Solvent-dependent formation of two different Cd(II) complexes with p-BrC6H4C(S)NHP(O)(OiPr)2 (HL). Crystallographic modification of Cd(HL)2L2

Article abstract of DOI:10.1016/j.poly.2010.02.031

Reaction of the potassium salt of N-(diisopropoxyphosphoryl)-p-bromothiobenzamide p-BrC₆H₄C(S)NHP(O)(OiPr)₂ (HL) with Cd(II) cations in freshly dried and distilled EtOH leads exclusively to the complex [Cd(p-BrC₆H₄C(S)NH₂-S)(L-O,S)₂] ([Cd(L^I)L₂]), while the same reaction in H₂O leads to the complex [Cd(HL-O)₂(L-O,S)₂] ([Cd(HL)₂L₂]). The corresponding reactions with Zn(II) always lead to the complex [Zn(L-O,S)₂] ([ZnL₂]) regardless of the solvent. The crystal structure of [Cd(HL)₂L₂]^.2/3H₂O reveals to be a polymorph to the previously reported anhydrous [Cd(HL)₂L₂].

Full text of DOI:10.1016/j.poly.2010.02.031

Polyhedron 29 (2010) 1837–1841
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Polyhedron
journal homepage: www.elsevier.com/locate/poly
Solvent-dependent formation of two different Cd(II) complexes with
p-BrC6H4C(S)NHP(O)(OiPr)2 (HL). Crystallographic modification of Cd(HL)2L2
a
a,
b
c
Damir A. Safin a, , Maria G. Babashkina , Axel Klein , Heinrich Nöth , Michael Bolte ,
*
*
Dmitriy B. Krivolapov d
a Institut für Anorganische Chemie, Universität zu Köln, Greinstrasse 6, D-50939 Köln, Germany
b Department Chemie und Biochemie der Universität München, Butenandtstrasse 5-13 (Haus D), D-81377 München, Germany
c Institut für Anorganische Chemie J.-W.-Goethe-Universität, Frankfurt/Main, Germany
d A.E. Arbuzov Institute of Organic and Physical Chemistry, Arbuzov Street 8, 420088, Kazan, Russian Federation
a r t i c l e i n f o
a b s t r a c t
Article history:
Reaction of the potassium salt of N-(diisopropoxyphosphoryl)-p-bromothiobenzamide p-BrC6H4C(S)NHP
(O)(OiPr)2 (HL) with Cd(II) cations in freshly dried and distilled EtOH leads exclusively to the complex
[Cd(p-BrC6H4C(S)NH2-S)(L-O,S)2] ([Cd(LI)L2]), while the same reaction in H2O leads to the complex
[Cd(HL-O)2(L-O,S)2] ([Cd(HL)2L2]). The corresponding reactions with Zn(II) always lead to the
complex [Zn(L-O,S)2] ([ZnL2]) regardless of the solvent. The crystal structure of [Cd(HL)2L2].2/3H2O
reveals to be a polymorph to the previously reported anhydrous [Cd(HL)2L2].
Received 29 December 2009
Accepted 22 February 2010
Available online 25 February 2010
Keywords:
Cadmium
Ó 2010 Elsevier Ltd. All rights reserved.
Coordination mode
Crystal structure
Polymorphism
Thiobenzamide ligand
Zinc
½Cd2L4ꢀ þ 3 H2O ! ½CdL2ꢀ þ p-BrC6H4CðSÞNH2
1. Introduction
þ ðiPrOÞ2PðOÞOH þ CdðOHÞ2 þ HL
ð2Þ
In the course of our research on transition metal complexes of
N-(thio)phosphorylated (thio)amides and (thio)ureas RC(X)NHP-
(Y)(OiPr)2 (X, Y = O, S) [1–23] we recently reported on the coordi-
nation of N-(diisopropoxyphosphoryl)-p-bromothiobenzamide
p-BrC6H4C(S)NHP(O)(OiPr)2 (HL) towards Zn(II) and Cd(II) [23].
Reaction of the potassium salt of the benzamide with Zn(II) and
Cd(II) cations in aqueous EtOH leads to the three different com-
plexes: [Zn(L-O,S)2] ([ZnL2]), [Cd(HL-O)2(L-O,S)2] ([Cd(HL)2L2])
and [Cd(p-BrC6H4C(S)NH2-S)(L-O,S)2] ([Cd(LI)L2]). The structures
of these three compounds were investigated by single crystal
X-ray diffraction analysis, IR, 1H and 31P{1H} NMR spectroscopy,
MALDI TOF spectrometry, and microanalysis. Concerning the for-
mation of the p-bromothiobenzamine ligand LI we assumed the a
priori formation of a binuclear complex (Eq. (1)) and the subse-
quent hydrolysis of the bridging ligand L (Eq. (2)).
Both the structure of a similar binuclear complex [Cd2Z4]
(HZ = PhC(S)NHP(O)(OiPr)2) (Chart 1) and its tendency to form
monomers in solution was reported earlier [10].
Since usually ligands of this type are rather stable towards
hydrolysis we got interested under which conditions this reaction
can be observed or inhibited. To this end we carried out the reac-
tion of deprotonated HL towards Zn(II) and Cd(II) in dried and dis-
tilled EtOH and alternatively in H2O, on which we want to report in
this contribution.
2. Experimental
2.1. Physical measurements
2 CdðCH3COOÞ2 þ 4 KL ! ½Cd2L4ꢀ þ 4 CH3COOK
½Cd2L4ꢀ $ 2½CdL2ꢀ
ð1aÞ
ð1bÞ
NMR spectra in CDCl3 were obtained on a Bruker Avance
300 MHz spectrometer at 25 °C. 1H and 31P{1H} NMR spectra were
recorded at 299.948 and 121.420 MHz, respectively. Chemical
shifts are reported with reference to SiMe4 (1H) and 85% H3PO4
* Corresponding authors. Tel.: +78 432315397.
E-mail addresses: damir.safin@ksu.ru (D.A. Safin), axel.klein@uni-koeln.de
(A. Klein).
(
31P{1H}). Elemental analyses were performed on a CHNS HEKA-
tech EuroEA 3000 analyzer.
0277-5387/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.poly.2010.02.031

1838
D.A. Safin et al. / Polyhedron 29 (2010) 1837–1841
synthesis in distilled EtOH; 0.292 g (71%) for the synthesis in
H2O. M.p. 142–143 °C. (Anal. Calc. for C26H36Br2N2O6P2S2Zn: C,
N
O
O
P
37.90; H, 4.40; N, 3.40. Found: C, 37.84; H, 4.42; N, 3.34%). mmax
/
cmꢁ1 1546 (SCN), 1150 (P@O), 1001 (POC). dH 8.16 (4H, m, 3J 8.3,
o-H, C6H4), 7.51 (4H, m, 3J 8.3, m-H, C6H4), 4.73 (4H, d sept, 3J
6.1, OCH), 1.36 (12H, d, 3J 5.7, CH3), 1.34 (12H, d, 3J 5.8, CH3). dP 6.8.
N
S
O
S
P
Cd
S
O
O
O
O
O
O
Cd
P
O
P
S
N
O
2.3.2. [Cd(LI)L2] and [Cd(HL)2L2]
O
N
A suspension of HL (0.38 g, 1 mmol) in distilled EtOH or H2O
(10 mL) was admixed to a solution of potassium hydroxide
(0.056 g, 1 mmol) in distilled EtOH or H2O. To this mixture a solu-
tion of Cd(CH3COO)2 (0.115 g, 0.5 mmol) in distilled EtOH or H2O
(10 mL) was added dropwise under vigorous stirring. The mixture
was stirred at room temperature for further 5 h and left overnight.
The complex was obtained by extraction of the reaction mixture
using several portions of dichloromethane and drying the collected
extracts using anhydrous MgSO4. After filtration the solvent was
removed in vacuo. The colorless precipitate was recrystallized from
a dichloromethane/n-hexane mixture.
Chart 1.
2.2. Crystal structure determination and refinement
X-ray diffraction data were collected on a Enraf Nonius Kappa
CCD diffractometer equipped with a rotating anode generator
and with Cu K
a radiation. The structures were solved by direct
methods using the SHELXS [24] program and refined first isotropi-
cally and then anisotropically using SHELXL97 [24]. Hydrogen atoms
were added to the structure model on calculated positions and
were refined as rigid atoms. H atoms bonded to N were freely re-
fined. The figure was generated using the program MERCURY [25].
2.3.3. [Cd(LI)L2]
Yield 0.315 g (87%). M.p. 129–130 °C. (Anal. Calc. for
C33H42Br3CdN3O6P2S3: C, 36.46; H, 3.89; N, 3.87. Found: C, 36.38;
H, 3.95; N, 3.82%).
m
max/cmꢁ1 3283, 3108, 1666 (NH2), 1515
(SCN), 1154 (P@O), 1003 (POC). dH 9.63 (1H, br. s, NH2), 8.15 (4H,
m, 3J 8.4, o-H, C6H4, L), 7.69 (2H, m, 3J 8.3, o-H, C6H4), 7.57 (2H,
m, 3J 8.5, m-H, C6H4), 7.49 (4H, m, 3J 8.4, m-H, C6H4, L), 7.36 (1H,
s, NH2), 4.74 (4H, d sept, 3J 6.1 Hz, OCH), 1.33 (24H, d, 3J 6.2,
CH3). dP 4.4.
2.3. Synthesis
2.3.1. [ZnL2]
A suspension of HL (0.38 g, 1 mmol) in distilled EtOH or H2O
(10 mL) was admixed to a solution of potassium hydroxide
(0.056 g, 1 mmol) in distilled EtOH or H2O. To this mixture a solu-
tion of ZnCl2 (0.068 g, 0.5 mmol) in distilled EtOH or H2O (10 mL)
was added dropwise under vigorous stirring. The mixture was stir-
red at room temperature for further 5 h and left overnight. The
complex was obtained by extraction of the reaction mixture using
several portions of dichloromethane and drying of the combined
extracts using anhydrous MgSO4. After filtration the solvent was
removed in vacuo. The colorless precipitate was recrystallized from
a dichloromethane/n-hexane mixture. Yield: 0.395 g (96%) for the
2.3.4. [Cd(HL)2L2]
Yield 0.334 g (82%). M.p. 114 °C. (Anal. Calc. for
C52H74Br4CdN4O12P4S4: C, 38.28; H, 4.57; N, 3.43. Found: C,
38.39; H, 4.50; N, 3.48%). m
max/cmꢁ1 3164 (NH), 1531 (SCN), 1249
(P@O, HL), 1158 (P@O, L), 1006 (POC). dH 9.03 (2H, d, 3J 10.0,
NH), 8.15 (4H, m, 3J 7.7, o-H, C6H4, L), 7.79 (4H, m, 3J 8.2, o-H,
C6H4, HL), 7.50 (4H, m, 3J 7.9, m-H, C6H4, L), 7.49 (4H, m, 3J 8.0,
m-H, C6H4, HL), 4.79 (4H, d sept, 3J 6.2, OCH, HL), 4.63 (4H, d sept,
Br
O
N
Zn
N
O
O
N
P
O
Br
P
Br
H
N
O
1) + KOH
2) + 1/2 Cd(II)
1) + KOH
2) + 1/2 Zn(II)
S
O
S
O
O
Br
S
Cd
O
P
NH2
EtOH
EtOH or H2O
O
P
S
S
O
S
O
O
N
P
HL
O
O
1) + KOH
2) + 1/2 Cd(II)
Br
Br
H2O
[ZnL2]
[Cd(LI)L2]
Br
S
O
O
N
P
H
Br
O
O
O
P
N
O
S
Cd
S
O
N
P
O
O
O
O
Br
H
P
S
N
O
Br
[Cd(HL)2L2]
Scheme 1.

D.A. Safin et al. / Polyhedron 29 (2010) 1837–1841
1839
Fig. 1. Thermal ellipsoid representation of the two independent molecules found in the crystal structure of [Cd(HL)2L2]ꢂ2/3H2O. Ellipsoids are drawn at the 30% probability
level. Hydrogen atoms except of NH and H2O were omitted for clarity.
3J 6.1, OCH, L), 1.33 (12H, d, 3J 6.1, CH3, L), 1.29 (36H, br s, CH3, L +
HL). dP 4.5 (2P, L), ꢁ5.5 (2P, HL).
any case, the high selectivity and yield is remarkable and deserves
further investigation.
In addition to the recently reported crystal structure of the
complex [Cd(HL)2L2] [23] we can report here on a second crys-
tallographic modification of this complex. Crystals of
[Cd(HL)2L2]ꢂ2/3H2O were obtained by slow evaporation of the
solvent from dichloromethane/n-hexane solution. The structure
of [Cd(HL)2L2]ꢂ2/3H2O is shown in Fig. 1, while the crystal and
structure refinement data are given in Table 1. Isostructural
complexes of Co(II) [8,17], Ni(II) [21] with HZ, and Cd(II), Ni(II)
with HL [9,10] have been synthesized and structurally character-
ized previously.
3. Results and discussion
The ligand HL was prepared as described previously [22,23].
The syntheses of the complexes were carried out as follows: the li-
gand was deprotonated in situ using KOH, followed by reaction
with salts of the corresponding metals in freshly dried and distilled
EtOH or alternatively in H2O. The compounds obtained are crystal-
line solids, which are soluble in most polar solvents. The structures
of these compounds were investigated by single crystal X-ray dif-
fraction analysis, IR, 1H and 31P{1H} NMR spectroscopy and micro-
analysis. It was established that all three complexes are identical to
those described by us earlier [23]: [ZnL2], [Cd(LI)L2], and
[Cd(HL)2L2].
The corresponding reaction using ZnCl2 both in distilled EtOH or
H2O leads exclusively to the complex [ZnL2], in both cases. The
same reaction using anhydrous Cd(CH3COO)2 leads to the exclusive
formation of either [Cd(LI)L2] (in freshly dried and distilled EtOH)
or [Cd(HL)2L2] (in H2O) (Scheme 1). The corresponding synthesis
in aqueous EtOH leads to a mixture of the complexes [Cd(LI)L2]
and [Cd(HL)2L2] [23]. Interestingly, the formation of the similar
Cd(II) complex [Cd(HZ)2Z2], described by us earlier [10], does not
depend on the solvent. Reaction of the potassium salt KZ with
Cd(CH3COO)2 both in freshly dried and distilled EtOH or H2O leads
to the exclusive formation of [Cd(HZ)2Z2]. Thus, both the nature of
the solvent (EtOH or H2O) and the influence of the p-Br substituent
affect considerably the complex formation.
In addition to the selectivity of the reaction we were interested
to find evidence for the formation of the binuclear complex follow-
ing Eq. (1b). However, neither in the isolated products nor in the
reaction mixture we found evidence for such species (NMR). At
the moment we have no explanation for this failure, since from
the related ligand HZ such a binuclear complex was isolated and
characterized (Chart 1). The p-Br (electron-demanding) substituent
seems thus to be responsible for this very different behavior. Thus,
in further experiments we will investigate this phenomenon by
further variation of this substituent. Furthermore, we will explore
the reaction under strict anhydrous conditions, since it is clear, that
the water necessary for the hydrolysis of the ligand (in dried EtOH)
was formed in the deprotonation reaction HL + KOH ? H2O + KL. In
As the anhydrous polymorph [Cd(HL)2L2] the complex
ꢀ
[Cd(HL)2L2]ꢂ2/3H2O crystallizes in the triclinic space group P1.
The asymmetric unit of the complex contains one and a half
independent complex molecules as one (Cd2, Molecule B) sits
on a centre of inversion whilst another (Cd1, Molecule A) is in
general space. Additionally one water molecule is found in the
unit cell. In the structure of [Cd(HL)2L2] [23] one complex mol-
ecule is located in a special position at the symmetry centre. The
cell of the new polymorph is thus markedly larger. The overall
Table 1
Crystal data and data collection details for [Cd(HL)2L2].2/3H2O.
Formula
(C52H74Br4CdN4O12P4S4)3ꢂ2H2O
Formula weight
Crystal system
Space group
a (Å)
b (Å)
c (Å)
4929.96
triclinic
ꢀ
P1
12.7023(4)
12.8401(3)
34.8219(12)
98.860(2)
95.901(2)
101.500(2)
5446.5(3)
1
a
(°)
b (°)
c
(°)
V (Å3)
Z
Dcalc (g cm–3
T (K)
)
1.503
293(2)
F (0 0 0)
l
2486
7.346
(mmꢁ1
)
Reflections collected
Unique reflections
40 301
13 861
Observed reflections (Rint
R indices (all data)
)
6234 (0.0124)
R1 = 0.0736, wR2 = 0.1671

1840
D.A. Safin et al. / Polyhedron 29 (2010) 1837–1841
Table 2
Table 3
Selected bond lengths (Å) and bond angles (°) for [Cd(HL)2L2].2/3H2O.
Hydrogen bond lengths (Å) and angles (°) for [Cd(HL)2L2]ꢂ2/3H2O.
Molecule A
Molecule B
D–Hꢂ ꢂ ꢂA
d(D–H)
d(Hꢂ ꢂ ꢂA)
d(Dꢂ ꢂ ꢂA)
\(DHA)
Bond lengths
Cd(1)–S(1)
Cd(1)–S(2)
Cd(1)–O(1)
Cd(1)–O(2)
Cd(1)–O(3)
Cd(1)–O(4)
S(1)–C(1)
S(2)–C(2)
S(3)–C(3)
S(4)–C(4)
P(1)–O(1)
P(1)–N(1)
P(2)–O(2)
P(2)–N(2)
P(3)–O(3)
P(3)–N(3)
P(4)–O(4)
P(4)–N(4)
N(1)–C(1)
N(2)–C(2)
N(3)–C(3)
N(4)–C(4)
N(3)–H(1)ꢂ ꢂ ꢂO(1)
0.86
0.86
0.86
2.21
2.03
2.10
3.022(9)
2.841(9)
2.923(9)
158
156
159
2.555(3)
2.571(3)
2.332(5)
2.286(5)
2.391(6)
2.405(6)
1.719(9)
1.736(10)
1.611(10)
1.619(11)
1.470(6)
1.595(8)
1.475(6)
1.569(9)
1.462(7)
1.660(8)
1.470(7)
1.660(8)
1.314(12)
1.326(13)
1.355(12)
1.372(13)
Cd(2)–S(5)
Cd(2)–O(13)
Cd(2)–O(14)
S(5)–C(53)
S(6)–C(54)
P(5)–O(13)
P(5)–N(9)
P(6)–O(14)
P(6)–N(10)
N(9)–C(53)
N(10)–C(54)
2.566(2)
2.291(5)
2.391(5)
1.731(10)
1.627(10)
1.461(6)
1.569(8)
1.471(6)
1.660(7)
1.301(12)
1.357(12)
N(4)–H(2)ꢂ ꢂ ꢂO(2)
N(10)–H(75)ꢂ ꢂ ꢂO(13)#1
Symmetry transformations used to generate equivalent atoms: #1 ꢁ1 ꢁ x, ꢁ2 ꢁ y,
ꢁz.
4. Conclusions
In summary, the reaction of the in situ deprotonated thiobenza-
mide ligand p-BrC6H4C(S)NHP(O)(OiPr)2 (HL) with ZnCl2 both in
freshly dried and distilled EtOH or H2O leads exclusively to the for-
mation of the complex [ZnL2], while the same reaction with anhy-
drous Cd(CH3COO)2 leads to the exclusive formation of either
[Cd(LI)L2] in dried and distilled EtOH, or [Cd(HL)2L2] in H2O. The
corresponding synthesis in aqueous EtOH gave a mixture of the
complexes [Cd(LI)L2] and [Cd(HL)2L2]. The water for the hydrolysis
of the ligand, allowing the formation of [Cd(LI)L2], comes from the
deprotonation reaction. It was assumed that a binuclear complex
[Cd2L4] is responsible for the activation of the normally hydroly-
sis-stable ligand, but no evidence could be found for such a com-
plex (which has previously been observed for the phenyl
derivative C6H5C(S)NHP(O)(OiPr)2). Further investigations will
thus be necessary to elucidate this interesting hydrolysis reaction,
one approach will be the variation of the para substituent, which
might play an important role.
Bond angles
S(1)–Cd(1)–S(2)
S(1)–Cd(1)–O(1)
S(1)–Cd(1)–O(2)
S(1)–Cd(1)–O(3)
S(1)–Cd(1)–O(4)
S(2)–Cd(1)–O(1)
S(2)–Cd(1)–O(2)
S(2)–Cd(1)–O(3)
S(2)–Cd(1)–O(4)
O(1)–Cd(1)–O(2)
O(1)–Cd(1)–O(3)
O(1)–Cd(1)–O(4)
O(2)–Cd(1)–O(3)
O(2)–Cd(1)–O(4)
O(3)–Cd(1)–O(4)
Cd(1)–S(1)–C(1)
Cd(1)–S(2)–C(2)
O(1)–P(1)–N(1)
O(2)–P(2)–N(2)
O(3)–P(3)–N(3)
O(4)–P(4)–N(4)
Cd(1)–O(1)–P(1)
Cd(1)–O(2)–P(2)
Cd(1)–O(3)–P(3)
Cd(1)–O(4)–P(4)
P(1)–N(1)–C(1)
P(2)–N(2)–C(2)
P(3)–N(3)–C(3)
P(4)–N(4)–C(4)
S(1)–C(1)–N(1)
S(2)–C(2)–N(2)
S(3)–C(3)–N(3)
S(4)–C(4)–N(4)
178.20(9)
87.97(14)
92.12(14)
90.01(14)
89.99(16)
91.19(14)
88.79(14)
88.43(14)
91.61(16)
177.46(19)
92.85(19)
89.7(2)
89.69(19)
87.75(19)
177.4(2)
107.4(3)
110.2(3)
122.9(4)
122.1(4)
108.2(4)
107.5(4)
122.5(3)
128.1(3)
136.4(3)
136.2(3)
132.7(6)
134.4(6)
129.9(7)
130.8(7)
128.8(7)
128.4(7)
124.7(7)
125.2(8)
S(5)–Cd(2)–O(13)
S(5)–Cd(2)–O(14)
S(5)–Cd(2)–S(5)#1
S(5)–Cd(2)–O(13)#1
S(5)–Cd(2)–O(14)#1
O(13)–Cd(2)–O(14)
O(13)–Cd(2)–S(5)#1
O(13)–Cd(2)–O(13)#1
O(13)–Cd(2)–O(14)#1
O(14)–Cd(2)–S(5)#1
O(14)–Cd(2)–O(13)#1
O(14)–Cd(2)–O(14)#1
S(5)#1–Cd(2)–O(14)#1
S(5)#1–Cd(2)–O(14)#1
O(13)#1–Cd(2)–O(14)#1
Cd(2)–S(5)–C(53)
O(13)–P(5)–N(9)
88.75(14)
90.14(14)
180.00
91.25(14)
89.87(14)
89.38(19)
91.25(14)
180.00
90.62(19)
89.87(14)
90.62(19)
180.00
88.75(14)
90.14(14)
89.38(19)
109.4(3)
121.9(4)
107.3(3)
127.0(3)
137.2(3)
135.3(6)
130.5(6)
127.5(7)
124.4(7)
5. Supplementary data
CCDC 766627 ([Cd(HL)2L2]ꢂ2/3H2O) contains the supplemen-
tary crystallographic data. These data can be obtained free of
charge via http://www.ccdc.cam.ac.uk/conts/retrieving.html, or
from the Cambridge Crystallographic Data Centre, 12 Union Road,
Cambridge CB2 1EZ, UK; fax: (+44) 1223-336-033; or email:
deposit@ccdc.cam.ac.uk.
O(14)–P(6)–N(10)
Cd(2)–O(13)–P(5)
Cd(2)–O(14)–P(6)
P(5)–N(9)–C(53)
P(6)–N(10)–C(54)
S(5)–C(53)–N(9)
Acknowledgments
S(6)–C(54)–N(10)
This work was supported by the Russian Science Support Foun-
dation. DAS and MGB thank DAAD for the scholarships (Fors-
chungsstipendien 2008/2009).
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quality of both crystal structures makes us confident, that both
structure solution and refinement are correct for the new poly-
morph. The bonding parameters are essentially the same for
the two independent molecules in [Cd(HL)2L2]ꢂ2/3H2O (Table 2)
and also very similar to the recently reported values in
[Cd(HL)2L2] [23]. There are two intramolecular N–Hꢂ ꢂ ꢂO bonds
between the oxygen atom of the P@O group of the anionic li-
gand L and the hydrogen atom of the NH fragment of the neutral
ligand HL in the crystal of [Cd(HL)2L2]ꢂ2/3H2O. The hydrogen
bond parameters are shown in Table 3. Similar hydrogen bond-
ing was also observed in the crystal of the anhydrous polymorph
[Cd(HL)2L2] [23].

D.A. Safin et al. / Polyhedron 29 (2010) 1837–1841
1841
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