Topological analysis of the three-dimensional coordination polymer poly[(μ₄-azido)[μ₄-5-(pyridin-4-yl)tetra­zolido]disilver(I)]

The title compound, [Ag₂(C₆H₄N₄)(N₃₎]_n, was obtained under hydro­thermal conditions at 433 K. The asymmetric unit of the ortho­rhom­bic space group (Pna2₁) consists of two Ag⁺ cations, an anionic 5-(pyridin-4-yl)tetra­zolide (4-ptz⁻) ligand and an anionic azide ligand. Both Ag⁺ centres are coordinated by four N atoms, forming a distorted tetra­hedral coordination environment. When all the component ions are viewed as 4-connected nodes, the whole three-dimensional network can be regarded topologically as a new kind of 4,4,4,4-connected net with the Schläfli symbol (4.8⁵)(4².8⁴)(4³.8³)₂.

S1. Introduction

In continuation of our investigations on the synthesis and properties of multifunctional pyridinyl­tetra­zole-based metal coordination complexes (Meng et al., 2012), we report here another 5-(pyridin-4-yl)-1Htetra­zole (4-ptz) complex of silver (I), (I). It was synthesized under hydro­thermal conditions using the rea­cta­nts pyridine-4-carbo­nitrile, sodium azide and silver nitrate in water for 3 d at 433 K. The formation mechanism can be explained with the well-known [2+3] cyclo-addition reaction of organic nitriles with azide anions in the presence of metal ions such as Zn²⁺, Cd²⁺,Cu⁺ and Ag⁺ as catalysts (Yang et al., 2009).

S2. Experimental

S2.1. Synthesis and crystallization

All the reagents and solvents were used as obtained without further purification. Equimolar amounts of Ag(NO₃) (0.2 mmol, 34.0 mg), pyridine-4-carbo­nitrile (0.2 mmol, 20.8 mg) and sodium azide (0.2 mmol, 13.0 mg) were mixed in water (10.0 ml) and stirred for half an hour under ambient conditions. The mixture was then sealed in a 20 ml Teflon-lined reaction vessel and heated at 433 K for 3 d. Subsequently, the reaction solution was cooled slowly to room temperature at a rate of 5 K h^-1. The final product was collected manually, washed with water and air-dried to obtain block-shaped crystals of (I) (yield 56%, based on silver nitrate, ca 23.2 mg).

S2.2. Second-harmonic generation (SHG)

A powder nonlinear second-harmonic generation (SHG) test of (I) was conducted using the Kurtz & Perry (1968) method. The SHG intensity data were obtained by placing a sieved (80–100 mesh) powder sample in an intense fundamental beam from a Q-switched Nd:YAG laser with wavelength 1064 nm. The output (at λ = 532 nm) was filtered first to remove the multiplier and was then displayed on an oscilloscope. This procedure was then repeated using the standard nonlinear optical (NLO) material (i.e. microcrystalline KDP), and the ratio of the second-harmonic intensity outputs was calculated successively. A solid-state photoluminescence measurement of the powder sample (I) was carried out on a Perkin–Elmer LS50B.

S2.3. Refinement

Crystal data, data collection and structure refinement details are summarized in Table 1. Systematic absences indicated two possible space groups, viz. Pna2₁ and Pnma. No suitable solution could be obtained in the space group Pnma, whereas a satisfactory solution and refinement was achieved in the polar space group Pna2₁. Convergence was reached with a Flack (1983) parameter of 0.50 (6), indicating inversion twinning in this noncentrosymmetric space group. ADDSYM in PLATON (Spek, 2009) did not detect higher (i.e. Pnma) symmetry.

H atoms bonded to C atoms were positioned geometrically, with aromatic C—H = 0.93 Å, and refined in a riding mode, with U_iso(H) = 1.2U_eq(C).

S3. Results and discussion

The asymmetric unit of poly[(µ₄-azido)[µ₄-5-(pyridin-4-yl)tetra­zolido]disilver(I), (I), consists of two Ag⁺ cations, one anionic 5-(pyridin-4-yl)tetra­zolide (4-ptz) ligand and one anionic azide ligand (Fig. 1), all in general positions. In the 4-ptz^- ligand, the plane of the tetra­zole ring is twisted slightly from that of the pyridine ring [dihedral angle = 17.7 (3)°], which is significantly smaller than the corresponding value of 45.0 (2)° in a copper analog (Meng et al., 2012). Both Ag⁺ cations atoms are coordinated by four N in a distorted tetra­hedral configuration. For atom Ag1, the Ag—N bond lengths range from 2.196 (5) to 2.478 (7) Å and the N—Ag—N angles range from 95.7 (2) to 140.82 (19)° (Table 2). The corresponding ranges for atom Ag2 are 2.224 (5)–2.666 (5) Å and 88.34 (19)–168.82 (18)°, indicating that the degree of distorsion from an ideal tetra­hedron around Ag2 is more extreme than that around Ag1. Comparing all Ag—N bonds, it can be seen that the Ag—N(tetra­zole) and Ag—N(pyridine) bond lengths have similar values and that they are shorter by ca 0.1 Å than the Ag—N(azide) bonds (Table 2).

The ionic components of (I) are linked into a three-dimensional coordination polymer which can be analyzed as follows. Firstly, the combination of the pyridine and 3-positioned N atom of the tetra­zole ring and atom Ag2 forms a one-dimensional coordination chain along the [011] direction under the symmetry operation of a glide plane at (0.25, 0, 0) (Fig. 2). By the symmetry operation of the twofold skew axis at [0.5,0,0], another one-dimensional chain is generated along the [011] direction (Fig. 2). Similarly, the 2- and 4-positioned N atoms of the tetra­zole ring and atom Ag1 are linked into another one-dimensional chain along the [100] direction by a glide plane at the (0, 0.25, 0) plane. Consequently, the 4-ptz^- anions and Ag⁺ cations are linked into a three-dimensional network based on these three symmetry operations. Secondly, the anionic azide ligands link the Ag⁺ cations, forming another one-dimensional chain along the [010] direction (Fig. 3). Based on the above, a three-dimensional network is formed by a combination of 4-ptz^- and azide ligands coordinating to Ag⁺ cations.

Considering the denticity of the ligands, 4-ptz^- can be regarded as tetra­dentate, with three tetra­zole and one pyridine N atoms coordinated to four different Ag⁺ cations in a µ₄- bridging mode, leaving one lateral tetra­zole N atom (N4) uncoordinated. The N1-coordinated Ag1 atom and N2-coordinated Ag2 atom are only 3.4502 (8) Å apart, wheras the N2-coordinated Ag2 atom and the N3-coordinated Ag1(x+1/2, -y+1/2, z) atom are separated by 4.3144 (8) Å. As for the azide ligand, it can also act as a tetra­dentate ligand, but in the µ₄-η_N6²:η_N8²- bridging mode from which each end-on N atom is coordinated to two Ag⁺ cations, with Ag1···Ag1(x, y+1, zz) and Ag2···Ag2(x, y+1, zz) distances of 3.651 (2) Å (Fig. 3). It is worth mentioning that this µ₄-bridging mode of the azide anion with silver is the first reported example, based on a search in the Cambridge Structural Database (CSD, Version 5.35, plus February 2014 update; Allen, 2002), where more than a 2000 hits are found related to azide anion coordination.

In the crystal packing, two weak inter­molecular C—H···N contacts were found in an analysis using PLATON (Spek, 2009) between pyridine atom C3 and azide atom N7 and between azide atoms C4 and N8 (Fig. 3 and Table 3). Additionally, a stronger π–π stacking inter­action between inversion-related pyridine rings was also observed, with a centroid-to-centroid distance of only 3.651 (2) Å (Fig. 3).

In order to further characterize the crystal structure of (I), we have tried to analyze and simplify the complex network using the program TOPOS (Blatov, 2006) by considering all the Ag⁺, 4-ptz^- and N₃^- ions as topologically independent 4,4,4,4-connected nodes. As mentioned above, we have firstly taken into account only the 4-ptz^- and Ag⁺ ions. The crystal structure can be simplified into a CdSO₄-like topological network (Fig. 4) by these two types of 4-connected nodes, with short and long Schläfli symbols of (6⁵.8) and [6.6.6.6.6 (2).*], respectively. By considering the azide ligand as the third 4-connected node, the CdSO₄-like network is further linked together, forming the final 4,4,4,4-connected three-dimensional topological network with a nodal stoichiometric ratio of 1:1:1:1 and a Schläfli symbol of (4.8⁵)(4².8⁴)(4³.8³)₂ (Fig. 5).

In view of a potential photoluminescence properties of this pyridinyl­tetra­zole-based metal polymer, complex (I) was studied with the exciting wavelength set at 355 nm. The result indicates that it could be a potentially good blue-coloured fluorescent material, with a maximum emission peak at about 452 nm which might be attributed to intra-ligand π–π* orbital charge transfer (MLCT).

The nonlinear SHG property of complex (I) is mainly due to its acentric polar crystal structure in the space group Pna2₁. The SHG experiment indicates that (I) has a SHG intensity that is ca 0.8 times as strong as that of potassium di­hydrogen phosphate (KDP), which indicates some of the powder samples in bulk compound (I) are probably acentric symmetrical structure giving a certain directional polarity, although the crystal selected for X-ray diffraction experiment has been an inversion-twinned sample.

Related literature

For related literature, see: Allen (2002); Blatov (2006); Flack (1983); Kurtz & Perry (1968); Meng et al. (2012); Spek (2009); Yang et al. (2009).

Computing details

Data collection: SMART (Bruker, 2002); cell refinement: SMART (Bruker, 2002); data reduction: SAINT (Bruker, 2002); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: DIAMOND (Brandenburg, 2008); software used to prepare material for publication: SHELXTL (Sheldrick, 2008) and PLATON (Spek, 2009).

Figures

	Fig. 1. The molecular structure of (I), showing the atom-numbering scheme. Displacement ellipsoids are drawn at the 30% probability level and H atoms are shown as small spheres of arbitrary radii. [Symmetry codes: (i) x-1/2, -y+1/2, z; (ii) x, y-1, z; (iii) -x+1/2, y+1/2, z+1/2; (iv) x+1/2, -y+3/2, z; (v) x+1/2, -y+1/2, z; (vi) -x+1/2, y-1/2, z-1/2.]
	Fig. 2. Part of the crystal structure of (I), showing the formation of a three-dimensional network with Ag^+ cations and 4-ptz anions only, where the pale-blue and pale-red regions represent the [011] and [011] chains formed by Ag1 and 4-ptz, respectively. The pale-red region represents the [100] coordination chain formed by Ag2 and 4-ptz.
	Fig. 3. Part of the crystal packing, showing (a) the formation of the three-dimensional coordination network, with C—H···N interactions shown as red dashed lines, (b) the π–π interactions between inversion-related pyridine rings and (c) the coordination linkage between metal Ag^+ ion and the azide anion. [Symmetry code: (i) x, y+1, z.]
	Fig. 4. (a) The topological network with only Ag^+ cations and 5-(pyridin-4-yl)-tetrazolide anions shown as 2- and 4-connected nodes, respectively. (b) The simplified CdSO4-like topological network in which the 2-connected nodes have been omitted.
	Fig. 5. The final simplified topological network with all the component ions being regarded as 4-connected nodes.

Poly[(µ₄-azido)[µ₄-5-(pyridin-4-yl)tetrazolido]disilver(I)]

Crystal data

[Ag2(C6H4N4)(N3)]	F(000) = 760
Mr = 403.91	Dx = 2.902 Mg m^−3
Orthorhombic, Pna21	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: P 2c -2n	Cell parameters from 2417 reflections
a = 12.6308 (9) Å	θ = 3.2–28.2°
b = 3.6509 (3) Å	µ = 4.22 mm^−1
c = 20.0499 (15) Å	T = 298 K
V = 924.58 (12) Å^3	Block, colourless
Z = 4	0.12 × 0.10 × 0.10 mm

Data collection

Bruker SMART APEX CCD area-detector diffractometer	2244 independent reflections
Radiation source: fine-focus sealed tube	2040 reflections with I > 2σ(I)
Graphite monochromator	Rint = 0.035
phi and ω scans	θmax = 28.3°, θmin = 2.0°
Absorption correction: multi-scan (SADABS; Sheldrick, 2008)	h = −16→11
Tmin = 0.622, Tmax = 0.678	k = −4→4
6266 measured reflections	l = −25→26

Refinement

Refinement on F^2	Secondary atom site location: difference Fourier map
Least-squares matrix: full	Hydrogen site location: inferred from neighbouring sites
R[F^2 > 2σ(F^2)] = 0.038	H-atom parameters constrained
wR(F^2) = 0.084	w = 1/[σ^2(Fo^2) + (0.0269P)^2 + 1.5374P] where P = (Fo^2 + 2Fc^2)/3
S = 1.14	(Δ/σ)max < 0.001
2244 reflections	Δρmax = 1.16 e Å^−3
146 parameters	Δρmin = −0.80 e Å^−3
1 restraint	Absolute structure: Flack (1983), 1074 Friedel pairs
Primary atom site location: structure-invariant direct methods	Absolute structure parameter: 0.50 (6)

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. The final converged flack parameter is of 0.50 (6), it is not confident for us to derive absolute structure of the compound. From this parameter, it can be concluded the crystal selected for X-ray diffraction should be inversion-related twin.

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

	x	y	z	Uiso*/Ueq
Ag1	0.14715 (4)	0.1949 (2)	0.25579 (2)	0.03993 (18)
Ag2	0.35349 (5)	0.3370 (2)	0.365545 (18)	0.0452 (2)
C1	0.3029 (4)	0.0240 (15)	0.0929 (3)	0.0184 (10)
C2	0.3620 (5)	−0.1344 (17)	0.0415 (3)	0.0258 (12)
H2	0.4333	−0.1891	0.0476	0.031*
C3	0.3134 (6)	−0.2071 (16)	−0.0177 (3)	0.0268 (13)
H3	0.3529	−0.3218	−0.0508	0.032*
C4	0.1554 (5)	0.0208 (19)	0.0186 (3)	0.0271 (13)
H4	0.0845	0.0752	0.0109	0.033*
C5	0.1962 (4)	0.0939 (16)	0.0808 (3)	0.0204 (11)
H5	0.1531	0.1886	0.1143	0.024*
C6	0.3563 (4)	0.1144 (18)	0.1564 (3)	0.0208 (12)
N1	0.3077 (4)	0.1875 (13)	0.2138 (2)	0.0224 (10)
N2	0.3869 (4)	0.2617 (15)	0.2572 (3)	0.0268 (9)
N3	0.4779 (4)	0.2295 (15)	0.2248 (3)	0.0319 (12)
N4	0.4607 (4)	0.1349 (16)	0.1613 (3)	0.0292 (11)
N5	0.2115 (4)	−0.1238 (13)	−0.0316 (2)	0.0254 (10)
N6	0.1501 (5)	0.7132 (17)	0.3355 (3)	0.0398 (14)
N7	0.0702 (6)	0.6986 (13)	0.3666 (3)	0.0335 (12)
N8	−0.0078 (5)	0.6836 (16)	0.3982 (3)	0.0419 (15)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Ag1	0.0171 (2)	0.0705 (4)	0.0322 (3)	0.0044 (2)	0.0039 (3)	0.0028 (3)
Ag2	0.0393 (4)	0.0775 (5)	0.0189 (3)	−0.0048 (3)	0.0063 (3)	−0.0114 (4)
C1	0.017 (3)	0.022 (3)	0.016 (2)	−0.003 (2)	0.002 (2)	0.0015 (19)
C2	0.022 (3)	0.031 (3)	0.025 (3)	0.000 (2)	0.006 (2)	−0.003 (2)
C3	0.040 (4)	0.023 (3)	0.017 (3)	0.011 (3)	0.005 (3)	−0.004 (2)
C4	0.015 (3)	0.042 (4)	0.024 (3)	−0.001 (3)	−0.002 (2)	−0.001 (3)
C5	0.016 (3)	0.028 (3)	0.018 (3)	0.000 (2)	0.003 (2)	0.0029 (19)
C6	0.019 (3)	0.024 (3)	0.019 (3)	0.001 (2)	−0.001 (2)	0.005 (2)
N1	0.013 (2)	0.033 (3)	0.021 (2)	0.002 (2)	−0.0006 (19)	−0.0014 (19)
N2	0.018 (2)	0.044 (2)	0.018 (2)	0.000 (2)	−0.003 (2)	−0.006 (2)
N3	0.018 (2)	0.056 (3)	0.021 (2)	−0.003 (2)	0.001 (2)	−0.008 (2)
N4	0.013 (2)	0.052 (3)	0.022 (2)	−0.002 (2)	0.0037 (19)	−0.004 (2)
N5	0.027 (3)	0.031 (3)	0.018 (2)	−0.003 (2)	−0.002 (2)	−0.0001 (19)
N6	0.037 (4)	0.047 (3)	0.035 (3)	0.004 (3)	0.004 (3)	0.002 (3)
N7	0.041 (4)	0.027 (3)	0.033 (3)	0.002 (2)	−0.010 (3)	0.004 (2)
N8	0.039 (4)	0.048 (4)	0.038 (4)	−0.005 (3)	0.001 (3)	−0.003 (3)

Geometric parameters (Å, º)

Ag1—N1	2.196 (5)	C4—N5	1.340 (8)
Ag1—N3^i	2.244 (5)	C4—C5	1.375 (8)
Ag1—N6^ii	2.377 (7)	C4—H4	0.9300
Ag1—N6	2.478 (7)	C5—H5	0.9300
Ag2—N2	2.231 (6)	C6—N4	1.325 (7)
Ag2—N5^iii	2.224 (5)	C6—N1	1.331 (8)
Ag2—N8^iv	2.561 (7)	N1—N2	1.353 (7)
Ag2—N8^v	2.667 (7)	N2—N3	1.325 (7)
C1—C5	1.394 (8)	N3—N4	1.337 (7)
C1—C2	1.398 (8)	N3—Ag1^v	2.244 (5)
C1—C6	1.477 (8)	N5—Ag2^vi	2.224 (5)
C2—C3	1.364 (9)	N6—N7	1.187 (10)
C2—H2	0.9300	N6—Ag1^vii	2.377 (7)
C3—N5	1.351 (9)	N7—N8	1.174 (9)
C3—H3	0.9300	N8—Ag2^viii	2.561 (6)
N1—Ag1—N3^i	140.82 (19)	C4—C5—C1	119.0 (5)
N1—Ag1—N6^ii	103.5 (2)	C4—C5—H5	120.5
N3^i—Ag1—N6^ii	107.0 (2)	C1—C5—H5	120.5
N1—Ag1—N6	104.05 (19)	N4—C6—N1	112.6 (5)
N3^i—Ag1—N6	95.7 (2)	N4—C6—C1	122.0 (5)
N6^ii—Ag1—N6	97.5 (2)	N1—C6—C1	125.4 (5)
N5^iii—Ag2—N2	168.82 (18)	C6—N1—N2	104.7 (5)
N5^iii—Ag2—N8^iv	88.34 (19)	C6—N1—Ag1	139.4 (4)
N5^iii—Ag2—N8^v	93.48 (19)	N2—N1—Ag1	115.7 (4)
N8^v—Ag2—N8^iv	88.56 (19)	N3—N2—N1	108.0 (5)
N2—Ag2—N8^iv	101.77 (19)	N3—N2—Ag2	130.7 (4)
N2—Ag2—N8^v	91.55 (19)	N1—N2—Ag2	120.7 (4)
C5—C1—C2	117.6 (5)	N2—N3—N4	110.4 (5)
C5—C1—C6	123.5 (5)	N2—N3—Ag1^v	132.8 (4)
C2—C1—C6	118.9 (5)	N4—N3—Ag1^v	116.8 (4)
C3—C2—C1	118.8 (6)	C6—N4—N3	104.3 (5)
C3—C2—H2	120.6	C4—N5—C3	115.9 (5)
C1—C2—H2	120.6	C4—N5—Ag2^vi	121.8 (4)
N5—C3—C2	124.4 (6)	C3—N5—Ag2^vi	121.9 (4)
N5—C3—H3	117.8	N7—N6—Ag1^vii	111.9 (5)
C2—C3—H3	117.8	N7—N6—Ag1	106.9 (5)
N5—C4—C5	124.0 (5)	Ag1^vii—N6—Ag1	97.5 (2)
N5—C4—H4	118.0	N8—N7—N6	178.9 (8)
C5—C4—H4	118.0	N7—N8—Ag2^viii	113.8 (5)
C5—C1—C2—C3	0.9 (9)	Ag1—N1—N2—Ag2	5.0 (5)
C6—C1—C2—C3	−178.0 (5)	N5^iii—Ag2—N2—N3	−162.3 (8)
C1—C2—C3—N5	2.6 (10)	N8^iv—Ag2—N2—N3	43.3 (5)
N5—C4—C5—C1	1.7 (10)	N5^iii—Ag2—N2—N1	7.8 (12)
C2—C1—C5—C4	−3.0 (8)	N8^iv—Ag2—N2—N1	−146.6 (4)
C6—C1—C5—C4	175.9 (6)	N1—N2—N3—N4	−0.2 (7)
C5—C1—C6—N4	−162.2 (6)	Ag2—N2—N3—N4	170.9 (4)
C2—C1—C6—N4	16.7 (9)	N1—N2—N3—Ag1^v	179.4 (4)
C5—C1—C6—N1	17.0 (10)	Ag2—N2—N3—Ag1^v	−9.6 (8)
C2—C1—C6—N1	−164.1 (6)	N1—C6—N4—N3	−0.7 (7)
N4—C6—N1—N2	0.6 (7)	C1—C6—N4—N3	178.6 (6)
C1—C6—N1—N2	−178.7 (6)	N2—N3—N4—C6	0.5 (7)
N4—C6—N1—Ag1	−175.8 (5)	Ag1^v—N3—N4—C6	−179.1 (4)
C1—C6—N1—Ag1	4.9 (11)	C5—C4—N5—C3	1.6 (9)
N3^i—Ag1—N1—C6	−27.4 (8)	C5—C4—N5—Ag2^vi	−171.4 (5)
N6^ii—Ag1—N1—C6	113.1 (7)	C2—C3—N5—C4	−3.9 (9)
N6—Ag1—N1—C6	−145.4 (6)	C2—C3—N5—Ag2^vi	169.2 (5)
N3^i—Ag1—N1—N2	156.4 (4)	N1—Ag1—N6—N7	−170.4 (5)
N6^ii—Ag1—N1—N2	−63.0 (4)	N3^i—Ag1—N6—N7	43.7 (5)
N6—Ag1—N1—N2	38.5 (4)	N6^ii—Ag1—N6—N7	−64.4 (6)
C6—N1—N2—N3	−0.2 (6)	N1—Ag1—N6—Ag1^vii	74.0 (2)
Ag1—N1—N2—N3	177.2 (4)	N3^i—Ag1—N6—Ag1^vii	−71.9 (2)
C6—N1—N2—Ag2	−172.4 (4)	N6^ii—Ag1—N6—Ag1^vii	180.0

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

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
C4—H4···N8^ix	0.93	2.61	3.235 (9)	125
C3—H3···N7^x	0.93	2.60	3.500 (8)	164

Symmetry codes: (ix) −x, −y+1, z−1/2; (x) −x+1/2, y−3/2, z−1/2.