A new two-dimensional Cd^II coordination polymer based on 1,3-bis­(2-methyl-1H-imidazol-1-yl)benzene and 1,3-phenyl­enedi­acetic acid: synthesis, crystal structure and physical properties

A novel two-dimensional Cd^II coordination framework, poly[[[μ-1,3-bis­(2-methyl-1H-imidazol-1-yl)benzene-κ²N:N′](μ-1,3-phenyl­enedi­acetato-κ⁴O,O′:O′′,O′′′)cadmium(II)] dihydrate], {[Cd(C₁₀H₈O₄)(C₁₄H₁₄N₄)]·2H₂O}_n or {[Cd(PDA)(1,3-BMIB)]·2H₂O}_n [1,3-BMIB is 1,3-bis­(2-methyl-1H-imidazol-1-yl)benzene and H₂PDA is 1,3-phenyl­enedi­acetic acid], has been prepared and characterized using IR, elemental analysis, thermal analysis and single-crystal X-ray diffraction, the latter revealing that the compound is a (4,4) grid coordination polymer with layers oriented parallel to the bc crystal planes. In the crystal, adjacent layers are further connected by O—H...O and C—H...O hydrogen bonds, forming a three-dimensional structure in the solid state. In addition, the compound exhibits strong fluorescence emissions and shows photocatalytic activity for the degradation of methyl­ene blue in the solid state at room temperature.

Computing details

Data collection: APEX3 (Bruker, 2018); cell refinement: SAINT (Bruker, 2018); data reduction: SAINT (Bruker, 2018); program(s) used to solve structure: SHELXS2014 (Sheldrick, 2015a); program(s) used to refine structure: SHELXL2014 (Sheldrick, 2015b); molecular graphics: DIAMOND (Brandenburg, 2005); software used to prepare material for publication: SHELXL2014 (Sheldrick, 2015b).

Poly[[[µ-1,3-bis(2-methyl-1H-imidazol-1-yl)benzene-κ²N:N'](µ-1,3-phenylenediacetato-κ⁴O,O':O'',O''')cadmium(II)] dihydrate]

Crystal data

[Cd(C10H8O4)(C14H14N4)]·2H2O	Z = 2
Mr = 578.89	F(000) = 588
Triclinic, P1	Dx = 1.561 Mg m^−3
a = 9.643 (8) Å	Mo Kα radiation, λ = 0.71073 Å
b = 10.934 (9) Å	Cell parameters from 9712 reflections
c = 11.787 (9) Å	θ = 2.1–27.8°
α = 95.776 (15)°	µ = 0.93 mm^−1
β = 94.925 (16)°	T = 296 K
γ = 91.207 (17)°	Block, colorless
V = 1231.4 (17) Å^3	0.21 × 0.19 × 0.17 mm

Data collection

Bruker APEXII CCD area detector diffractometer	3414 reflections with I > 2σ(I)
phi and ω scans	Rint = 0.054
Absorption correction: multi-scan (SADABS; Bruker, 2008)	θmax = 26.0°, θmin = 1.9°
Tmin = 0.822, Tmax = 0.853	h = −11→11
10600 measured reflections	k = −13→13
4813 independent reflections	l = −14→14

Refinement

Refinement on F^2	0 restraints
Least-squares matrix: full	Hydrogen site location: mixed
R[F^2 > 2σ(F^2)] = 0.047	H-atom parameters constrained
wR(F^2) = 0.119	w = 1/[σ^2(Fo^2) + (0.057P)^2] where P = (Fo^2 + 2Fc^2)/3
S = 0.96	(Δ/σ)max < 0.001
4813 reflections	Δρmax = 0.53 e Å^−3
318 parameters	Δρmin = −1.25 e Å^−3

Special details

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 (Ų)

	x	y	z	Uiso*/Ueq
C1	0.5720 (6)	0.5366 (5)	0.3536 (5)	0.0492 (14)
C2	0.4889 (6)	0.6532 (5)	0.3630 (5)	0.0566 (15)
H2A	0.3977	0.6324	0.3858	0.068*
H2B	0.5348	0.7099	0.4240	0.068*
C3	0.4687 (5)	0.7199 (4)	0.2561 (4)	0.0384 (12)
C4	0.4411 (5)	0.8452 (4)	0.2692 (4)	0.0387 (12)
H4	0.4420	0.8847	0.3429	0.046*
C5	0.4124 (5)	0.9138 (4)	0.1770 (4)	0.0373 (11)
C6	0.4132 (5)	0.8547 (5)	0.0664 (5)	0.0441 (13)
H6	0.3934	0.8979	0.0033	0.053*
C7	0.4437 (5)	0.7317 (5)	0.0515 (4)	0.0467 (13)
H7	0.4480	0.6936	−0.0222	0.056*
C8	0.4679 (5)	0.6635 (5)	0.1440 (4)	0.0441 (13)
H8	0.4838	0.5798	0.1314	0.053*
C9	0.3755 (5)	1.0466 (5)	0.1959 (5)	0.0514 (14)
H9A	0.2932	1.0527	0.2378	0.062*
H9B	0.3525	1.0768	0.1222	0.062*
C10	0.4919 (6)	1.1288 (4)	0.2617 (5)	0.0430 (13)
C11	0.8077 (5)	0.4551 (4)	0.1327 (4)	0.0409 (12)
H11	0.7610	0.5255	0.1563	0.049*
C12	0.8605 (5)	0.4340 (4)	0.0319 (4)	0.0372 (11)
H12	0.8563	0.4846	−0.0272	0.045*
C13	0.9025 (5)	0.2775 (4)	0.1342 (4)	0.0371 (11)
C14	0.9934 (5)	0.2616 (4)	−0.0594 (4)	0.0326 (11)
C15	0.9205 (5)	0.2427 (4)	−0.1669 (4)	0.0367 (11)
H15	0.8280	0.2647	−0.1782	0.044*
C16	0.9904 (5)	0.1897 (4)	−0.2570 (4)	0.0359 (11)
C17	1.1256 (5)	0.1554 (5)	−0.2422 (4)	0.0440 (13)
H17	1.1704	0.1210	−0.3040	0.053*
C18	1.1954 (5)	0.1726 (5)	−0.1333 (5)	0.0463 (13)
H18	1.2863	0.1468	−0.1218	0.056*
C19	1.1303 (5)	0.2277 (4)	−0.0419 (4)	0.0406 (12)
H19	1.1781	0.2417	0.0303	0.049*
C20	0.8581 (5)	0.0650 (5)	−0.4251 (4)	0.0443 (13)
H20	0.8669	−0.0132	−0.4012	0.053*
C21	0.7859 (6)	0.0949 (5)	−0.5218 (4)	0.0476 (13)
H21	0.7358	0.0397	−0.5762	0.057*
C22	0.8798 (5)	0.2656 (5)	−0.4339 (4)	0.0398 (12)
C23	0.9483 (6)	0.1562 (5)	0.1687 (5)	0.0543 (15)
H23A	1.0296	0.1681	0.2220	0.081*
H23B	0.9698	0.1043	0.1022	0.081*
H23C	0.8749	0.1179	0.2040	0.081*
C24	0.9242 (6)	0.3968 (5)	−0.4025 (5)	0.0607 (16)
H24A	0.8982	0.4435	−0.4653	0.091*
H24B	1.0234	0.4026	−0.3854	0.091*
H24C	0.8797	0.4288	−0.3365	0.091*
Cd1	0.69218 (4)	0.31120 (3)	0.33160 (3)	0.03812 (14)
N1	0.8321 (4)	0.3580 (4)	0.1972 (3)	0.0386 (10)
N2	0.9233 (4)	0.3206 (3)	0.0334 (3)	0.0319 (9)
N3	0.9157 (4)	0.1731 (4)	−0.3693 (3)	0.0396 (10)
N4	0.7984 (4)	0.2198 (4)	−0.5267 (3)	0.0412 (10)
O1	0.5250 (4)	0.4471 (3)	0.2838 (4)	0.0656 (12)
O2	0.6833 (4)	0.5293 (4)	0.4142 (3)	0.0666 (12)
O3	0.6054 (4)	1.1358 (4)	0.2180 (3)	0.0596 (11)
O4	0.4722 (4)	1.1873 (4)	0.3549 (3)	0.0663 (11)
O5	0.2362 (5)	0.3928 (4)	0.2076 (4)	0.0873 (15)
H5A	0.3217	0.4150	0.2193	0.131*
H5B	0.2154	0.3541	0.2632	0.131*
O6	0.2129 (5)	0.2639 (5)	0.4184 (4)	0.0955 (16)
H6B	0.2263	0.3282	0.4654	0.143*
H6A	0.2912	0.2305	0.4136	0.143*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
C1	0.052 (4)	0.054 (4)	0.048 (3)	0.016 (3)	0.025 (3)	0.020 (3)
C2	0.072 (4)	0.059 (4)	0.042 (3)	0.026 (3)	0.012 (3)	0.009 (3)
C3	0.033 (3)	0.045 (3)	0.037 (3)	0.003 (2)	0.003 (2)	0.003 (2)
C4	0.039 (3)	0.041 (3)	0.034 (3)	0.007 (2)	−0.003 (2)	−0.007 (2)
C5	0.024 (2)	0.038 (3)	0.049 (3)	−0.004 (2)	−0.001 (2)	0.004 (2)
C6	0.037 (3)	0.046 (3)	0.049 (3)	−0.010 (2)	−0.004 (2)	0.010 (3)
C7	0.053 (3)	0.051 (3)	0.034 (3)	−0.007 (3)	0.001 (2)	−0.005 (2)
C8	0.044 (3)	0.037 (3)	0.051 (3)	0.004 (2)	0.004 (3)	−0.001 (2)
C9	0.048 (3)	0.044 (3)	0.059 (4)	0.001 (3)	−0.007 (3)	0.004 (3)
C10	0.051 (3)	0.026 (3)	0.053 (3)	0.000 (2)	−0.002 (3)	0.010 (2)
C11	0.041 (3)	0.035 (3)	0.047 (3)	−0.002 (2)	0.010 (2)	0.005 (2)
C12	0.036 (3)	0.038 (3)	0.038 (3)	0.005 (2)	0.002 (2)	0.006 (2)
C13	0.037 (3)	0.040 (3)	0.035 (3)	−0.001 (2)	0.003 (2)	0.003 (2)
C14	0.034 (3)	0.033 (3)	0.031 (3)	0.002 (2)	0.004 (2)	0.001 (2)
C15	0.033 (3)	0.041 (3)	0.035 (3)	0.005 (2)	−0.001 (2)	0.001 (2)
C16	0.041 (3)	0.037 (3)	0.028 (3)	0.003 (2)	−0.001 (2)	0.002 (2)
C17	0.042 (3)	0.052 (3)	0.039 (3)	0.009 (3)	0.013 (2)	−0.002 (2)
C18	0.036 (3)	0.049 (3)	0.053 (3)	0.011 (2)	0.005 (3)	0.001 (3)
C19	0.037 (3)	0.046 (3)	0.036 (3)	0.002 (2)	−0.004 (2)	−0.001 (2)
C20	0.058 (3)	0.034 (3)	0.040 (3)	0.007 (2)	0.003 (3)	−0.001 (2)
C21	0.059 (3)	0.042 (3)	0.038 (3)	0.011 (3)	0.000 (3)	−0.008 (2)
C22	0.043 (3)	0.047 (3)	0.030 (3)	0.005 (2)	0.008 (2)	0.005 (2)
C23	0.065 (4)	0.049 (3)	0.055 (3)	0.017 (3)	0.019 (3)	0.021 (3)
C24	0.076 (4)	0.048 (3)	0.057 (4)	−0.008 (3)	−0.009 (3)	0.011 (3)
Cd1	0.0474 (3)	0.0384 (2)	0.0285 (2)	0.00463 (16)	0.00428 (16)	0.00203 (15)
N1	0.038 (2)	0.041 (2)	0.037 (2)	0.0024 (19)	0.0071 (19)	0.0007 (19)
N2	0.034 (2)	0.034 (2)	0.028 (2)	0.0013 (17)	0.0035 (17)	0.0031 (17)
N3	0.049 (3)	0.038 (2)	0.031 (2)	0.004 (2)	0.0011 (19)	0.0017 (19)
N4	0.055 (3)	0.041 (2)	0.028 (2)	0.011 (2)	0.002 (2)	0.0043 (18)
O1	0.061 (3)	0.043 (2)	0.088 (3)	0.009 (2)	−0.004 (2)	−0.006 (2)
O2	0.076 (3)	0.069 (3)	0.051 (2)	0.034 (2)	−0.010 (2)	−0.005 (2)
O3	0.054 (3)	0.070 (3)	0.051 (2)	−0.013 (2)	0.008 (2)	−0.007 (2)
O4	0.066 (3)	0.069 (3)	0.060 (3)	0.006 (2)	0.013 (2)	−0.017 (2)
O5	0.070 (3)	0.120 (4)	0.066 (3)	−0.017 (3)	−0.009 (2)	−0.001 (3)
O6	0.065 (3)	0.108 (4)	0.106 (4)	0.002 (3)	0.009 (3)	−0.024 (3)

Geometric parameters (Å, º)

C1—O2	1.246 (6)	C16—C17	1.366 (7)
C1—O1	1.263 (7)	C16—N3	1.445 (6)
C1—C2	1.520 (7)	C17—C18	1.390 (7)
C2—C3	1.520 (7)	C17—H17	0.9300
C2—H2A	0.9700	C18—C19	1.385 (7)
C2—H2B	0.9700	C18—H18	0.9300
C3—C4	1.397 (6)	C19—H19	0.9300
C3—C8	1.400 (7)	C20—C21	1.356 (7)
C4—C5	1.393 (6)	C20—N3	1.375 (6)
C4—H4	0.9300	C20—H20	0.9300
C5—C6	1.395 (7)	C21—N4	1.376 (6)
C5—C9	1.502 (7)	C21—H21	0.9300
C6—C7	1.380 (7)	C22—N4	1.337 (6)
C6—H6	0.9300	C22—N3	1.358 (6)
C7—C8	1.387 (7)	C22—C24	1.490 (7)
C7—H7	0.9300	C23—H23A	0.9600
C8—H8	0.9300	C23—H23B	0.9600
C9—C10	1.529 (7)	C23—H23C	0.9600
C9—H9A	0.9700	C24—H24A	0.9600
C9—H9B	0.9700	C24—H24B	0.9600
C10—O4	1.245 (6)	C24—H24C	0.9600
C10—O3	1.253 (6)	Cd1—N4^i	2.219 (4)
C11—C12	1.334 (7)	Cd1—N1	2.256 (4)
C11—N1	1.379 (6)	Cd1—O1	2.288 (4)
C11—H11	0.9300	Cd1—O3^ii	2.322 (4)
C12—N2	1.393 (6)	Cd1—O2	2.490 (4)
C12—H12	0.9300	Cd1—O4^ii	2.543 (4)
C13—N1	1.329 (6)	N4—Cd1^iii	2.219 (4)
C13—N2	1.350 (6)	O3—Cd1^iv	2.322 (4)
C13—C23	1.491 (6)	O4—Cd1^iv	2.543 (4)
C14—C19	1.383 (6)	O5—H5A	0.8500
C14—C15	1.389 (6)	O5—H5B	0.8500
C14—N2	1.439 (5)	O6—H6B	0.8500
C15—C16	1.391 (6)	O6—H6A	0.8500
C15—H15	0.9300
O2—C1—O1	120.9 (5)	C17—C18—H18	119.8
O2—C1—C2	120.6 (6)	C14—C19—C18	119.2 (4)
O1—C1—C2	118.5 (5)	C14—C19—H19	120.4
C3—C2—C1	116.6 (4)	C18—C19—H19	120.4
C3—C2—H2A	108.1	C21—C20—N3	106.5 (4)
C1—C2—H2A	108.1	C21—C20—H20	126.8
C3—C2—H2B	108.1	N3—C20—H20	126.8
C1—C2—H2B	108.1	C20—C21—N4	109.2 (5)
H2A—C2—H2B	107.3	C20—C21—H21	125.4
C4—C3—C8	117.0 (4)	N4—C21—H21	125.4
C4—C3—C2	118.2 (4)	N4—C22—N3	109.3 (4)
C8—C3—C2	124.7 (5)	N4—C22—C24	126.7 (4)
C5—C4—C3	123.1 (4)	N3—C22—C24	124.0 (4)
C5—C4—H4	118.5	C13—C23—H23A	109.5
C3—C4—H4	118.5	C13—C23—H23B	109.5
C4—C5—C6	118.3 (4)	H23A—C23—H23B	109.5
C4—C5—C9	120.8 (5)	C13—C23—H23C	109.5
C6—C5—C9	120.8 (5)	H23A—C23—H23C	109.5
C7—C6—C5	119.6 (5)	H23B—C23—H23C	109.5
C7—C6—H6	120.2	C22—C24—H24A	109.5
C5—C6—H6	120.2	C22—C24—H24B	109.5
C6—C7—C8	121.5 (5)	H24A—C24—H24B	109.5
C6—C7—H7	119.2	C22—C24—H24C	109.5
C8—C7—H7	119.2	H24A—C24—H24C	109.5
C7—C8—C3	120.4 (5)	H24B—C24—H24C	109.5
C7—C8—H8	119.8	N4^i—Cd1—N1	114.26 (15)
C3—C8—H8	119.8	N4^i—Cd1—O1	145.18 (16)
C5—C9—C10	113.7 (4)	N1—Cd1—O1	94.83 (16)
C5—C9—H9A	108.8	N4^i—Cd1—O3^ii	98.13 (15)
C10—C9—H9A	108.8	N1—Cd1—O3^ii	91.32 (15)
C5—C9—H9B	108.8	O1—Cd1—O3^ii	99.96 (15)
C10—C9—H9B	108.8	N4^i—Cd1—O2	103.33 (14)
H9A—C9—H9B	107.7	N1—Cd1—O2	93.06 (14)
O4—C10—O3	122.0 (5)	O1—Cd1—O2	54.14 (14)
O4—C10—C9	120.3 (5)	O3^ii—Cd1—O2	154.00 (14)
O3—C10—C9	117.8 (5)	N4^i—Cd1—O4^ii	88.39 (15)
C12—C11—N1	110.5 (4)	N1—Cd1—O4^ii	141.31 (14)
C12—C11—H11	124.7	O1—Cd1—O4^ii	79.21 (15)
N1—C11—H11	124.7	O3^ii—Cd1—O4^ii	53.13 (13)
C11—C12—N2	105.7 (4)	O2—Cd1—O4^ii	112.59 (14)
C11—C12—H12	127.2	C13—N1—C11	106.1 (4)
N2—C12—H12	127.2	C13—N1—Cd1	125.4 (3)
N1—C13—N2	110.0 (4)	C11—N1—Cd1	122.4 (3)
N1—C13—C23	124.9 (4)	C13—N2—C12	107.7 (4)
N2—C13—C23	125.1 (4)	C13—N2—C14	127.8 (4)
C19—C14—C15	121.3 (4)	C12—N2—C14	124.5 (4)
C19—C14—N2	120.6 (4)	C22—N3—C20	108.1 (4)
C15—C14—N2	118.0 (4)	C22—N3—C16	124.9 (4)
C14—C15—C16	117.7 (4)	C20—N3—C16	126.8 (4)
C14—C15—H15	121.1	C22—N4—C21	106.9 (4)
C16—C15—H15	121.1	C22—N4—Cd1^iii	131.2 (3)
C17—C16—C15	122.2 (4)	C21—N4—Cd1^iii	121.8 (3)
C17—C16—N3	120.2 (4)	C1—O1—Cd1	96.9 (3)
C15—C16—N3	117.7 (4)	C1—O2—Cd1	87.9 (3)
C16—C17—C18	119.1 (5)	C10—O3—Cd1^iv	96.5 (3)
C16—C17—H17	120.5	C10—O4—Cd1^iv	86.4 (3)
C18—C17—H17	120.5	H5A—O5—H5B	108.0
C19—C18—C17	120.5 (5)	H6B—O6—H6A	107.1
C19—C18—H18	119.8
O2—C1—C2—C3	118.2 (6)	C12—C11—N1—Cd1	−152.9 (3)
O1—C1—C2—C3	−62.4 (7)	N1—C13—N2—C12	−1.0 (5)
C1—C2—C3—C4	−156.7 (5)	C23—C13—N2—C12	177.5 (5)
C1—C2—C3—C8	27.1 (8)	N1—C13—N2—C14	−179.3 (4)
C8—C3—C4—C5	0.6 (7)	C23—C13—N2—C14	−0.8 (8)
C2—C3—C4—C5	−175.9 (5)	C11—C12—N2—C13	1.4 (5)
C3—C4—C5—C6	−0.9 (7)	C11—C12—N2—C14	179.8 (4)
C3—C4—C5—C9	176.5 (5)	C19—C14—N2—C13	−58.1 (7)
C4—C5—C6—C7	−0.7 (7)	C15—C14—N2—C13	123.4 (5)
C9—C5—C6—C7	−178.2 (5)	C19—C14—N2—C12	123.8 (5)
C5—C6—C7—C8	2.8 (8)	C15—C14—N2—C12	−54.7 (6)
C6—C7—C8—C3	−3.1 (8)	N4—C22—N3—C20	1.8 (6)
C4—C3—C8—C7	1.4 (7)	C24—C22—N3—C20	−179.2 (5)
C2—C3—C8—C7	177.7 (5)	N4—C22—N3—C16	−173.1 (4)
C4—C5—C9—C10	62.7 (7)	C24—C22—N3—C16	5.9 (8)
C6—C5—C9—C10	−119.9 (5)	C21—C20—N3—C22	−1.2 (6)
C5—C9—C10—O4	−119.8 (5)	C21—C20—N3—C16	173.5 (4)
C5—C9—C10—O3	61.0 (6)	C17—C16—N3—C22	−108.4 (6)
N1—C11—C12—N2	−1.3 (5)	C15—C16—N3—C22	70.7 (6)
C19—C14—C15—C16	−0.8 (7)	C17—C16—N3—C20	77.7 (7)
N2—C14—C15—C16	177.7 (4)	C15—C16—N3—C20	−103.2 (6)
C14—C15—C16—C17	0.8 (7)	N3—C22—N4—C21	−1.6 (6)
C14—C15—C16—N3	−178.4 (4)	C24—C22—N4—C21	179.4 (5)
C15—C16—C17—C18	0.7 (8)	N3—C22—N4—Cd1^iii	177.4 (3)
N3—C16—C17—C18	179.9 (5)	C24—C22—N4—Cd1^iii	−1.5 (8)
C16—C17—C18—C19	−2.3 (8)	C20—C21—N4—C22	0.9 (6)
C15—C14—C19—C18	−0.7 (7)	C20—C21—N4—Cd1^iii	−178.3 (3)
N2—C14—C19—C18	−179.2 (4)	O2—C1—O1—Cd1	3.9 (6)
C17—C18—C19—C14	2.2 (8)	C2—C1—O1—Cd1	−175.5 (4)
N3—C20—C21—N4	0.2 (6)	O1—C1—O2—Cd1	−3.6 (5)
N2—C13—N1—C11	0.2 (5)	C2—C1—O2—Cd1	175.8 (4)
C23—C13—N1—C11	−178.3 (5)	O4—C10—O3—Cd1^iv	−15.3 (5)
N2—C13—N1—Cd1	152.8 (3)	C9—C10—O3—Cd1^iv	163.9 (4)
C23—C13—N1—Cd1	−25.7 (7)	O3—C10—O4—Cd1^iv	13.9 (5)
C12—C11—N1—C13	0.7 (6)	C9—C10—O4—Cd1^iv	−165.3 (4)

Symmetry codes: (i) x, y, z+1; (ii) x, y−1, z; (iii) x, y, z−1; (iv) x, y+1, z.

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
C19—H19···O5^v	0.93	2.54	3.360 (7)	147
C6—H6···O3^vi	0.93	2.60	3.354 (7)	139
O6—H6B···O2^vii	0.85	2.12	2.945 (6)	164
O6—H6A···O4^ii	0.85	1.98	2.793 (6)	160
O5—H5B···O6	0.85	2.17	3.000 (7)	167
O5—H5A···O1	0.85	2.05	2.883 (6)	166
C8—H8···O1	0.93	2.43	3.051 (7)	124

Symmetry codes: (ii) x, y−1, z; (v) x+1, y, z; (vi) −x+1, −y+2, −z; (vii) −x+1, −y+1, −z+1.