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Acta Cryst.(2002). E58, o1177±o1179 DOI: 10.1107/S160053680201766X B. Sridharet al. C6H14N3O3+ClO4ÿ

o1177

organic papers

Acta Crystallographica Section E

Structure Reports Online

ISSN 1600-5368

L

-Citrullinium perchlorate

B. Sridhar,aN. Srinivasan,b

Bjoern Dalhuscand R. K.

Rajarama*

aDepartment of Physics, Madurai Kamaraj

University, Madurai 625 021, India,

bDepartment of Physics, Thiagarajar College,

Madurai 625 009, India, andcDepartment

of Chemistry, University of Oslo, Blindern, N-0315 Oslo, Norway

Correspondence e-mail: sshiya@yahoo.com

Key indicators

Single-crystal X-ray study

T= 105 K

Mean(C±C) = 0.001 AÊ

Rfactor = 0.023

wRfactor = 0.068

Data-to-parameter ratio = 32.1

For details of how these key indicators were automatically derived from the article, see http://journals.iucr.org/e.

#2002 International Union of Crystallography Printed in Great Britain ± all rights reserved

In the title compound, C6H14N3O3+ClO4ÿ, the citrullinium

residue forms a strong OÐH O hydrogen bonds with the

terminal O atom of a symmetry-related residue. This residue

has a gauche I-trans-trans-trans conformation. The crystal

structure is stabilized by an NÐH O hydrogen-bonding

network. The perchlorate anion is linked to the cation,

forming chains along theaaxis.

Comment

Citrulline amino acid is found in the urea cycle. The crystal

structures of l-citrulline hydrochloride (Naganathan &

Venkatesan, 1971), l-citrulline hydrochloride and l

-homo-citrulline hydrochloride (Ashidaet al., 1972), andl-citrulline (Toffoliet al., 1987) have been reported. In the present study,

the crystal structure determination of l-citrullinium

perchlorate, (I), was undertaken.

The asymmetric unit of the unit cell of (I) contains a citrullinium cation and a perchlorate anion (Fig. 1). The unsymmetrical CÐO bond distances [1.2189 (8) and 1.3151 (8) AÊ] and the OÐCÐC bond angles [122.81 (6) and

111.63 (5)] clearly con®rm the protonation of the carboxyl

group. Generally, the citrulline residue has three planar

groups,viz. the carboxyl group, the aliphatic group and the

carbamylamino group or urea unit (Naganathan &

Venka-tesan, 1971). The backbone conformation angle 1 (O1AÐ

C1ÐC2ÐN1) indicates a cis conformation [7.52 (8)]. The

deviation of the -amino N atom from the mean carboxyl

plane is 0.149 (1) AÊ. This tendency to twist about the CÐN

bond is found in various amino acids (Lakshminarayananet

al., 1967). The straight-chain conformation angle 1 (N1Ð

C2ÐC3ÐC4) is gauche I [68.99 (7)], while 2 (C2ÐC3Ð

C4ÐC5) istrans[ÿ177.39 (6)]. The other two conformation

angles3(C3ÐC4ÐC5ÐN2) and4(C4ÐC5ÐN2ÐC6) are

also bothtrans[ÿ179.47 (6) and 162.71 (7)]. The

conforma-tion angles51(C5ÐN2ÐC6ÐO1C) and52(C5ÐN2ÐC6Ð

N3) are 174.51 (6) andÿ4.72 (11), respectively. The aliphatic

chain has a fully extended planar conformation (Table 1). The average ClÐO bond distances and OÐClÐO bond

angles are 1.4450 (6) and 109.47 (4), respectively, con®rming

a nearly tetrahedral symmetry. The perchlorate anion plays a vital role in hydrogen bonding, stabilizing the crystal structure.

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o1178

B. Sridharet al. C6H14N3O3+ClO4ÿ Acta Cryst.(2002). E58, o1177±o1179

The carboxyl O atom of the citrulline residue forms a strong

OÐH O hydrogen bond with the terminal O atom of a

symmetry-related residue (Table 2). The-,"- and-N atoms

(N1, N2 and N3) of the citrullinium residue form NÐH O

hydrogen bonds with the O atoms of the perchlorate anion. In

addition, the -N atom forms an intermolecular NÐH O

hydrogen bond with the terminal O atoms (Fig. 3). A class I hydrogen-bonding pattern is observed in the present structure,

having three two-center hydrogen bonds (Jeffrey & Saenger, 1991). Atom O4 of the perchlorate anion links the citrullinium

residues through NÐH O hydrogen bonds in a chain

running along the a axis [O4i H1AÐN1ÐH1B O4ii;

symmetry codes: (i)ÿx+ 2,y+1

2,ÿz+12; (ii)ÿx+ 1,y+ 1/2,

ÿz+1

2]. The citrullinium residues are packed as corrugated

sheets in theabplane, interconnected by OÐH O hydrogen

bonding (Fig. 3).

Experimental

The title compound was crystallized by slow evaporation from an aqueous solution of citrulline and perchloric acid in a stoichiometric ratio of 1:1.

Crystal data

C6H14N3O3+ClO4ÿ

Mr= 275.65

Orthorhombic,P212121

a= 5.1113 (1) AÊ b= 11.3497 (2) AÊ c= 19.3853 (3) AÊ V= 1124.57 (3) AÊ3

Z= 4

Dx= 1.628 Mg mÿ3

Dm= 1.615 Mg mÿ3

Dmmeasured by ¯otation in a

mixture of carbon tetrachloride and xylene

MoKradiation Cell parameters from 7473

re¯ections

= 2.1±37.5 = 0.37 mmÿ1

T= 105 (2) K Block, colorless 0.700.450.30 mm

Data collection

Bruker SMART CCD diffractometer

!scans

Absorption correction: multi-scan

(SADABS; Sheldrick, 1996)

Tmin= 0.77,Tmax= 0.89

25355 measured re¯ections

5866 independent re¯ections 5757 re¯ections withI> 2(I) Rint= 0.017

max= 37.5

h=ÿ8!8

k=ÿ19!18

l=ÿ32!33

Re®nement

Re®nement onF2

R[F2> 2(F2)] = 0.023

wR(F2) = 0.068

S= 1.03 5866 re¯ections 183 parameters

H atoms treated by a mixture of independent and constrained re®nement

w= 1/[2(F

o2) + (0.047P)2

+ 0.1185P]

whereP= (Fo2+ 2Fc2)/3

(/)max= 0.001

max= 0.44 e AÊÿ3

min=ÿ0.48 e AÊÿ3

Extinction correction:SHELXL97 Extinction coef®cient: 0.0217 (18) Absolute structure: Flack (1983) Flack parameter = 0.02 (3)

Figure 3

Packing diagram of (I), viewed down the c axis. H atoms have been omitted for clarity.

Figure 2

Packing diagram of (I), viewed down theaaxis.

Figure 1

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Table 1

Selected geometric parameters (AÊ,).

O1AÐC1 1.2189 (8) O1BÐC1 1.3151 (8)

O1AÐC1ÐC2ÐN1 7.52 (8)

N1ÐC2ÐC3ÐC4 68.99 (7)

C2ÐC3ÐC4ÐC5 ÿ177.39 (6) C3ÐC4ÐC5ÐN2 ÿ179.47 (6)

C4ÐC5ÐN2ÐC6 162.71 (7)

C5ÐN2ÐC6ÐO1C 174.51 (6) C5ÐN2ÐC6ÐN3 ÿ4.72 (11)

Table 2

Hydrogen-bonding geometry (AÊ,).

DÐH A DÐH H A D A DÐH A

O1BÐH1 O1Ci 0.70 (2) 1.84 (2) 2.5292 (8) 171 (2)

N1ÐH1A O4ii 0.846 (17) 2.234 (17) 3.0226 (9) 155 (2)

N1ÐH1B O4iii 0.926 (15) 2.162 (16) 2.9699 (9) 145 (1)

N1ÐH1C O1Civ 0.918 (15) 1.906 (15) 2.7986 (8) 164 (1)

N2ÐH2A O2iii 0.837 (15) 2.583 (14) 3.3562 (10) 154 (1)

N3ÐH3C O4v 0.818 (13) 2.326 (13) 3.1205 (8) 164 (1)

N3ÐH3D O1vi 0.833 (15) 2.292 (15) 3.0952 (9) 162 (1)

Symmetry codes: (i) 1ÿx;yÿ1

2;12ÿz; (ii) 2ÿx;12‡y;21ÿz; (iii) 1ÿx;12‡y;12ÿz; (iv) 3

2ÿx;2ÿy;zÿ12; (v)12‡x;32ÿy;1ÿz; (vi) 1‡x;y;z.

All H atoms were located from a difference Fourier map. Those on the N and O atoms were re®ned freely, but the remainder were placed in idealized positions and were re®ned as riding on their parent atoms. 2484 Fridel pairs were measured and used.

Data collection:SMART(Bruker, 1998); cell re®nement:SAINT

(Bruker, 1998); data reduction: SAINT; program(s) used to solve

structure: SHELXS97 (Sheldrick, 1997); program(s) used to re®ne structure: SHELXL97 (Sheldrick, 1997); molecular graphics:

PLATON (Spek, 1999); software used to prepare material for publication:SHELXL97.

BS thanks the Council of Scienti®c & Industrial Research (CSIR), Government of India, for ®nancial assistance and RKR thanks the Department of Science and Technology (DST), Government of India, for ®nancial support. Financial support from UGC is acknowledged.

References

Ashida, T., Funakoshi, K., Tsukihara, T., Ueki, T. & Kakudo, M. (1972).Acta Cryst.B28, 1367±1374.

Bruker (1998).SMARTandSAINT. Bruker AXS Inc., Madison, Wisconsin, USA.

Flack, H. D. (1983).Acta Cryst.A39, 876±881.

Jeffrey, G. A. & Saenger, W. (1991). Hydrogen Bonding in Biological

Structures. Berlin, Heidelberg, New York: Springer-Verlag.

Johnson, C. K. (1976).ORTEPII. Report ORNL-5138. Oak Ridge National Laboratory, Tennessee, USA.

Lakshminarayanan, A. V., Sashisekaran, V. & Ramachandran, G. N. (1967). In

Conformation of Biopolymers, edited by G. N. Ramachandran. London:

Academic Press.

Naganathan, P. S. & Venkatesan, K. (1971).Acta Cryst.B27, 1079±1085. Sheldrick, G. M. (1996).SADABS. University of GoÈttingen, Germany. Sheldrick, G. M. (1997). SHELXL97 and SHELXS97. University of

GoÈttingen, Germany.

Spek, A. L. (1999). PLATON for Windows. Utrecht University, The Netherlands.

Toffoli, P., Khodadad, P., Rodier, N. & Astoin, J. (1987).Acta Cryst.C43, 945± 947.

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Acta Cryst. (2002). E58, o1177–o1179

supporting information

Acta Cryst. (2002). E58, o1177–o1179 [doi:10.1107/S160053680201766X]

L

-Citrullinium perchlorate

B. Sridhar, N. Srinivasan, Bjoern Dalhus and R. K. Rajaram

S1. Comment

Citrulline amino acid is found in the urea cycle. The crystal structures of L-citrulline hydrochloride (Naganathan &

Ventatesan, 1971), L-citrulline hydrochloride and L-homocitrulline hydrochloride (Ashida et al., 1972), and L-citrulline

(Toffoli et al., 1987) have been reported. In the present study, the crystal structure determination of L-citrullinium

perchlorate, (I), was undertaken.

The asymmetric part of the unit cell of (I) contains a citrullinium residue and a perchlorate anion (Fig. 1). The

unsymmetrical C—O bond distances [1.2189 (8) and 1.3151 (8) Å] and the O—C—C bond angles [122.81 (6) and

111.63 (5)°], clearly confirm the protonation of the carboxyl group. Generally, the citrulline residue has three planar

groups, viz. the carboxyl group, the aliphatic group and the carbamylamino group or urea fraction (Naganathan &

Venkatesan, 1971). The backbone conformation angle ψ1 (O1A—C1—C2—N1) is in a cis conformation [7.52 (8)°]. The

deviation of the α-amino N atom from the mean carboxyl plane is 0.149 (1) Å. This tendency of twisting about the C—N

bond is found in various amino acids (Lakshminarayanana et al., 1967). The straight-chain conformation angle χ1 (N1—

C2—C3—C4) is gauche I [68.99 (7)°], while χ2 (C2—C3—C4—C5) is trans [−177.39 (6)°]. The other two conformation

angles χ3 (C3—C4—C5—N2) and χ4 (C4—C5—N2—C6) are also both trans [−179.47 (6) and 162.71 (7)°]. The

conformation angles χ51 (C5—N2—C6—O1C) and χ52 (C5—N2—C6—N3) are 174.51 (6) and −4.72 (11)°, respectively.

The aliphatic chain has a fully extended planar conformation (Table 1).

The average Cl—O bond distances and O—Cl—O bond angles are 1.4450 (6) and 109.47 (4)°, respectively, confirming

a nearly tetrahedral symmetry. The perchlorate anion plays a vital role in hydrogen bonding and stabilize the crystal

structure. The carboxyl O atom of the citrulline residue forms a strong O—H···O hydrogen bond with its

symmetry-related terminal O atom (Table 2). The α-, ε- and η-N atoms (N1, N2 and N3) of the citrullinium residue form N—H···O

hydrogen bonds with the O atoms of the perchlorate anion. In addition, the α-N atom forms an intermolecular N—H···O

hydrogen bond with the terminal O atoms (Fig. 3). A class-I hydrogen-bonding pattern is observed in the present

structure, having three two-centered hydrogen bonding (Jeffrey & Saenger, 1991). Atom O4 of the perchlorate anion links

the citrullinium residues through Nα—H···O hydrogen bonds in a chain running along the a axis [O4i···H1A—N1—

H1B···O4ii; symmetry codes: (i) −x + 2, y + 1/2, −z + 1/2; (ii) (-x + 1, y + 1/2, −z + 1/2]. The citrullinium residues are

packed as corrugated sheets in the ab plane, interconnected by O—H···O hydrogen bonding (Fig. 3).

S2. Experimental

The title compound was crystallized by slow evaporation from an aqueous solution of citrulline and perchloric acid in a

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Acta Cryst. (2002). E58, o1177–o1179

S3. Refinement

All H atoms were located from a difference Fourier map. Those on the N and O atoms were refined freely but the

remainder were placed in idealized positions and were refined as riding on their parent atoms. 2484 Fridel pairs were

[image:5.610.129.486.137.397.2]

observed.

Figure 1

The molecular structure of the title compound, showing the atom-numbering scheme and 50% probability displacement

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[image:6.610.168.444.67.512.2]

Acta Cryst. (2002). E58, o1177–o1179

Figure 2

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[image:7.610.127.484.71.265.2]

Acta Cryst. (2002). E58, o1177–o1179

Figure 3

Packing diagran of (I), viewed down the c axis. H atoms have been omitted for clarity.

L-citrullinium perchlorate

Crystal data

C6H14N3O3+·ClO4−

Mr = 275.65

Orthorhombic, P212121

a = 5.1113 (1) Å b = 11.3497 (2) Å c = 19.3853 (3) Å V = 1124.57 (3) Å3

Z = 4 F(000) = 576

Dx = 1.628 Mg m−3

Dm = 1.615 Mg m−3

Dm measured by flotation in a mixture of carbon

tetrachloride and xylene Mo radiation, λ = 0.71074 Å Cell parameters from 7473 reflections θ = 2.1–37.5°

µ = 0.37 mm−1

T = 105 K Block, colorless 0.70 × 0.45 × 0.30 mm

Data collection

Bruker SMART CCD diffractometer

Radiation source: fine-focus sealed tube Graphite monochromator

Detector resolution: 8.33 pixels mm-1

ω scans

Absorption correction: multi-scan (SADABS; Sheldrick, 1996) Tmin = 0.77, Tmax = 0.89

25355 measured reflections 5866 independent reflections 5757 reflections with I > 2σ(I) Rint = 0.017

θmax = 37.5°, θmin = 2.8°

h = −8→8 k = −19→18 l = −32→33

Refinement

Refinement on F2

Least-squares matrix: full R[F2 > 2σ(F2)] = 0.023

wR(F2) = 0.068

S = 1.03 5866 reflections 183 parameters 0 restraints

Primary atom site location: structure-invariant direct methods

Secondary atom site location: difference Fourier map

Hydrogen site location: inferred from neighbouring sites

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Acta Cryst. (2002). E58, o1177–o1179

w = 1/[σ2(F

o2) + (0.047P)2 + 0.1185P]

where P = (Fo2 + 2Fc2)/3

(Δ/σ)max = 0.001

Δρmax = 0.44 e Å−3

Δρmin = −0.48 e Å−3

Extinction correction: SHELXL97, Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4

Extinction coefficient: 0.0217 (18)

Absolute structure: Flack H D (1983), Acta Cryst. A39, 876-881

Absolute structure parameter: 0.02 (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 (Å2)

x y z Uiso*/Ueq

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Acta Cryst. (2002). E58, o1177–o1179

O1C 0.61697 (11) 1.04561 (5) 0.42560 (3) 0.01511 (9) N3 0.95763 (13) 0.91948 (5) 0.43683 (3) 0.01455 (9) H3C 0.928 (3) 0.9158 (11) 0.4782 (7) 0.015 (3)* H3D 1.061 (3) 0.8712 (12) 0.4194 (7) 0.017 (3)*

Atomic displacement parameters (Å2)

U11 U22 U33 U12 U13 U23

Cl 0.01302 (6) 0.01252 (6) 0.01715 (6) −0.00012 (4) −0.00106 (5) 0.00302 (5) O1 0.0203 (3) 0.0156 (2) 0.0431 (4) 0.0045 (2) −0.0010 (3) −0.0050 (2) O2 0.0398 (4) 0.0334 (3) 0.0162 (2) −0.0069 (3) −0.0066 (2) 0.0089 (2) O3 0.0118 (2) 0.0219 (2) 0.0421 (3) 0.00055 (18) 0.0028 (2) 0.0052 (3) O4 0.0177 (2) 0.01562 (19) 0.0169 (2) −0.00234 (17) −0.00082 (17) 0.00517 (16) O1A 0.01091 (19) 0.0185 (2) 0.01366 (19) −0.00150 (15) −0.00222 (14) 0.00144 (16) O1B 0.0187 (2) 0.01492 (19) 0.0186 (2) −0.00439 (18) −0.00520 (16) 0.00358 (17) C1 0.0104 (2) 0.0129 (2) 0.00953 (19) −0.00050 (17) 0.00038 (16) −0.00213 (17) C2 0.0096 (2) 0.0141 (2) 0.00929 (19) −0.00007 (17) −0.00008 (16) −0.00218 (16) N1 0.0143 (2) 0.0173 (2) 0.01111 (19) −0.00463 (18) −0.00152 (17) 0.00068 (16) C3 0.0110 (2) 0.0154 (2) 0.0099 (2) 0.00152 (18) −0.00155 (16) −0.00241 (17) C4 0.0144 (3) 0.0229 (3) 0.0092 (2) 0.0058 (2) −0.00150 (18) −0.00383 (19) C5 0.0130 (3) 0.0203 (3) 0.0095 (2) 0.0038 (2) −0.00060 (17) −0.00397 (18) N2 0.0146 (2) 0.0148 (2) 0.00829 (18) 0.00438 (18) 0.00008 (16) −0.00087 (15) C6 0.0119 (2) 0.0115 (2) 0.0095 (2) 0.00113 (17) 0.00010 (16) −0.00131 (16) O1C 0.0169 (2) 0.0173 (2) 0.01119 (19) 0.00716 (16) 0.00089 (15) −0.00252 (15) N3 0.0167 (2) 0.0169 (2) 0.01003 (19) 0.00580 (19) −0.00088 (18) −0.00009 (16)

Geometric parameters (Å, º)

Cl—O1 1.4327 (7) C3—H3A 0.9700 Cl—O2 1.4360 (7) C3—H3B 0.9700 Cl—O3 1.4407 (6) C4—C5 1.5135 (9) Cl—O4 1.4706 (5) C4—H4A 0.9700 O1A—C1 1.2189 (8) C4—H4B 0.9700 O1B—C1 1.3151 (8) C5—N2 1.4578 (9) O1B—H1 0.70 (2) C5—H5A 0.9700 C1—C2 1.5164 (9) C5—H5B 0.9700 C2—N1 1.4946 (9) N2—C6 1.3443 (8) C2—C3 1.5308 (8) N2—H2A 0.837 (15) C2—H2 0.9800 C6—O1C 1.2670 (8) N1—H1A 0.846 (17) C6—N3 1.3436 (9) N1—H1B 0.926 (15) N3—H3C 0.818 (13) N1—H1C 0.918 (15) N3—H3D 0.833 (15) C3—C4 1.5279 (9)

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Acta Cryst. (2002). E58, o1177–o1179

O2—Cl—O4 108.92 (4) C5—C4—H4A 109.2 O3—Cl—O4 107.82 (4) C3—C4—H4A 109.2 C1—O1B—H1 110.3 (17) C5—C4—H4B 109.2 O1A—C1—O1B 125.52 (6) C3—C4—H4B 109.2 O1A—C1—C2 122.81 (6) H4A—C4—H4B 107.9 O1B—C1—C2 111.63 (5) N2—C5—C4 109.21 (5) N1—C2—C1 108.25 (5) N2—C5—H5A 109.8 N1—C2—C3 111.02 (5) C4—C5—H5A 109.8 C1—C2—C3 114.19 (5) N2—C5—H5B 109.8 N1—C2—H2 107.7 C4—C5—H5B 109.8 C1—C2—H2 107.7 H5A—C5—H5B 108.3 C3—C2—H2 107.7 C6—N2—C5 125.34 (6) C2—N1—H1A 106.2 (12) C6—N2—H2A 116.0 (10) C2—N1—H1B 111.5 (9) C5—N2—H2A 118.4 (10) H1A—N1—H1B 110.4 (15) O1C—C6—N3 120.30 (6) C2—N1—H1C 111.4 (9) O1C—C6—N2 120.32 (6) H1A—N1—H1C 113.7 (15) N3—C6—N2 119.38 (6) H1B—N1—H1C 103.9 (13) C6—N3—H3C 117.6 (10) C4—C3—C2 113.42 (5) C6—N3—H3D 121.3 (9) C4—C3—H3A 108.9 H3C—N3—H3D 118.8 (13) C2—C3—H3A 108.9

O1A—C1—C2—N1 7.52 (8) C2—C3—C4—C5 −177.39 (6) O1B—C1—C2—N1 −174.58 (5) C3—C4—C5—N2 −179.47 (6) O1A—C1—C2—C3 131.73 (6) C4—C5—N2—C6 162.71 (7) O1B—C1—C2—C3 −50.37 (7) C5—N2—C6—O1C 174.51 (6) N1—C2—C3—C4 68.99 (7) C5—N2—C6—N3 −4.72 (11) C1—C2—C3—C4 −53.72 (8)

Hydrogen-bond geometry (Å, º)

D—H···A D—H H···A D···A D—H···A

O1B—H1···O1Ci 0.70 (2) 1.84 (2) 2.5292 (8) 171 (2)

N1—H1A···O4ii 0.846 (17) 2.234 (17) 3.0226 (9) 155 (2)

N1—H1B···O4iii 0.926 (15) 2.162 (16) 2.9699 (9) 145 (1)

N1—H1C···O1Civ 0.918 (15) 1.906 (15) 2.7986 (8) 164 (1)

N2—H2A···O2iii 0.837 (15) 2.583 (14) 3.3562 (10) 154 (1)

N3—H3C···O4v 0.818 (13) 2.326 (13) 3.1205 (8) 164 (1)

N3—H3D···O1vi 0.833 (15) 2.292 (15) 3.0952 (9) 162 (1)

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

Figure

Figure 1
Figure 2
Figure 3

References

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