Tetra­aqua­bis[3-(2-pyridylsulfan­yl)propionato N-oxide]nickel(II)

Ravindran Durai Nayagam, B.; Jebas, S.R.; Edward Rajkumar, J.P.; Schollmeyer, D.

doi:10.1107/S1600536809011283

metal-organic compounds

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ISSN: 2056-9890

Volume 65| Part 4| April 2009| Page m470

doi:10.1107/S1600536809011283

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Tetraaquabis[3-(2-pyridylsulfanyl)propionato N-oxide]nickel(II)

B. Ravindran Durai Nayagam,^a ^* Samuel Robinson Jebas,^b J. P. Edward Rajkumar ^c and Dieter Schollmeyer ^d

^aDepartment of Chemistry, Popes College, Sawyerpuram 628 251, Tamilnadu, India, ^bDepartment of Physics, Karunya University, Karunya Nagar, Coimbatore 641 114, India, ^cDepartment of Physics, Popes College, Sawyerpuram 628 251, Tamilnadu, India, and ^dInstitut für Organische Chemie, Universität Mainz, Duesbergweg 10-14, 55099 Mainz, Germany
^*Correspondence e-mail: b_ravidurai@yahoo.com

(Received 25 March 2009; accepted 26 March 2009; online 31 March 2009)

In the centrosymmetric title compound, [Ni(C₈H₈NO₃S)₂(H₂O)₄], the Ni^II ion, which lies on an inversion centre, is six coordinated by four water molecules and two propionate O atoms from two 2-pyridylsulfanylpropionate N-oxide ligands, forming a slightly distorted octahedral geometry. An intramolecular O—H⋯O hydrogen bond stabilizes the molecular conformation. The crystal packing is consolidated by intermolecular O—H⋯O and C—H⋯O hydrogen bonding.

Related literature

For the biological activities of N-oxide derivatives, see: Bovin et al. (1992 ); Katsuyuki et al. (1991 ). Leonard et al. (1955 ); Lobana & Bhatia (1989 ); Symons & West (1985 ). For related literature, see: Jebas et al. (2005 ); Ravindran et al. (2008 ).

Experimental

Crystal data

[Ni(C₈H₈NO₃S)₂(H₂O)₄]
M_r = 527.20
Triclinic, $[P \overline 1]$
a = 4.8155 (5) Å
b = 8.7650 (10) Å
c = 12.9560 (15) Å
α = 86.400 (2)°
β = 79.501 (2)°
γ = 84.929 (2)°
V = 534.98 (10) Å³
Z = 1
Mo Kα radiation
μ = 1.16 mm⁻¹
T = 173 K
0.35 × 0.28 × 0.07 mm

Data collection

Bruker SMART APEXII CCD diffractometer
Absorption correction: multi-scan (SADABS; Sheldrick, 2008 ) T_min = 0.405, T_max = 0.492 (expected range = 0.759–0.922)
9627 measured reflections
2615 independent reflections
2501 reflections with I > 2σ(I)
R_int = 0.017

Refinement

R[F² > 2σ(F²)] = 0.020
wR(F²) = 0.055
S = 1.05
2615 reflections
142 parameters
H-atom parameters constrained
Δρ_max = 0.40 e Å⁻³
Δρ_min = −0.36 e Å⁻³

Table 1
Selected bond lengths (Å)

Ni1—O13	2.0488 (8)
Ni1—O15	2.0644 (8)
Ni1—O14	2.0898 (8)
N1—O7	1.3154 (13)

Table 2
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
O14—H14A⋯O12	0.81	1.84	2.6248 (12)	162
O14—H14B⋯O15ⁱ	0.81	2.27	2.9517 (12)	142
O14—H14B⋯O13ⁱ	0.81	2.64	3.2316 (12)	131
O15—H15A⋯O7ⁱⁱ	0.82	1.83	2.6469 (12)	172
O15—H15B⋯O13ⁱⁱⁱ	0.83	1.83	2.6570 (11)	172
C4—H4⋯O12^iv	0.95	2.48	3.2044 (17)	133
C6—H6⋯O14^v	0.95	2.46	3.2515 (16)	140
C10—H10B⋯O12^vi	0.99	2.42	3.3910 (14)	167
Symmetry codes: (i) x-1, y, z; (ii) x-1, y, z+1; (iii) -x+2, -y+1, -z+1; (iv) -x+1, -y+2, -z; (v) x+1, y, z-1; (vi) x+1, y, z.

Data collection: APEX2 (Bruker, 2008 ); cell refinement: APEX2; data reduction: APEX2; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXTL and PLATON (Spek, 2009 ).

Supporting information

Crystal structure: contains datablocks global, I. DOI: 10.1107/S1600536809011283/bt2915sup1.cif

Structure factors: contains datablock I. DOI: 10.1107/S1600536809011283/bt2915Isup2.hkl

checkCIF report

Comment top

N-oxides and their derivatives show a broad spectrum of biological activity such as antifungal, antimicrobial and antibacterial activities (Lobana & Bhatia, 1989; Symons & West, 1985). These compounds are also found to be involved in DNA strand scission under physiological conditions (Katsuyuki et al., 1991; Bovin et al., 1992). Pyridine N–oxides bearing a sulfur group in position two display significant antimicrobial activity (Leonard et al., 1955). In view of the importance of N–oxides, we have previously reported the crystal structures of N–oxide derivatives (Jebas et al., 2005; Ravindran et al., 2008). As an extension of our work on N–oxide derivatives, we report here the crystal structure of the title compound.

The asymmetric unit comprises of half molecule of the title compound, the other half is symmetry generated [symmetry code: -x + 1,-y + 1,-z + 1]. The Ni^II ion which lies on an inversion centre is six coordinated by four water molecules and two propianoto oxygen atoms from two 2-pyridylsulfanylpropionato N-oxide ligands forming a slightly distorted octahedral geometry. The bond lengths and angles agree well with the N–oxide derivatives reported earlier (Jebas et al., 2005)

Intramolecular O—H···O hydrogen bonding influences the conformation of the molecule. The crystal packing (Fig. 2) is consolidated by intermolecular O—H···O and C—H···O hydrogen bonding together with intramolecular S···O = 2.6968 (10) Å; O···O = 2.6248 (12) Å, intermolecular O···Oⁱ = 2.6469 (12) Å; O···Oⁱⁱ = 2.6570 (12) Å and O···Oⁱⁱⁱ = 2.9455 (12) Å [symmetry code: (i):1 + x,y,-1 + z; (ii) 2 - x,1 - y,1 - z; (iii) 1 - x,1 - y,1 - z] short contacts. The molecules are stacked along the a axis.

Related literature top

For the biological activities of N-oxide derivatives, see: Bovin et al. (1992); Katsuyuki et al. (1991). Leonard et al. (1955); Lobana & Bhatia (1989); Symons & West (1985). For related literature, see: Jebas et al. (2005); Ravindran et al. (2008).

Experimental top

A mixture of the potassium salt of 3(1-oxo-pyridinine- 2-sulfanyl)propionic acid (0.237 g,1 mmol) and Nickel (II) chloride (0.13 g, 0.5 mmol), in water (20 ml) was heated at 333k with continous stirring for one hour. The solution was kept aside for slow evaporation. After two weeks, green colored crystals were obtained.

Computing details top

Data collection: APEX2 (Bruker, 2008); cell refinement: APEX2 (Bruker, 2008); data reduction: APEX2 (Bruker, 2008); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: SHELXTL (Sheldrick, 2008); software used to prepare material for publication: SHELXTL (Sheldrick, 2008) and PLATON (Spek, 2009).

Figures top

	Fig. 1. The molecular structure of the title compound, showing 50% probability displacement ellipsoids and the atom numbering scheme. Symmetry code: -x + 1,-y + 1,-z + 1.
	Fig. 2. The crystal packing of the title compound, viewed down the a axis. Molecules are stacked along the a axis.

Tetraaquabis[3-(2-pyridylsulfanyl)propionato N-oxide]nickel(II) top

Crystal data top

[Ni(C₈H₈NO₃S)₂(H₂O)₄]	Z = 1
M_r = 527.20	F(000) = 274
Triclinic, P1	D_x = 1.636 Mg m⁻³
Hall symbol: -P 1	Mo Kα radiation, λ = 0.71069 Å
a = 4.8155 (5) Å	Cell parameters from 6946 reflections
b = 8.765 (1) Å	θ = 2.3–28.2°
c = 12.9560 (15) Å	µ = 1.16 mm⁻¹
α = 86.400 (2)°	T = 173 K
β = 79.501 (2)°	Plate, green
γ = 84.929 (2)°	0.35 × 0.28 × 0.07 mm
V = 534.98 (10) Å³

Data collection top

Bruker SMART APEXII CCD diffractometer	2615 independent reflections
Radiation source: sealed Tube	2501 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.017
CCD scan	θ_max = 28.2°, θ_min = 1.6°
Absorption correction: multi-scan (SADABS; Sheldrick, 2008)	h = −6→6
T_min = 0.405, T_max = 0.492	k = −11→11
9627 measured reflections	l = −17→17

Refinement top

Refinement on F²	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F² > 2σ(F²)] = 0.020	Hydrogen site location: inferred from neighbouring sites
wR(F²) = 0.055	H-atom parameters constrained
S = 1.05	w = 1/[σ²(F_o²) + (0.0294P)² + 0.2035P] where P = (F_o² + 2F_c²)/3
2615 reflections	(Δ/σ)_max < 0.001
142 parameters	Δρ_max = 0.40 e Å⁻³
0 restraints	Δρ_min = −0.36 e Å⁻³

Crystal data top

[Ni(C₈H₈NO₃S)₂(H₂O)₄]	γ = 84.929 (2)°
M_r = 527.20	V = 534.98 (10) Å³
Triclinic, P1	Z = 1
a = 4.8155 (5) Å	Mo Kα radiation
b = 8.765 (1) Å	µ = 1.16 mm⁻¹
c = 12.9560 (15) Å	T = 173 K
α = 86.400 (2)°	0.35 × 0.28 × 0.07 mm
β = 79.501 (2)°

Data collection top

Bruker SMART APEXII CCD diffractometer	2615 independent reflections
Absorption correction: multi-scan (SADABS; Sheldrick, 2008)	2501 reflections with I > 2σ(I)
T_min = 0.405, T_max = 0.492	R_int = 0.017
9627 measured reflections

Refinement top

R[F² > 2σ(F²)] = 0.020	0 restraints
wR(F²) = 0.055	H-atom parameters constrained
S = 1.05	Δρ_max = 0.40 e Å⁻³
2615 reflections	Δρ_min = −0.36 e Å⁻³
142 parameters

Special details top

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 (Å²) top

	x	y	z	U_iso*/U_eq
Ni1	0.5000	0.5000	0.5000	0.01405 (6)
N1	1.2267 (2)	0.77325 (12)	−0.20598 (8)	0.0197 (2)
C2	1.0696 (2)	0.79224 (13)	−0.10776 (8)	0.0167 (2)
C3	0.8347 (3)	0.89789 (14)	−0.09608 (9)	0.0212 (2)
H3	0.7229	0.9119	−0.0285	0.025*
C4	0.7627 (3)	0.98279 (15)	−0.18222 (10)	0.0254 (3)
H4	0.6027	1.0555	−0.1741	0.031*
C5	0.9266 (3)	0.96061 (16)	−0.28069 (10)	0.0291 (3)
H5	0.8794	1.0180	−0.3406	0.035*
C6	1.1567 (3)	0.85564 (16)	−0.29110 (10)	0.0274 (3)
H6	1.2685	0.8403	−0.3586	0.033*
O7	1.44740 (19)	0.67209 (11)	−0.21521 (7)	0.0264 (2)
S8	1.19805 (6)	0.67388 (3)	−0.01121 (2)	0.01821 (7)
C9	0.9142 (2)	0.70873 (14)	0.09994 (8)	0.0176 (2)
H9A	0.8880	0.8194	0.1136	0.021*
H9B	0.7344	0.6765	0.0848	0.021*
C10	0.9942 (2)	0.61698 (13)	0.19529 (8)	0.0177 (2)
H10A	1.0268	0.5071	0.1796	0.021*
H10B	1.1730	0.6511	0.2100	0.021*
C11	0.7643 (2)	0.63642 (13)	0.29187 (8)	0.0159 (2)
O12	0.55540 (18)	0.72858 (11)	0.28796 (7)	0.02341 (18)
O13	0.80467 (17)	0.55410 (10)	0.37393 (6)	0.01937 (17)
O14	0.26378 (17)	0.70791 (10)	0.47964 (6)	0.01973 (17)
H14A	0.3264	0.7271	0.4179	0.030*
H14B	0.0952	0.7007	0.4849	0.030*
O15	0.71999 (16)	0.60521 (10)	0.59458 (6)	0.01836 (17)
H15A	0.6207	0.6252	0.6514	0.028*
H15B	0.8596	0.5522	0.6092	0.028*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Ni1	0.01095 (10)	0.01935 (11)	0.01075 (10)	0.00213 (7)	−0.00158 (7)	0.00208 (7)
N1	0.0229 (5)	0.0205 (5)	0.0145 (4)	−0.0001 (4)	−0.0011 (4)	−0.0011 (4)
C2	0.0191 (5)	0.0181 (5)	0.0127 (5)	−0.0037 (4)	−0.0019 (4)	−0.0001 (4)
C3	0.0223 (5)	0.0220 (6)	0.0178 (5)	0.0001 (4)	−0.0012 (4)	0.0002 (4)
C4	0.0277 (6)	0.0222 (6)	0.0259 (6)	0.0029 (5)	−0.0068 (5)	0.0018 (5)
C5	0.0399 (7)	0.0273 (6)	0.0199 (6)	0.0009 (5)	−0.0090 (5)	0.0055 (5)
C6	0.0377 (7)	0.0291 (6)	0.0135 (5)	0.0006 (5)	−0.0017 (5)	0.0019 (5)
O7	0.0259 (4)	0.0308 (5)	0.0188 (4)	0.0075 (4)	0.0012 (3)	−0.0025 (3)
S8	0.01735 (13)	0.02289 (15)	0.01281 (13)	0.00103 (10)	−0.00070 (10)	0.00202 (10)
C9	0.0165 (5)	0.0228 (5)	0.0121 (5)	−0.0006 (4)	−0.0004 (4)	0.0017 (4)
C10	0.0159 (5)	0.0229 (5)	0.0131 (5)	−0.0003 (4)	−0.0016 (4)	0.0021 (4)
C11	0.0147 (5)	0.0205 (5)	0.0127 (5)	−0.0024 (4)	−0.0027 (4)	0.0011 (4)
O12	0.0214 (4)	0.0286 (5)	0.0168 (4)	0.0067 (3)	−0.0004 (3)	0.0053 (3)
O13	0.0137 (4)	0.0298 (4)	0.0129 (4)	0.0020 (3)	−0.0015 (3)	0.0052 (3)
O14	0.0164 (4)	0.0241 (4)	0.0169 (4)	0.0030 (3)	−0.0013 (3)	0.0019 (3)
O15	0.0147 (4)	0.0256 (4)	0.0139 (4)	0.0030 (3)	−0.0024 (3)	−0.0009 (3)

Geometric parameters (Å, º) top

Ni1—O13ⁱ	2.0488 (8)	C5—H5	0.9500
Ni1—O13	2.0488 (8)	C6—H6	0.9500
Ni1—O15ⁱ	2.0644 (8)	S8—C9	1.8165 (11)
Ni1—O15	2.0644 (8)	C9—C10	1.5216 (15)
Ni1—O14	2.0898 (8)	C9—H9A	0.9900
Ni1—O14ⁱ	2.0898 (8)	C9—H9B	0.9900
N1—O7	1.3154 (13)	C10—C11	1.5195 (15)
N1—C6	1.3579 (16)	C10—H10A	0.9900
N1—C2	1.3687 (14)	C10—H10B	0.9900
C2—C3	1.3889 (16)	C11—O12	1.2395 (14)
C2—S8	1.7405 (11)	C11—O13	1.2818 (13)
C3—C4	1.3823 (17)	O14—H14A	0.8142
C3—H3	0.9500	O14—H14B	0.8100
C4—C5	1.3877 (19)	O15—H15A	0.8216
C4—H4	0.9500	O15—H15B	0.8268
C5—C6	1.3685 (19)

O13ⁱ—Ni1—O13	180.0	C6—C5—H5	120.2
O13ⁱ—Ni1—O15ⁱ	88.53 (3)	C4—C5—H5	120.2
O13—Ni1—O15ⁱ	91.47 (3)	N1—C6—C5	120.67 (11)
O13ⁱ—Ni1—O15	91.47 (3)	N1—C6—H6	119.7
O13—Ni1—O15	88.53 (3)	C5—C6—H6	119.7
O15ⁱ—Ni1—O15	180.0	C2—S8—C9	100.23 (5)
O13ⁱ—Ni1—O14	88.50 (3)	C10—C9—S8	108.10 (8)
O13—Ni1—O14	91.50 (3)	C10—C9—H9A	110.1
O15ⁱ—Ni1—O14	90.66 (3)	S8—C9—H9A	110.1
O15—Ni1—O14	89.34 (3)	C10—C9—H9B	110.1
O13ⁱ—Ni1—O14ⁱ	91.50 (3)	S8—C9—H9B	110.1
O13—Ni1—O14ⁱ	88.50 (3)	H9A—C9—H9B	108.4
O15ⁱ—Ni1—O14ⁱ	89.34 (3)	C11—C10—C9	111.71 (9)
O15—Ni1—O14ⁱ	90.66 (3)	C11—C10—H10A	109.3
O14—Ni1—O14ⁱ	180.0	C9—C10—H10A	109.3
O7—N1—C6	121.11 (10)	C11—C10—H10B	109.3
O7—N1—C2	117.75 (10)	C9—C10—H10B	109.3
C6—N1—C2	121.15 (10)	H10A—C10—H10B	107.9
N1—C2—C3	118.82 (10)	O12—C11—O13	124.27 (10)
N1—C2—S8	112.99 (8)	O12—C11—C10	119.85 (10)
C3—C2—S8	128.19 (9)	O13—C11—C10	115.87 (9)
C4—C3—C2	120.43 (11)	C11—O13—Ni1	126.08 (7)
C4—C3—H3	119.8	Ni1—O14—H14A	98.9
C2—C3—H3	119.8	Ni1—O14—H14B	114.4
C3—C4—C5	119.25 (12)	H14A—O14—H14B	107.5
C3—C4—H4	120.4	Ni1—O15—H15A	111.7
C5—C4—H4	120.4	Ni1—O15—H15B	113.8
C6—C5—C4	119.69 (12)	H15A—O15—H15B	105.3

O7—N1—C2—C3	−179.71 (10)	N1—C2—S8—C9	−171.71 (9)
C6—N1—C2—C3	0.08 (17)	C3—C2—S8—C9	7.83 (12)
O7—N1—C2—S8	−0.12 (14)	C2—S8—C9—C10	−178.14 (8)
C6—N1—C2—S8	179.66 (10)	S8—C9—C10—C11	−178.64 (8)
N1—C2—C3—C4	−0.39 (18)	C9—C10—C11—O12	−5.56 (15)
S8—C2—C3—C4	−179.90 (10)	C9—C10—C11—O13	175.00 (10)
C2—C3—C4—C5	0.4 (2)	O12—C11—O13—Ni1	20.56 (17)
C3—C4—C5—C6	−0.1 (2)	C10—C11—O13—Ni1	−160.03 (7)
O7—N1—C6—C5	179.99 (12)	O15—Ni1—O13—C11	−120.55 (9)
C2—N1—C6—C5	0.2 (2)	O14—Ni1—O13—C11	−31.25 (10)
C4—C5—C6—N1	−0.2 (2)

Symmetry code: (i) −x+1, −y+1, −z+1.

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O14—H14A···O12	0.81	1.84	2.6248 (12)	162
O14—H14B···O15ⁱⁱ	0.81	2.27	2.9517 (12)	142
O14—H14B···O13ⁱⁱ	0.81	2.64	3.2316 (12)	131
O15—H15A···O7ⁱⁱⁱ	0.82	1.83	2.6469 (12)	172
O15—H15B···O13^iv	0.83	1.83	2.6570 (11)	172
C4—H4···O12^v	0.95	2.48	3.2044 (17)	133
C6—H6···O14^vi	0.95	2.46	3.2515 (16)	140
C10—H10B···O12^vii	0.99	2.42	3.3910 (14)	167

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

Experimental details

Crystal data
Chemical formula	[Ni(C₈H₈NO₃S)₂(H₂O)₄]
M_r	527.20
Crystal system, space group	Triclinic, P1
Temperature (K)	173
a, b, c (Å)	4.8155 (5), 8.765 (1), 12.9560 (15)
α, β, γ (°)	86.400 (2), 79.501 (2), 84.929 (2)
V (Å³)	534.98 (10)
Z	1
Radiation type	Mo Kα
µ (mm⁻¹)	1.16
Crystal size (mm)	0.35 × 0.28 × 0.07

Data collection
Diffractometer	Bruker SMART APEXII CCD diffractometer
Absorption correction	Multi-scan (SADABS; Sheldrick, 2008)
T_min, T_max	0.405, 0.492
No. of measured, independent and observed [I > 2σ(I)] reflections	9627, 2615, 2501
R_int	0.017
(sin θ/λ)_max (Å⁻¹)	0.664

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.020, 0.055, 1.05
No. of reflections	2615
No. of parameters	142
H-atom treatment	H-atom parameters constrained
Δρ_max, Δρ_min (e Å⁻³)	0.40, −0.36

Computer programs: APEX2 (Bruker, 2008), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), SHELXTL (Sheldrick, 2008) and PLATON (Spek, 2009).

Selected bond lengths (Å) top

Ni1—O13	2.0488 (8)	Ni1—O14	2.0898 (8)
Ni1—O15	2.0644 (8)	N1—O7	1.3154 (13)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O14—H14A···O12	0.81	1.84	2.6248 (12)	162
O14—H14B···O15ⁱ	0.81	2.27	2.9517 (12)	142
O14—H14B···O13ⁱ	0.81	2.64	3.2316 (12)	131
O15—H15A···O7ⁱⁱ	0.82	1.83	2.6469 (12)	172
O15—H15B···O13ⁱⁱⁱ	0.83	1.83	2.6570 (11)	172
C4—H4···O12^iv	0.95	2.48	3.2044 (17)	133
C6—H6···O14^v	0.95	2.46	3.2515 (16)	140
C10—H10B···O12^vi	0.99	2.42	3.3910 (14)	167

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

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Volume 65| Part 4| April 2009| Page m470

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