Chemical Physics Letters 501 (2011) 351–357

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Chemical Physics Letters

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Crystal structure and electronic properties of three phenylpropionic acidderivatives: A combined X-ray powder diffraction and quantum mechanical study

Uday Das a,c, Basab Chattopadhyay b, Monika Mukherjee b, A.K. Mukherjee c,⇑a Department of Physics, Haldia Govt. College, Haldia, Purba Medinipur, WB, Indiab Department of Solid State Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, Indiac Department of Physics, Jadavpur University, Jadavpur, Kolkata 700032, India

a r t i c l e i n f o

Article history:Received 18 October 2010In final form 1 December 2010Available online 3 December 2010

0009-2614/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.cplett.2010.12.002

⇑ Corresponding author. Fax: +91 33 24146484.E-mail address: akm_ju@rediffmail.com (A.K. Muk

a b s t r a c t

Crystal structures of three derivatives of phenylpropionic acid, 2, 3 and 4 with hydroxyl, methyl andmethoxy substitutions at the 2, 4 positions have been determined from X-ray powder diffraction dataand their electronic structures were calculated at the DFT level. The optimized molecular geometriesagree closely to that obtained from the crystallographic analysis. Intermolecular O–H. . .O hydrogenbonds generate R2

2(8) rings, which are further connected through O–H. . .O and C–H. . .O hydrogen bondsinto two-dimensional framework in 2 and 4, and a step like architecture in 3. The HOMO–LUMO energygap (>4.0 eV) indicates a high kinetic stability of the three compounds.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Phenylpropionic acid (1) derivatives exhibit a strong bindingability to peroxidases, which catalyze the oxidation of a numberof organic and inorganic substrates [1]. Pharmaceutical composi-tions containing such compounds can inhibit the chemotactic acti-vation of neutrophils (PMN leukocytes) induced by the interactionof Interleukin-8 with CXCR1 and CXCR2 membrane receptors [2].The propionate metabolites are also useful in the treatment of neu-trophil-dependent pathologies such as psoriasis, ulcerative colitis,melanoma, chronic obstructive pulmonary disease, rheumatoidarthritis and idiopathic fibrosis [3–5]. In vivo and in vitro experi-ments have demonstrated that phenylpropionates and hydroxy-phenylpropionates can act as synergistic activators of the MhpRtranscriptional regulators from Escherichia coli [6]. In this respect,knowledge of the molecular as well as the electronic structuresof phenylpropionic acid derivatives is of key importance for a bet-ter understanding of their structure–activity correlation.

Although single crystal X-ray diffractometry is the most widelyused technique for atomic level structural characterization ofmolecular compounds, an intrinsic limitation of this approach isthe requirement to grow single crystals of appropriate size andquality that make them amenable to structure analysis. In suchcases, X-ray powder diffraction offers an alternative option forstructure elucidation. Unlike the single-crystal diffractometrywhich reveals crystal structure corresponding to a particular singlecrystal selected for data collection, X-ray powder diffraction pro-

ll rights reserved.

herjee).

vides structural information of the bulk material. With the recentdevelopments in the direct-space approaches for structure solution[7–9], ab initio crystal structure analysis of organic molecular solidscan now be accomplished from X-ray powder diffraction data.Crystal structures of several molecular compounds have beendetermined from laboratory X-ray powder data using the direct-space approaches [10–13].

The present letter reports the structural characterization ofthree phenylpropionic acid derivatives, 3-(2-hydroxy-4-methyl-phenyl)propionic acid (2), 3-(2-methoxyphenyl)propionic acid (3)and 3-(3-methoxyphenyl)propionic acid (4) using X-ray powderdiffraction data, along with the DFT calculation to study the molec-ular geometry and the electronic structure. An investigation ofclose intermolecular interactions in the molecules of 2–4 via Hirsh-feld surface analysis is also presented.

2. Materials and methods

2.1. Synthesis

A solution of 7-methyl-2H-chromene-2-one (1 g, 0.00625 mol)in 15 ml of EtOH was hydrogenated in presence of palladium char-coal at 60 psi in a Parr apparatus for 30 h. The reaction mixture wasfiltered and concentrated in vacuum to produce 3-(2-hydroxy-4-methylphenyl)propionic acid (2) as colorless crystalline powder.Yield 0.78 g (67%). Anal. Calc. for C10H12O3: C 66.65, H 6.71, O26.64%; Found C 67.1, H 6.5, O 26.4%. The compounds, 3-(2-methoxyphenyl)propionic acid (3) and 3-(4-methoxyphenyl)pro-pionic acid (4) were purchased from Aldrich, NY, USA (CAS Nos.6342-77-4 and 1929-29-9), and were used without furtherpurification.

352 U. Das et al. / Chemical Physics Letters 501 (2011) 351–357

2.2. X-ray structure analysis

X-ray powder diffraction data of 2–4 were recorded at ambienttemperature (20 �C) on a Bruker D8 Advance diffractometer oper-ating in the reflection mode and using CuKa radiation (k =1.5418 Å). The indexing of powder patterns of 2–4 carried outusing the program NTREOR [14] indicates monoclinic unit cellswith a = 22.909(3), b = 4.883(7), c = 8.237(4) Å and b = 99.13(4)�[M(20)=17, F(20) = 35 (0.011605, 50)] for 2, a = 7.803(5), b =19.596(9), c = 6.724(8) Å and b = 114.46(7)� [M(20) = 34, F(20) = 54(0.007507, 50] for 3, and a = 12.277(4), b = 7.686(3), c = 11.159(4) Å,and b = 114.26(6)� [M(20) = 19, F(20) = 31 (0.010247, 63)] for 4.Statistical analysis of powder patterns using the EXPO 2004 [15]indicated the most probable space groups as P21/n for 2 and 4,and P21/c for 3, which were used for structure solution. The fullpattern decomposition was performed in each case with EXPO2004 following the Le Bail algorithm and using a pearson VII peakprofile function. Good quality Le Bail fits, Rp = 0.0490 in 2,Rp = 0.0377 in 3 and Rp = 0.0404 in 4 indicated the correctness ofchoice of unit cells and the estimates of profile parameters for sub-sequent structure solution. The structures were solved by globaloptimization of structural models in direct-space based on a MonteCarlo search using the simulated annealing technique (in paralleltempering mode), as implemented in the program FOX [16]. Theinitial molecular geometry input in FOX was determined by theMOPAC 5.0 program [17].

The atomic coordinates obtained from FOX were used as thestarting model for the Rietveld refinement using the program GSASprogram package [18] with an EXPGUI [19] interface. The latticeparameters, background coefficients and profile parameters wererefined initially followed by the refinement of positional coordi-nates of all non-hydrogen atoms with soft constraints on bondlengths and bond angles, and planar restraints on the phenyl moi-eties. The background was described by the shifted Chebyshevfunction of first kind with 10 points regularly distributed overthe entire 2h range. A fixed isotropic displacement parameter of0.04 Å2 for all non-hydrogen atoms was maintained. Hydrogenatoms were placed in the calculated positions with a common Biso

value of 0.06 Å2. In the final stage of refinement, preferred orienta-tion correction was applied using the generalized spherical har-monic model, and the order of spherical harmonics necessary todescribe the preferred orientation was 12 in 2 (texture in-dex = 1.32), 8 in 3 (texture index 1.16), and 10 in 4 (texture index1.16). Relevant crystal data for 2–4 are summarized in Table 1.

Table 1Crystal data and structure refinement parameters of compounds 2–4.

Compound C10H12O3 (2) C10H12O3 (3) C10H12O3 (4)

Molecular weight 180.20 180.20 180.20Temperature (K) 293 293 293Crystal system Monoclinic Monoclinic MonoclinicSpace group P21/n P21/c P21/na (Å) 22.8913(14) 7.8243(4) 12.2392(5)b (Å) 4.8938(27) 19.6301(7) 7.6727(2)c (Å) 8.2326(6) 6.7344(4) 11.1229(2)b (�) 99.227(3) 114.385(4) 114.168(3)Volume (Å3) 910.33(14) 942.08(6) 952.97(3)Z 4 4 4Dcalc (g cm�3) 1.315 1.270 1.256Wavelength (Å) 1.5418 1.5418 1.54182h Interval (�) 5–100 5–100 5–100No. of parameters 64 55 58No. of data points 4624 4780 4780No. of restraints 51 47 46Rp 0.0527 0.0541 0.0476Rwp 0.0748 0.0780 0.0662R(F2) 0.1425 0.1009 0.0669v2 5.514 3.204 1.416

2.3. Computational details

Density functional theory (DFT) calculations have been carriedout in the solid state (periodic) for all three samples with theDMOL3 code [20] in the framework of a generalized-gradientapproximation (GGA) [21]. The geometry optimization was per-formed using the BLYP [22,23] density functional and a double nu-meric plus polarization (DNP) basis set. The atomic coordinateswere taken from the final X-ray refinement cycle. Geometry opti-mizations were carried out without any structural constraints.The net charges of atoms and dipoles, and the electronic structuresof isolated molecules of 2–4 were calculated at the BLYP level.

2.4. Hirshfeld surface analysis

Hirshfeld surfaces [24–26] and the associated 2D-fingerprintplots [27–29] were calculated using CrystalExplorer [30], which ac-cepts a structure input file in the CIF format. Bond lengths tohydrogen atoms were set to typical neutron values (C–H =1.083 Å). For each point on the Hirshfeld isosurface, two distancesde, the distance from the point to the nearest nucleus external tothe surface, and di, the distance to the nearest nucleus internal tothe surface, are defined. The normalized contact distance (dnorm)based on de and di is given by

dnorm ¼ðdi � rvdw

i Þrvdw

i

þ ðde � rvdwe Þ

rvdwe

;

where rvdwi and rvdw

e being the van der Waals radii of the atoms.The value of dnorm is negative or positive depending on intermolec-ular contacts being shorter or longer than the van der Waals sepa-rations. The parameter dnorm displays a surface with a red–white–blue color scheme, where bright red spots highlight shorter con-tacts, white areas represent contacts around the van der Waalsseparation, and blue regions are devoid of close contacts.

3. Results and discussion

3.1. Crystal and molecular structure

The compounds, 2–4, consist of terminal aromatic ring–n–ali-phatic string–carboxyl group sequence (Figure 1). The essential dif-ference among the compounds is the position of substituents in the3-phenylpropionic acid (1) skeleton (hydroxyl and methyl groupsat the 2 and 4 positions in 2, methoxy group at the 2 and 4 posi-tions in 3 and 4, respectively). The Rietveld plots and molecularviews with atom labelling scheme for 2–4 are shown in Figures 2and 3i. The molecules in 2–4 have fully extended propionic sidechain in a trans configuration; the torsion angles C6–C7–C8–C9 as-

Figure 1. Chemical diagrams of compounds 1–4.

Figure 2. Final Rietveld plots and molecular views of (i) C10H12O3 (2) and (ii) C10H12O3 (3) with atom numbering scheme. Red crosses: observed pattern, green curve:calculated pattern, magenta curve: difference curve. The intensity in the high-angle region has been multiplied by a factor of 5. (For interpretation of the references to colourin this figure legend, the reader is referred to the web version of this article.)

U. Das et al. / Chemical Physics Letters 501 (2011) 351–357 353

sumes a value of �166.8(2)�, �170.7(2)� and 175.4(2)� in 2, 3 and4, respectively. While the carboxyl group (C9, O11, O12) in 3 iscoplanar with the phenyl ring (C1–C6), they are almost perpendic-ular to each other in 4; the dihedral angles between the relevantleast-squares planes in 3 and 4 are 14.2(2)� and 85.9(2)�, respec-tively. Similar nearly parallel or virtually perpendicular orientationbetween the carboxyl acid group and the aromatic fragment hasbeen reported for analogous compounds, in which the terminalaromatic ring and the carboxyl group are linked via aliphaticchains of varying length [31–33]. In compound 2, however, the car-boxyl group is twisted about the C7–C8 bond with respect to thephenyl ring by 38.1(1)�.

The molecular-orbital optimization with DMol3 also corrobo-rates an almost trans configuration of aliphatic chain in 2–4, andthe overall molecular geometry in the compounds as establishedby the quantum mechanical calculations agrees well with that ob-tained from the X-ray structure analysis. The r.m.s. deviations be-tween the geometrically optimized bond lengths and bond angles

(Table 2) and the corresponding crystallographically determinedvalues are 0.02 Å, 1.6� in 2, 0.02 Å, 0.8� in 3 and 0.02 Å, 0.7� in 4.The differences between the molecular energies of the X-ray ana-lyzed and DFT analyzed structures of 2, 3, and 4 are 8.91, 7.98.and 10.93 eV, respectively.

The molecular packing in 2–4 exhibits intermolecular O–H. . .Ohydrogen bonds and C–H. . .O interactions (Supplementary Table1). The carboxylic acid group in 2–4 forms O–H. . .O hydrogenbonded dimer with O12. . .O11 distances of 2.617(4)–2.761(5) Åin an R2

2(8) graph-set motif [34,35], which links the molecules intopairs around the inversion centers in the unit cell. The adjacent di-mers in 2 are linked through intermolecular O13–H13. . .O13hydrogen bonds to form two dimensional corrugated sheets paral-lel to the (1 1 0) plane (Figure 4). Additional reinforcements withinthe two dimensional sheets in 2 are provided by C–H. . .O hydrogenbonds (Supplementary Table 1). The dimeric units in 3 are joinedthrough intermolecular C–H. . .O hydrogen bonds forming a steplike framework (Supplementary Figure 1). In 4, however, the inter-

Figure 3. (i) Final Rietveld plots and molecular view of C10H12O3 (4) with atom numbering scheme. Red crosses: observed pattern, green curve: calculated pattern, magentacurve: difference curve. The intensity in the high-angle region has been multiplied by a factor of 5. (ii) A view of two dimensional supramolecular architecture in C10H12O3(4)based on R2

2(8) and R66(44) synthons. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

354 U. Das et al. / Chemical Physics Letters 501 (2011) 351–357

connection of R22(8) rings via C–H. . .O hydrogen bonds generates a

two dimensional supramolecular architecture based on R22(8) and

R66(44) synthons (Figure 3ii).

The Hirshfeld surfaces of 2–4 are illustrated in Figure 5i, show-ing surfaces that have been mapped over dnorm range of �0.5 to1.5 Å. The dominant interactions between carboxylic O–H and Oatoms forming dimeric species in 2–4 can be seen in the Hirshfeldsurface plots as the bright red areas (marked a and b) in Figure 5i.The phenolic O13. . .O13 interactions in the Hirshfeld surface of 2are encircled in Figure 5i. The light red spots labelled as c and d

in Figure 5i are due to C. . .H contacts. Other visible spots in Figure5i correspond to H. . .H interactions. In the 2D fingerprint plots, Fig-ure 5ii, prominent pairs of sharp spikes of almost equal length inthe region 1.7 < de + di < 2.7 Å are characteristics of nearly equalO(donor). . .O(acceptor) distances and a cyclic hydrogen bondedR2

2(8) synthon [36]. The points in the (di, de) regions of (1.3,1.0 Å) and (1.0, 1.3 Å) in the fingerprint plot of 2 (Figure 5ii) aredue to O13–H13. . .O13 interactions. The upper spike a in Figure5ii corresponds to the donor spike (carboxylic H-atoms interactingwith O-atoms of the COOH groups), the lower spike b being an

Table 2Comparison of selected bond lengths (Å) and bond angles (�) for compounds 2–4 as determined from X-ray structure analysis and DFT calculations.

Bonds/angles C10H12O3 (2) C10H12O3 (3) C10H12O3 (4)

X-ray DFT X-ray DFT X-ray DFT

C5–O13 1.369(1) 1.405 1.372(2) 1.381C6–C7 1.510(1) 1.508 1.507(2) 1.517 1.510(1) 1.515C7–C8 1.541(1) 1.563 1.537(2) 1.536 1.531(1) 1.541C8–C9 1.521(1) 1.506 1.508(2) 1.510 1.501(1) 1.508C9–O12 1.296(2) 1.342 1.301(2) 1.338 1.300(1) 1.338C9–O11 1.200(1) 1.240 1.221(2) 1.243 1.219(1) 1.243C3–C10 1.511(1) 1.492O13–C10 1.393(2) 1.449 1.421(1) 1.446C4–C5–C6 120.5(2) 121.6 120.7(2) 120.8 121.2(1) 121.4C4–C5–O13 120.8(2) 120.1 125.1(2) 123.8C6–C5–O13 118.8(2) 118.1 114.2(2) 115.4C5–C6–C1 120.0(2) 116.5 118.5(2) 118.0 118.2(1) 117.8C5–C6–C7 119.9(2) 123.2 119.6(2) 118.6 120.7(1) 121.1C6–C7–C8 118.8(2) 116.3 114.5(1) 115.4 112.8(1) 112.7C7–C8–C9 113.3(2) 115.0 112.9(2) 114.9 113.2(1) 113.8C8–C9–O11 125.3(2) 123.3 122.7(2) 121.9 122.6(2) 123.6C8–C9–O12 115.4(2) 114.5 115.4(2) 115.1 113.9(1) 113.0O11–C9–O12 119.3(2) 122.3 121.8(2) 123.0 122.5(2) 123.4

Figure 4. Perspective view of crystal packing in C10H12O3 (2) along [0 1 0] direction.

U. Das et al. / Chemical Physics Letters 501 (2011) 351–357 355

acceptor spike (O-atoms from propionic acids interacting with theH atoms of –COOH groups). The wings in the region di = 1.0 Å tode = 1.6 Å and de = 1.0 Å to di = 1.6 Å, marked as c and d in Figure5ii, in the fingerprint plots correspond to C. . .H interactions in 2–4. The difference between the molecular interactions in 2–4 interms of H. . .H interactions is reflected in the distribution of scat-tered points in high (di, de) region of the fingerprint plots Figure 5ii,and the appearance of small wings around di = 1.3 to de = 2.0 Å anddi = 2.0 to de = 1.3 Å in 4 (marked with circles in Figure 5ii). The rel-ative contributions of different interactions to the Hirshfeld sur-faces in 2–4 were calculated (Supplementary Figure 2), whichindicated that the H. . .H interactions contributed about 50%(47.7% in 2, 52.9% in 3 and 50.1% in 4) to the Hirshfeld surface area,while the remaining 50% contribution were mostly due to O. . .H(31.1% in 2, 26.5% in 3 and 30.4% in 4) and C. . .H (19.4% in 2,16.4% in 3 and 15.3% in 4) interactions.

The hydrogen bonded R22(8) synthon energy in 2–4 was calcu-

lated with the program DMol3 at the BLYP level using the DNP ba-sis set. The synthons were built from the refined structures of 2–4with O–H bond lengths set to typical neutron value (O–H = 0.983 Å). The synthon energy(DE) was estimated as the differ-

ence between O–H. . .O bonded dimers in 2–4 and the sum of theisolated monomer energies, and the calculation only included theintermolecular O–H. . .O hydrogen bonds not the C–H. . .O interac-tions [37]. The essentially same DE values of �6.0, �6.1 and�7.0 kcal/mol for compounds 2, 3 and 4, respectively, reveal therobust nature of R2

2(8) synthon.

3.2. Electronic structure

The net charges of atoms and dipoles, and the molecular orbitalenergies of 2–4 calculated at the BLYP level (Supplementary Table2) indicate that all oxygen atoms in the molecules bear negativecharges, while the carbon atoms of the phenyl ring having substi-tutions (C3, C5, and C6 in 2, C5 and C6 in 3, C3 and C6 in 4), themethoxy group (C10 in 3 and 4) and the carboxylic group (C9) bearpositive charges. The remaining carbon atoms in the moleculesbear negative charges. The large electron densities at the oxygenatom (O13) of hydroxyl in 2 and methoxy (in 3 and 4) groups sug-gest possible protonation of the associated atoms. Due to thischarge redistribution, the dipole of the molecules of 2–4 becomes1.07, 1.40 and 0.97 a.u., respectively.

Figure 5. (i) Hirshfeld surfaces and (ii) 2D fingerprint plots of compounds 2–4. A colouring scheme red (contacts < vdW separations)-white (contacts �vdW separations)-blue(contacts > vdW separations) used in (i). Close contacts are divided into four regions: a is H. . .O, b is O. . .H, c is H. . .C and d is C. . .H. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.)

356 U. Das et al. / Chemical Physics Letters 501 (2011) 351–357

The HOMO (highest occupied molecular orbital) and LUMO(lowest unoccupied molecular orbital) wave functions in mole-cules of 2–4 exhibit some differences, and the charge densitiesfor the HOMO and LUMO (Figure 6 and Supplementary Figure 3)indicate very little charge accumulation on the hydrogen atoms.The HOMO in 2–4 is primarily localized on the oxygen atom of hy-droxyl (O13 in 2) and methoxy (O13 in 3 and 4) groups as well ason the C–C bonds of the phenyl ring (C2–C3 and C5–C6 in 2, C2–C3,C4–C5 and C5–C6 in 3, C1–C6, C2–C3 and C3–C4 in 4) showingbonding–antibonding patterns characteristic of phenyl ring sys-tems. The aliphatic side chain in 2–4 had hardly any HOMO popu-lation. The LUMO in the molecules is populated on the carboxylicgroup atoms (C9, O11 and O12) with only minor population local-ized on the carbon atoms of the phenyl ring (C1–C6) in 2–4.

Figure 6. Charge density of HOMO (a) and LUMO (b) orbitals in compounds 2 and 3calculated by DFT method [surfaces in yellow (+) and blue (�)]. (For interpretationof the references to colour in this figure legend, the reader is referred to the webversion of this article.)

The orbital energy level analysis at the BLYP level shows EHOMO

and ELUMO values of �5.25 and �1.21 eV for 2, �5.13 and �0.96 eVfor 3, and �5.08 and �1.04 eV for 4, respectively. The HOMO–LUMO energy separation has been used as an indicator of kineticstability of the molecule [12]. A large HOMO–LUMO gap impliesa high kinetic stability and low chemical reactivity, because it isenergetically unfavourable to add electrons to a high-lying LUMOor to extract electrons from a low-lying HOMO [38]. The HOMO–LUMO energy gaps in 2, 3 and 4 are 4.04, 4.17 and 4.04 eV, respec-tively. The differences between the orbital energies correspondingto HOMO-1 and HOMO-2 of 0.34 eV in 2, 0.19 eV in 3 and 0.16 eVin 4 are much larger than 0.05 eV, which indicate that the HOMO-1and HOMO-2 energy levels in 2–4 are non-degenerate. Similar con-clusion can be drawn from the LUMO + 1 and LUMO + 2 orbital en-ergy calculations in compounds 2–4.

4. Conclusion

The crystal and molecular structures of three propionic acidderivatives, 3-(2-hydroxy-4-methylphenyl)propionic acid (2),3-(2-methoxyphenyl)propionic acid (3) and 3-(3-methoxy-phenyl)propionic acid (4), have been solved using laboratoryX-ray powder diffraction data. The molecular geometry and theelectronic structures of 2–4 have been analysed by the DFT calcu-lations. The molecular conformations of the compounds as estab-lished by the X-ray analysis agree well with that obtained fromthe quantum mechanical calculations. A comparison of close inter-molecular interactions in compounds 2–4 using the Hirshfeld sur-face analysis revealed that about 50% of total contribution ofdifferent interactions to the Hirshfeld surface area can be attrib-uted to the H. . .H interactions, while the remaining 50% contribu-tion corresponds to the O. . .H and C. . .H interactions.Intermolecular O–H. . .O hydrogen bonds in 2–4 generate R2

2(8)synthons of nearly equal synthon energy, �6.0 to �7.0 eV. The

U. Das et al. / Chemical Physics Letters 501 (2011) 351–357 357

molecular packing in 2 and 3 exhibits a columnar structure builtwith O–H. . .O and C–H. . .O hydrogen bonds, whereas in 4, the com-bination of O–H. . .O and C–H. . .O generates a three dimensionalarchitecture based on two R2

2(8) rings and four C–H. . .O hydrogenbonds forming an R6

6(44) ring. The HOMO–LUMO energy separation(>4.0 eV) suggests high kinetic stability of 2–4.

Acknowledgements

Financial support from the University Grants Commission, NewDelhi, and The Department of Science and Technology, Govern-ment of India, New Delhi, through the DRS (SAP) and FIST pro-grams for purchasing the Bruker D8 Advance X-ray powderdiffractometer, is gratefully acknowledged. We thank Mr. DebayanSarkar, IACS, Kolkata, for helping in the synthesis of compound 2.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.cplett.2010.12.002.

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