)	C17—C18—C19	120.7 (3)
C2^i—C3—C4	118.28 (16)	C17—C18—H18	119.7
C5^i—C4—C5	119.3 (3)	C19—C18—H18	119.7
C5^i—C4—C3	120.33 (15)	N1—C19—C18	120.8 (4)
C5—C4—C3	120.33 (15)	N1—C19—H19	119.6
C6—C5—C4	118.4 (2)	C18—C19—H19	119.6
C6—C5—C8	121.7 (2)	C21—C20—N3	121.8 (3)
C4—C5—C8	119.9 (2)	C21—C20—H20	119.1
C5—C6—C7	121.2 (2)	N3—C20—H20	119.1
C5—C6—H6	119.4	C20—C21—C22	120.1 (3)
C7—C6—H6	119.4	C20—C21—H21	120.0
C2—C7—C6	122.8 (3)	C22—C21—H21	120.0
C2—C7—H7	118.6	N4—C22—C21	121.7 (3)
C6—C7—H7	118.6	N4—C22—C23	121.4 (3)
C13—C8—C9	118.3 (2)	C21—C22—C23	116.9 (3)
C13—C8—C5	121.7 (2)	C24—C23—C22	120.0 (3)
C9—C8—C5	120.0 (2)	C24—C23—H23	120.0
C10—C9—C8	120.43 (15)	C22—C23—H23	120.0
C10—C9—C8^i	120.43 (15)	C23—C24—N3	121.5 (3)
C8—C9—C8^i	119.1 (3)	C23—C24—H24	119.3
C9—C10—C11^i	117.85 (16)	N3—C24—H24	119.3
C9—C10—C11	117.85 (16)	C19—N1—C15	119.1 (3)
C11^i—C10—C11	124.3 (3)	C19—N1—H1	116.0
C12—C11—C10	119.7 (2)	C15—N1—H1	124.7
C12—C11—C14	115.7 (2)	C17—N2—H2A	120.0
C10—C11—C14	124.4 (2)	C17—N2—H2B	120.0
C11—C12—C13	122.0 (2)	H2A—N2—H2B	120.0
C11—C12—H12	119.0	C20—N3—C24	119.7 (3)
C13—C12—H12	119.0	C20—N3—H3	123.3
C8—C13—C12	121.4 (3)	C24—N3—H3	117.0
C8—C13—H13	119.3	C22—N4—H4A	120.0
C12—C13—H13	119.3	C22—N4—H4B	120.0
O4—C14—O3	125.0 (3)	H4A—N4—H4B	120.0
O4—C14—C11	116.9 (3)	H5A—O5—H5B	121.3
O3—C14—C11	117.8 (3)	H6A—O6—H6B	121.3
C16—C15—N1	123.8 (3)	H7A—O7—H7B	121.3
C16—C15—H15	118.1	H8A—O8—H8B	121.3
O1—C1—C2—C7	−51.5 (3)	C8—C9—C10—C11	−3.70 (17)
O2—C1—C2—C7	123.3 (3)	C8^i—C9—C10—C11	176.30 (17)
O1—C1—C2—C3	136.0 (2)	C9—C10—C11—C12	−1.4 (3)
O2—C1—C2—C3	−49.2 (3)	C11^i—C10—C11—C12	178.6 (3)
C7—C2—C3—C2^i	176.7 (3)	C9—C10—C11—C14	172.6 (2)
C1—C2—C3—C2^i	−11.0 (2)	C11^i—C10—C11—C14	−7.4 (2)
C7—C2—C3—C4	−3.3 (3)	C10—C11—C12—C13	4.4 (4)
C1—C2—C3—C4	169.0 (2)	C14—C11—C12—C13	−170.1 (3)
C2—C3—C4—C5^i	177.92 (17)	C9—C8—C13—C12	−3.1 (4)
C2^i—C3—C4—C5^i	−2.08 (17)	C5—C8—C13—C12	176.7 (2)
C2—C3—C4—C5	−2.08 (17)	C11—C12—C13—C8	−2.1 (5)
C2^i—C3—C4—C5	177.92 (17)	C12—C11—C14—O4	113.4 (3)
C5^i—C4—C5—C6	−173.7 (3)	C10—C11—C14—O4	−60.9 (4)
C3—C4—C5—C6	6.3 (3)	C12—C11—C14—O3	−61.8 (4)
C5^i—C4—C5—C8	6.16 (16)	C10—C11—C14—O3	123.9 (3)
C3—C4—C5—C8	−173.84 (16)	N1—C15—C16—C17	−0.4 (5)
C4—C5—C6—C7	−5.2 (4)	C15—C16—C17—N2	−180.0 (3)
C8—C5—C6—C7	175.0 (2)	C15—C16—C17—C18	0.6 (5)
C3—C2—C7—C6	4.7 (4)	N2—C17—C18—C19	−179.7 (4)
C1—C2—C7—C6	−168.3 (3)	C16—C17—C18—C19	−0.3 (5)
C5—C6—C7—C2	−0.3 (4)	C17—C18—C19—N1	−0.2 (6)
C6—C5—C8—C13	−12.5 (4)	N3—C20—C21—C22	−0.1 (5)
C4—C5—C8—C13	167.7 (2)	C20—C21—C22—N4	−177.2 (3)
C6—C5—C8—C9	167.3 (2)	C20—C21—C22—C23	1.3 (5)
C4—C5—C8—C9	−12.6 (3)	N4—C22—C23—C24	177.2 (3)
C13—C8—C9—C10	5.9 (3)	C21—C22—C23—C24	−1.3 (4)
C5—C8—C9—C10	−173.84 (16)	C22—C23—C24—N3	0.2 (5)
C13—C8—C9—C8^i	−174.1 (3)	C18—C19—N1—C15	0.3 (5)
C5—C8—C9—C8^i	6.16 (16)	C16—C15—N1—C19	0.0 (5)
C8—C9—C10—C11^i	176.30 (17)	C21—C20—N3—C24	−1.1 (5)
C8^i—C9—C10—C11^i	−3.70 (17)	C23—C24—N3—C20	1.0 (5)

Symmetry code: (i) −x+2, y, −z+3/2.

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
N1—H1···O8	0.90	1.77	2.641 (4)	163
N2—H2A···O4^ii	0.86	2.49	3.281 (3)	153
N2—H2B···O5^iii	0.86	2.08	2.936 (4)	174
N3—H3···O3^iv	0.90	1.82	2.706 (3)	170
N4—H4A···O2^v	0.86	2.04	2.895 (3)	174
N4—H4B···O6^vi	0.86	2.13	2.930 (4)	154
O5—H5A···O7^vii	0.82	2.00	2.791 (3)	160
O5—H5B···O4	0.82	1.91	2.720 (3)	167
O6—H6A···O7^viii	0.82	1.98	2.797 (4)	177
O6—H6B···O1	0.82	1.95	2.766 (3)	176
O7—H7A···O3^ix	0.82	1.93	2.742 (4)	174
O7—H7B···O1	0.82	1.90	2.704 (3)	167
O8—H8A···O2^v	0.82	1.88	2.686 (3)	169
O8—H8B···O5^x	0.82	2.04	2.783 (3)	151
C18—H18···O4^ii	0.93	2.49	3.318 (4)	148
C19—H19···O6^iii	0.93	2.54	3.368 (4)	148

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