A Preview of a Construction of a Crystal Lattice Based on Intermolecular Interactions

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Article A Preview of a Construction of a Crystal Lattice Based on Intermolecular Interactions

Vladimír Hejtmánek 1 , Martin Draˇcínský 2 and Jan Sýkora 1,*

1 Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojová 135, 165 02 Prague, Czech Republic; [email protected] 2 Institute of Organic Chemistry and Biochemistry, Czech Academy of Sciences, Flemingovo nám. 2, 166 10 Prague, Czech Republic; [email protected] * Correspondence: [email protected]; Tel.: +420-220-390-307

Received: 25 February 2019; Accepted: 14 March 2019; Published: 19 March 2019

Abstract: A general procedure of crystal packing reconstruction using a certain number of intermolecular interactions is introduced and demonstrated on the crystal structure of L-histidine·HCl·H2O. Geometric restrictions based on intermolecular interactions are used for formation of a molecular pair as a basic repetitive motif of the crystal packing. The geometric restrictions were applied gradually within a supervised procedure, narrowing the scope of possible arrangement of two adjacent molecules. Subsequently, a pair of histidine molecules was used for construction of a molecular chain. The chain formed contained translation information on histidine molecules in one dimension, which coincided with one of the cell parameters. Furthermore, the periodicity in the second and third dimensions can be accomplished by chain assembly into sheets (2D), and sheets can be arranged into the final 3D structure. For this purpose, the rest of the available intermolecular interactions could be used to control the mutual assembly of molecular chains and sheets. Complete molecular packing would enable derivation of standard crystallographic parameters that can be used for verification of the structural model obtained. However, the procedure described for construction of the whole 3D structure from molecular chains was not attempted, and is only briefly outlined here. The procedure described can be employed especially when standard crystallographic parameters are not available and traditional methods based on X-ray diffraction fail.

Keywords: NMR crystallography; intermolecular interactions; geometry restraints; crystal lattice

1. Introduction X-ray structure analysis has an irreplaceable role in the determination of crystal structures of organic compounds, as supported by almost a million entries in the Cambridge Structure Database (CSD) [1]. Regarding single crystal X-ray analysis, the final structure is obtained in several steps performed in a specific order [2]. The chosen single crystal is mounted in a diffractometer, and within a few minutes, several diffraction frames are acquired. Based on the reflection positions, unit cell parameters are calculated, revealing the crystal system. The accuracy of the cell parameters rises with the number of acquired reflections. Subsequently, a particular space group of the system is proposed. With a sufficient number of reflections, the unit cell parameters and the appropriate space group, the crystal structure can be solved. Nowadays, there are plenty of methods enabling structure solution. The original heavy atom methods [3] have been progressively substituted by direct methods [4]. The X-ray beam interacts with the electrons and diffracts on periodic electron density in a given direction. The initial structure solution always consists of a coarse skeleton of heavy atoms, as lighter atoms scatter weakly due to the low number of electrons. Light atoms have to be localized and refined at the very end of structure refinement. It has to be pointed out that even the positions of the lightest

Crystals 2019, 9, 159; doi:10.3390/cryst9030159 www.mdpi.com/journal/crystals Crystals 2019, 9, 159 2 of 11 atoms, the hydrogen atoms, can be localized from residual electron density maps using good-quality diffraction data. The position of the hydrogen atom indicates the location of its electron, which is pulled towards the bonding partner; therefore, it does not coincide with the position of the hydrogen nucleus. Having just one electron, the contribution of hydrogen atoms to the diffraction pattern is rather small. An alternative experimental technique for the characterization of solids is solid-state NMR spectroscopy (SS-NMR). NMR-based crystallography is a rapidly developing ﬁeld generating around one thousand research articles per year, according to Web of Science (WOS) [5]. In contrast to X-ray diffraction, hydrogen atoms are the most sensitive for NMR spectroscopy. Furthermore, SS-NMR does not require a long-range order in the studied materials, and is therefore suitable for characterization of disordered and amorphous samples [6–10]. X-ray diffraction and SS-NMR are thus complementary techniques in many respects. The progress of NMR crystallography in recent years can mostly be attributed to the development of fast and reliable computational techniques (mostly based on density functional theory) that enable the linking of experimentally observable parameters with structural models [11–13]. However, the main challenge of NMR crystallography is represented by direct structure elucidation. Although it was demonstrated that de novo crystal structure predictions are possible for NMR crystallography, all current procedures place high demands on computer time, because the NMR parameters of a large set of initially predicted crystal structures have to be calculated and compared with the experimental data [14,15]. Therefore, the majority of current NMR crystallography procedures use crystallographic parameters as an input, namely, cell lattice parameters and a space group, which are generally inaccessible by NMR methods. In this paper, the hypothetical possibility of reconstructing the molecular packing network and determining the crystallographic parameters using just NMR data is explored. Structural data of L-histidine·HCl·H2O are used for a case simulation, considering that cell parameters could not be determined from X-ray powder diffraction for some reason, which is quite common for monoclinic and triclinic systems. This particular system was chosen because the recent pioneering work of Nishiyama and coworkers has demonstrated that intermolecular distances can be obtained with SS-NMR experiments [16,17]. The newly developed approaches were demonstrated on a crystalline 1 sample of L-histidine·HCl·H2O. In addition to 10 intramolecular hydrogen interactions, the H DQ-SQ correlation spectrum collected in ref. [16] also revealed 7 intermolecular hydrogen-hydrogen interactions. Moreover, two intermolecular contacts between nitrogen and hydrogen atoms were observed in the 2D 14N(SQ)-1H(SQ) spectrum. Altogether, 9 intermolecular distances were described in ref. [16]. Considering that quantiﬁcation of intermolecular distances will be feasible experimentally in the future, the procedure for the derivation of standard crystallographic parameters from such data was attempted here. Solid state NMR can easily determine the number of non-equivalent molecules in the asymmetric unit (Z’)[18]. In accordance with the experiments, Z’ = 1 was used in this simulation. To focus on the intermolecular interactions and to simplify the structure prediction procedure, we did not intend to determine the conformation of the histidine molecule, and used its known conformation found in solid state.

2. Results and Discussion When examining the data found in ref. [16] closely, the most information on involvement in intermolecular interactions was obtained for the imidazole moiety of the histidine molecule (Table1). Three intermolecular hydrogen-hydrogen interactions were detected; namely, H5···H8’, H5···H7’ and H6···H8’ (for numbering see Figure1). Additionally, there were the two above-mentioned nitrogen-hydrogen interactions, H5···N7’ and N5···H7’. The rest of the intermolecular interactions described belong to the interaction between the imidazole moiety and the CH2 group (H8···H3) and 3 interactions of histidine with water molecules trapped in the crystal lattice (H1···w, H5···w, H8···w). Crystals 20192019,, 99,, 159x FOR PEER REVIEW 33 of of 11

Table 1. Intermolecular interactions detected in the crystal structure of L-histidine·HCl·H2O. Table 1. Intermolecular interactions detected in the crystal structure of L-histidine·HCl·H2O. Atoms Distance [Å] H3···H8’Atoms Distance2.90 [Å] H3H5···H7’···H8’ 3.15 2.90 H5H5···H8’···H7’ 3.25 3.15 H5···H8’ 3.25 H6···H8’ 3.03 H6···H8’ 3.03 H1···Hw 2.35 H1···Hw 2.35 H5H5···Hw···Hw 2.47 2.47 H8H8···Hw···Hw 3.09 3.09 H5H5···N7’···N7’ 3.44 3.44 N5N5···H7’···H7’ 3.84 3.84

Figure 1. Molecule of L-histidine with atom numbering according to Ref. [16]. Figure 1. Molecule of L-histidine with atom numbering according to Ref. [16]. Generally, for the construction of a 3D network of molecular packing, it is necessary to start with a singleGenerally, molecule offor proper the construction geometry. Toof a establish 3D network an initial of molecular molecule packing, of proper it geometry,is necessary intramolecular to start with a single molecule of proper geometry. To establish an initial molecule of proper geometry, distances can be used. In this work, the original coordinates found in L-histidine·HCl·H2O from neutronintramolecular diffraction distances are used can for be simplification used. In [th19is]. Subsequently,work, the original the geometry coordinates optimization found ofin theL- crystalhistidine·HCl·H structure2 wasO from performed neutron with diffraction the CASTEP are programused for [20simplification]. [19]. Subsequently, the geometryThe obtained optimization molecular of the geometrycrystal structure was used was for performed the creation with of the a molecularCASTEP program pair in the [20]. mutual arrangementThe obtained meeting molecular the geometric geometry restrictions was used based for on the available creation intermolecular of a molecular distances. pair in the The mutual initial molecule,arrangement labeled meeting as moleculethe geometric A in restrictions the following based text, on is available fixed in space.intermolecular The second distances. molecule The (moleculeinitial molecule, B) is thenlabeled moved as molecule into proximity A in the with following molecule text,A is andfixed arranged in space. in The 3D second space, molecule meeting the(molecule requirements B) is then given moved by the into intermolecular proximity with interactions molecule A listed and arranged above. To in define 3D space, the position meeting of the a singlerequirements atom of given molecule by the B intermolecular with respect to interactions molecule A,listed three above. independent To define distances the position between of a single this atom of and molecule three atoms B with in respect molecule to molecule A are necessary. A, three independent However, adistances set of three between distances this atom leads and to twothree possible atoms in solutions molecule related A are bynecessary. a plane However, of symmetry a set defined of three by distances the three leads atoms to two in molecule possible A.solutions When definingrelated by the a plane position of symmetry of a whole defined molecule by the B, athree second atoms set in of molecule three distances A. When is required.defining the position of a whole molecule B, a second set of three distances is required. For the case of L- For the case of L-histidine·HCl·H2O, three different distances were found, e.g., for hydrogen H5, namelyhistidine·HCl·Hd(H5···H7’)2O, three = 3.15 different Å, d(H5 distances···H8’) = 3.25were Å found,and d(H5 e.g.,··· forN7’) hydrogen = 3.44 Å H5, (Table namely1). Unfortunately, d(H5···H7’) = the3.15 threeÅ, d(H5···H8’) spheres defined = 3.25 byÅ and this d(H5···N7’) distance do = not3.44 intersect Å (Table in 1). a singleUnfortunately, point. A the similar three situation spheres wasdefined found by forthis atomdistance H8, do for not which intersect three in interactions a single point. were A alsosimilar given, situation namely was d(H3 found···H8’) for atom = 2.90 H8, Å, d(H5for which···H8’) three = 3.25 interactions Å and d(H6 were···H8’) also = given, 3.03 Å namely (Table1). d(H3···H8’) Obviously, = the 2.90 three Å, d(H5···H8’) distances originate = 3.25 Å from and interactionsd(H6···H8’) = with 3.03 at Å least(Table two 1). different Obviously, molecules the three in thedistances crystal originate lattice. Such from a interactions possibility always with at has least to betwo taken different into account.molecules Overall, in the crystal a specific lattice. position Such of a H5possibility or H8 toward always molecule has to be A taken cannot into be account. defined withOverall, certainty a specific using position the available of H5 data.or H8 toward molecule A cannot be defined with certainty using the availableTo narrow data. the space where molecule B can be located and to define its proper orientation, it was necessaryTo narrow to find the other space interactions where molecule that could B can bebe attributedlocated and to to a define single molecule.its proper orientation, Useful geometric it was necessary to find other interactions that could be attributed to a single molecule. Useful geometric restrictions were found in the imidazole moiety, namely a combination of three interatomic distances restrictions were found in the imidazole moiety, namely a combination of three interatomic distances between four nuclei N5–H5···H7’–N7’. The imidazole moiety is very convenient, as its geometry can between four nuclei N5 – H5···H7’ – N7’. The imidazole moiety is very convenient, as its geometry

Crystals 2019, 9, x FOR PEER REVIEW 4 of 11 CrystalsCrystals2019 2019, 9, 159, 9, x FOR PEER REVIEW 4 of4 11of 11 can be easily optimized, and the following steps can be also performed just with the imidazole fragmentcan be of easily the molecule. optimized, The andhydrogen the following atom H7’ steps of molecule can be Balso has performedto be placed just at a with distance the ofimidazole 3.15 Åbe from easilyfragment hydrogen optimized, of the H5. molecule. and This the constraintfollowing The hydrogen reduced steps atom can the be H7’possible also of performedmolecule space to B justahas sphere withto be thewithplaced imidazole a radiusat a distance of fragment 3.15 of Å 3.15 centeredof theÅ from molecule. on hydrogen H5. Simultaneously, The hydrogenH5. This constraint atom the second H7’ reduced of moleculeconstrai thent Bpossible keeps has to hydrogen bespace placed to a atH7’sphere a distanceat a with distance ofa radius 3.15 of Å3.44 of from 3.15 Å Å fromhydrogencentered nitrogen H5. on N5, This H5. reducing constraintSimultaneously, the reduced available the the secondspace possible from constrai space a spherent to keeps a sphereto a hydrogencircle with (Figure a radius H7’ 2). at of Eacha 3.15distance location Å centered of 3.44on Å α suchon H5.from a ring Simultaneously, nitrogen can be characterizedN5, reducing the second bythe constraintangle available defined keepsspace as hydrogenfrom H70 ’–H5–H7’a sphere H7’ at to angle, a a distance circle where (Figure of H7 3.440 ’2). Åis an fromEach arbitrarily nitrogenlocation on N5, reducing the available space from a sphere to a circle (Figure2). Each location on such a ring can chosensuch initial a ring position can be characterized of H7’. by angle α defined as H70’–H5–H7’ angle, where H70’ is an arbitrarily α be characterizedchosen initial byposition angle ofdeﬁned H7’. as H70’–H5–H7’ angle, where H70’ is an arbitrarily chosen initial position of H7’.

Figure 2. Definition of a sphere around H5 and its reduction to a circle. Figure 2. Definition of a sphere around H5 and its reduction to a circle. Figure 2. Definition of a sphere around H5 and its reduction to a circle. Having H7’ fixed at the circle around molecule A, nitrogen N7’ has to be placed at a distance of Having H7’ fixed at the circle around molecule A, nitrogen N7’ has to be placed at a distance of 3.84 Å from hydrogen H5. This constraint locates N7’ on a smaller circle around any chosen position 3.84 Å fromHaving hydrogen H7’ fixed H5. at This the constraintcircle around locates molecule N7’ on A, a nitrogen smaller circle N7’ has around to be any placed chosen at a positiondistance of of H7’. The graphical presentation of the possible location of N7’ resembles precession around the of H7’.3.84 TheÅ from graphical hydrogen presentation H5. This constraint of the possible locates location N7’ on ofa smaller N7’ resembles circle around precession any chosen around position the imaginary axis H5···H7’. The chosen orientation of molecule B can then be described by precession imaginaryof H7’. The axis graphical H5···H7’. presentation The chosen orientation of the possible of molecule location B of can N7’ then resembles be described precession by precession around the angle θ (N5–H5–H7’–N7’). Obviously, molecule B can also freely rotate around the H7’–N7’ bond. angleimaginaryθ (N5–H5–H7’–N7’). axis H5···H7’. Obviously,The chosen molecule orientation B canof molecule also freely B can rotate then around be described the H7’–N7’ by precession bond. This rotationθ defines the last torsion angle (β, H5–H7’–N7’–C6), which is necessary for a complete Thisangle rotation (N5–H5–H7’–N7’). defines the last torsion Obviously, angle molecule (β, H5–H7’–N7’–C6), B can also freely which rotate is necessary around the for H7’–N7’ a complete bond. description of the position and orientation of moleculeβ B towards original molecule A under the given descriptionThis rotation of the defines position the and last orientation torsion angle of molecule ( , H5–H7’–N7’–C6), B towards original which molecule is necessary A under for the a complete given geometric constraints (Figure 3). geometricdescription constraints of the position (Figure3 and). orientation of molecule B towards original molecule A under the given geometric constraints (Figure 3).

Figure 3. Definition of the three rotational angles resulting from geometric constraints; d(H5···H7’) = 3.15 Å, Figure 3. Definition of the three rotational angles resulting from geometric constraints; d(H5···H7’) = d(H5···N7’) = 3.44 Å and d(N5···H7’) = 3.84 Å. 3.15 FigureÅ, d(H5···N7’) 3. Definition = 3.44 of Å the and three d(N5···H7’) rotational = 3.84 angles Å. resulting from geometric constraints; d(H5···H7’) = Inspection3.15 Å, d(H5···N7’) of the 3D space= 3.44 definedÅ and d(N5···H7’) by the three = 3.84 angles Å. defined using a 5-degree step for each angle providesInspection over 370,000 of the possible3D space arrangements defined by the of two three histidine angles molecules.defined using Naturally, a 5-degree a significant step for part each of anglethese solutionsprovidesInspection canover beof immediatelythe370,000 3D space possible omitteddefined arrangements dueby the to mutual three of angles intersectiontwo definedhistidine of vanusing molecules. der a Waals5-degree Naturally, radii step of somefor aeach significantatoms.angle On providespart the other of these hand,over solutions 370,000 some of can thepossible be solutions immediat arrangements evinceely omitted the approachingof duetwo to histidine mutual of hydrogen intersection molecules. H8’ to of hydrogenNaturally, van der a WaalsH6significant to aradii distance of partsome of of 3.03 atoms.these Å, which solutions On the was othercan reported be hand, immediat as some anotherely of omitted observablethe solutions due intermolecular to evincemutual the intersection approaching interaction, of vanvide of der Waals radii of some atoms. On the other hand, some of the solutions evince the approaching of

Crystals 2019, 9, x FOR PEER REVIEW 5 of 11 CrystalsCrystals2019 2019, 9, 9 159, x FOR PEER REVIEW 5 of 511 of 11 hydrogen H8’ to hydrogen H6 to a distance of 3.03 Å, which was reported as another observable hydrogenintermolecular H8’ to interaction, hydrogen vide H6 suprato a .distance The additional of 3.03 conditionÅ, which ofwas d(H6···H8’) reported =as 3.03 another Å reduces observable the 3D supraintermolecularspace. The significantly additional interaction, to condition the surface vide of supra d(H6of the. The··· sphericalH8’) additional = 3.03 shape Å condition reduces depicted theof ind(H6···H8’) 3D Figure space 4. signiﬁcantly The= 3.03 number Å reduces toof thepossible the surface 3D ofspacearrangements the spherical significantly shapeusing to the depictedthe 5° surface grid indrops of Figure the to spherical several4. The number thousand.shape depicted of possible in Figure arrangements 4. The number using of the possible 5 ◦ grid dropsarrangements to several using thousand. the 5° grid drops to several thousand.

Figure 4. 2D projection of a spherical shape showing possible combinations of angles α, β and θ Figure 4. 2D projection of a spherical shape showing possible combinations of angles α, β and θ Figuremeeting 4. additional 2D projection condition of a sphericald(H6···H8’) shape = 3.03 sh Å.owing The depictedpossible contourcombinations plot was of anglesobtained α, forβ and angle θ meeting additional condition d(H6···H8’) = 3.03 Å. The depicted contour plot was obtained for angle θ meetingθ ranging additional from 130–360, condition 0–25 d(H6···H8’)deg; other values = 3.03 ofÅ. angleThe depicted θ lead to contour forbidden plot interatomic was obtained contacts. for angle θ rangingθ ranging from from 130–360, 130–360, 0–25 0–25 deg; deg; other other values values of of angle angle θlead lead to to forbidden forbidden interatomicinteratomic contacts. contacts. By systematically selecting one particular arrangement of the two histidine molecules after By systematically selecting one particular arrangement of the two histidine molecules after another anotherBy systematicallyand using it asselecting a basic one repetitive particular motif, arrangement it is possible of the to twostart histidine the formation molecules of larger after andanotherconglomerates. using and it as using a basicOne itof repetitive asthe a consequences basic motif, repetitive it of is the possiblemotif, symmetrical it tois startpossible arrangement the formationto start of the molecules of formation larger conglomerates.in theof crystallarger Oneconglomerates.lattice of the is consequencesthat every One of molecule the of consequences the must symmetrical have of the arrangement sysamemmetrical surroundings ofarrangement molecules as the of in others,molecules the crystal considering in latticethe crystal isone that everylatticeindependent molecule is that moleculemustevery have molecule in the the same asymmetric must surroundings have unit.the sameTherefore, as the surroundings others, multiplying considering as athe pair oneothers, of independenthistidine considering molecules molecule one inindependentand the asymmetricplacing molecule molecule unit. A Therefore,in into the the asymmetric position multiplying of unit. molecule aTherefore, pair B ofgenerates histidine multiplying another molecules a moleculepair of and histidine (molecule placing molecules molecule C). This A intoandstep the placing can position be moleculerepeated of molecule Ato intogenerate Bthe generates position a molecular of another molecule chain molecule ofB generatesan arbitrary (molecule another length. C). molecule This The stepmutual (molecule can arrangement be repeated C). This to generatestepof molecule can a molecularbe repeatedB to molecule chain to generate ofA is an in arbitrary athis molecular way length.transposed chain The of to mutualan the arbitrary spacing arrangement length. of molecule The of moleculemutual C to molecule arrangement B to molecule B and Aof isso in moleculeon this (Figure way B 5).to transposed molecule A to is the in this spacing way oftransposed molecule to C the to moleculespacing of Bmolecule and so on C to (Figure molecule5). B and so on (Figure 5).

FigureFigure 5. 5.Chain Chain generation generation principle principle described described using using 0–0–0 0 – 0 – combination 0 combination of angles;of angles; blue—original blue—original pair, green—pairFigurepair, green—pair 5. Chain copy generation placing copy placing molecule principle molecule A’ described into A’ B into position using B position 0 generating – 0 –generating 0 combination molecule molecule of C angles; (B’); C (B’); red—2 blue—original red—2nd pairnd pair copy placingpair,copy green—pairplacing molecule molecule A” copy into placingA’’ B’ into position moleculeB’ position generating A’ generating into moleculeB position molecule D generating (B”). D (B’’). molecule C (B’); red—2nd pair copy placing molecule A’’ into B’ position generating molecule D (B’’). AsAs a consequencea consequence of of crystal crystal lattice lattice periodicity,periodicity, certain arrangements arrangements provide provide linear linear chains chains with with identicalidenticalAs distances a distancesconsequence between between of crystal equivalent equivalent lattice atoms atoms periodicity, inin moleculesmolecules certain A Aarrangements and and C, C, while while theprovide the other other linear solutions solutions chains provide providewith helicalidentical structures. distances The between helical equivalent arrangement atoms can in molecules be acceptable A and but C,only while for the an other exact solutions 2-, 3-, 4- provide or 6-fold axis, as only such axes can be found in real crystal systems. The additional condition of chain linearity or deﬁned helicity further reduces the scope of possible solutions. The number of possible solutions Crystals 2019, 9, x FOR PEER REVIEW 6 of 11

Crystals 2019, 9, x FOR PEER REVIEW 6 of 11 helical structures. The helical arrangement can be acceptable but only for an exact 2-, 3-, 4- or 6-fold Crystalshelicalaxis, 2019as structures. only, 9, 159 such Theaxes helical can be arrangement found in real can crystal be acceptable systems. butThe only additional for an exactcondition 2-, 3-, of 4- chain or 6-fold linearity6 of 11 axis,or defined as only helicity such axes further can be reduces found in the real scope crystal of systems. possible The solutions. additional The condition number of of chain possible linearity solutions ordropped defined to helicity six using further 1-degree reduces incrementation the scope of possible of the solutions. three rotation The number angles. of One possible particular solutions solution droppeddropped(Figure to 6a)to six sixplaces using using hydrogen 1-degree1-degree H8’’ incrementationincrementation (molecule C)of of that thee a three threedistance rotation rotation of 3.25 angles. angles. Å from One One hydrogen particular particular H5solution solution(molecule (Figure(FigureA), in6 a) accordance6a) places places hydrogen hydrogen with the H8” H8’’ list (molecule (molecule of found C) C) atintermol at a a distance distanceecular of of 3.25 interactions3.25 Å Å from from hydrogen hydrogen(Table 1). H5 H5 For (molecule (molecule examples A), of inA),molecular accordance in accordance chains with theobtained,with list the of foundseelist Figureof intermolecularfound 6b,c. intermol ecular interactions interactions (Table 1(Table). For examples1). For examples of molecular of chainsmolecular obtained, chains see obtained, Figure6 seeb,c. Figure 6b,c.

FigureFigureFigure 6. 6. 6. Molecular MolecularMolecular chains chains generated generated generated for fo different forr different different sets sets of sets angles of angles of α angles, β ,α and, β,α andθ,; (βa, )θ 260 and; (a )– 260 θ143;( – a– ) 143140, 260–143–140, – ( b140,) 195 (b ) 195 (b–)– 120 195–120–240,120 – – 240, 240, ( c()c 180) (180c) – 180–150–240. –150 150 – 240.– 240.

InInIn summary, summary,summary, the the the arrangement arrangement arrangement of of histidine ofhistidine histidine molecules molecules molecules within within within a finala final a chain final chain waschain was accomplished wasaccomplished accomplished using sixusingusing constraints. sixsix constraints. constraints. Three distancesThree Three distances distances in the N5–H5in inthe the N5··· N5H7’–N7’ – H5···H7’ – H5···H7’ system, – N7’ – N7’ system, interactions system, interactions interactions H5···H8’ H5···H8’ and H5···H8’ H6 and···H8’, and andH6···H8’,H6···H8’, the condition and and the the ofcondition condition chain linearity of of chain chain providedlinearity linearity provided a uniqueprovided a solution. unique a unique solution. The solution. definition The Thedefinition of definition the basic of the repetitiveof basic the basic motifrepetitiverepetitive is an essentialmotif motif is is an an step essential essential for the step subsequentstep for for the the subseque subseque constructionnt constructionnt construction of the 3D of molecular the of 3Dthe molecular 3D packing. molecular packing. Such packing. a Such process Such wasaa processprocess not attempted waswas not not here; attempted attempted therefore, here; here; the therefore, following therefore, the paragraphs the following following serve paragraphs paragraphs just as aserve tentative serve just indicationasjust a astentative a oftentative how itindication couldindication be done. of of how how it it could could be be done. done. TheTheThe chain chain chain formed formed contains contains contains transl translationtranslationation information information information on histidineon on histidine histidine molecules molecules molecules in one in dimension. inone one dimension. dimension. This This translation might coincide with one of the cell parameters. To reveal the periodicity in the second and Thistranslation translation might might coincide coincide with with one one of the of the cell cell parameters. parameters. To Toreveal reveal the the periodicity periodicity in inthe the second second and third dimension, the chains have to be assembled into sheets (2D) and the sheets into the final 3D andthird third dimension, dimension, the the chains chains have have to to be be assembled assembled into into sheets (2D)(2D) andand thethesheets sheets into into the the final final 3D 3D structure. For this purpose, the rest of the available intermolecular interactions can be used in order structure.structure. For For this this purpose, purpose, the the rest rest of of the the available availabl intermoleculare intermolecular interactions interactions can can be be used used in in order order to control the mutual assembly of molecular chains. The so-far-unused intermolecular interaction toto control control the the mutual mutual assembly assembly of of molecular molecular chains. chains. The The so-far-unused so-far-unused intermolecular intermolecular interaction interaction between H3···H8’ can be utilized for arrangement of two adjacent chains. A sphere with a radius of between H3···H8’ can be utilized for arrangement of two adjacent chains. A sphere with a radius of 2.90between Å made H3···H8’ around can each be H8utilized atom forin thearrangement chain defines of twothe possibleadjacent location chains. ofA H3sphere atoms with from a radius the of 2.90 Å made around each H8 atom in the chain defines the possible location of H3 atoms from the second2.90 Å chainmade (Figure around 7a). each Van H8 der atom Waals in presentationthe chain defines reveals the a relativelypossible locationnarrow range of H3 in atoms the space from the second chain (Figure7a). Van der Waals presentation reveals a relatively narrow range in the space wheresecond the chain second (Figure chain 7a). can Van be placed der Waals (Figure presentation 7b). reveals a relatively narrow range in the space wherewhere the the second second chain chain can can be be placed placed (Figure (Figure7b). 7b).

Figure 7. Visualization of 2.90 Å spheres around hydrogen H8 in the molecular chain; (a) stick presentation, (b) space ﬁll presentation. Crystals 2019, 9, x FOR PEER REVIEW 7 of 11 Crystals 2019, 9, x FOR PEER REVIEW 7 of 11

Figure 7. Visualization of 2.90 Å spheres around hydrogen H8 in the molecular chain; (a) stick CrystalsFigure2019, 9 ,7. 159 Visualization of 2.90 Å spheres around hydrogen H8 in the molecular chain; (a) stick7 of 11 presentation, (b) space fill presentation. presentation, (b) space fill presentation. The axes of both chains have to be parallel in order to fulfill d(H3···H8’) = 2.90 Å for each pair of TheThe axes axes of of both both chains chains have have to to be be parallel parallel in in order order to to fulfill fulfill d(H3 d(H3···H8’)···H8’) = = 2.902.90 ÅÅ forfor eacheach pairpair of of atoms. However, there are two possible arrangements generated by rotation of the second chain axis atoms.atoms. However,However, there there are are two two possible possible arrangements arrangements generated generated by by rotation rotation of of the the second second chain chain axis axis by 180 degrees (Figure 8). Furthermore, when the second chain is located at d(H3···H8’) = 2.90 Å, it byby 180 180 degrees degrees (Figure (Figure8). 8). Furthermore, Furthermore, when when the the second second chain chain is located is located at d(H3 at d(H3···H8’)···H8’) = 2.90 = 2.90 Å, it Å, can it can be tilted around the axis defined by H3 atoms in the chain arbitrarily until the violation of van becan tilted be tilted around around the axis the definedaxis defined by H3 by atoms H3 atoms in the in chain the chain arbitrarily arbitrarily until until the violation the violation of van of dervan der Waals radii is reached. The position of the second chain is defined by d(H3···H8’) distance and Waalsder Waals radii isradii reached. is reached. The position The position of the secondof the second chain is chain defined is defined by d(H3 ···byH8’) d(H3···H8’) distance distance and angle andγ angle γ of the inclination. Furthermore, some space has to be left for the positioning of the water ofangle the inclination.γ of the inclination. Furthermore, Furthermore, some space some has space to be has left forto be the left positioning for the positioning of the water of molecule.the water molecule. Unfortunately, its location in the crystal lattice can be determined only approximately, as Unfortunately,molecule. Unfortunately, its location its in location the crystal in latticethe crystal can belattice determined can be determined only approximately, only approximately, as signals ofas signals of both its hydrogen atoms coincide in the published spectra. However, two of three bothsignals its hydrogenof both its atoms hydrogen coincide atoms in the coincide published in the spectra. published However, spectra. two However, of three intermolecular two of three intermolecular interactions attributed to the water molecule indicate its probable location. The interactionsintermolecular attributed interactions to the waterattributed molecule to the indicate water its molecule probable location.indicate Theits probable intersection location. of spheres The intersection of spheres defined by d(H5···Hw) = 2.47 Å and d(H8···Hw) = 3.09 Å localizes possible definedintersection by d(H5 of spheres···Hw) =defined 2.47 Å andby d(H5···Hw) d(H8···Hw) = =2.47 3.09 Å Å and localizes d(H8···Hw) possible = 3.09 placement Å localizes of the possible water placement of the water molecule among histidine molecules. moleculeplacement among of the histidine water molecule molecules. among histidine molecules.

Figure 8. Possible arrangements of two adjacent molecular chains generated by 180-degree rotation FigureFigure 8. 8.Possible Possible arrangements arrangementsof of two two adjacent adjacent molecular molecular chains chains generated generatedby by 180-degree 180-degree rotation rotation of the second chain (a) and (b). The chains are viewed end-on indicating possible location of a water ofof thethe secondsecond chain ( (aa)) and and (b (b).). The The chains chains are are viewed viewed end-on end-on indicating indicating possible possible location location of a water of a molecule. watermolecule. molecule.

ForFor the the 2D-sheet 2D-sheet generation, generation, the the chain chain pair pair is is mult multiplied,iplied, and and chain chain A A is is placed placed into into the the position position For the 2D-sheet generation, the chain pair is multiplied, and chain A is placed into the position ofof chain chain B, B, generating generating another another chai chain,n, similar similar to the to themanner manner described described for chain for chain generation generation from froma pair a of chain B, generating another chain, similar to the manner described for chain generation from a pair ofpair histidine of histidine molecules. molecules. Only Onlychains chains of certain of certain mutual mutual orientation orientation can provide can provide a linear a linear 2D-sheet 2D-sheet in of histidine molecules. Only chains of certain mutual orientation can provide a linear 2D-sheet in whichin which corresponding corresponding atoms atoms in inchain chain A Aand and C Care are equidistant. equidistant. The The 2D-sheet 2D-sheet formed formed contains contains which corresponding atoms in chain A and C are equidistant. The 2D-sheet formed contains translationtranslation information information in in the the second second dimension dimension (Fig (Figureure 9)9) which which might might coincide coincide with with one one of of the the cell cell translation information in the second dimension (Figure 9) which might coincide with one of the cell parameters.parameters. Perpendicular Perpendicular arrangements arrangements of of histid histidineine molecules molecules in in the the sheet sheet (chains (chains A A and and C, C, see see parameters. Perpendicular arrangements of histidine molecules in the sheet (chains A and C, see horizontalhorizontal lines lines in in Figure Figure 9)9) to to histidine histidine molecules molecules within within a a single single chain chain (molecules (molecules A A and and C) C) should should horizontal lines in Figure 9) to histidine molecules within a single chain (molecules A and C) should bebe tested tested preferentially, preferentially, as as perpendicular perpendicular arrangem arrangementsents can can provide provide perpendicular perpendicular cell cell parameters parameters in in be tested preferentially, as perpendicular arrangements can provide perpendicular cell parameters in thethe final ﬁnal step. step. the final step.

Figure 9. 2D-sheet generated by multiplication of a chosen pair of chains. The chains are viewed Figure 9. 2D-sheet generated by multiplication of a chosen pair of chains. The chains are viewed end- end-on.Figure 9. Green 2D-sheet spheres generated indicate by d(H3 multiplication···H8’) = 2.90 of Åa chos throughen pair the of sheet. chains. The chains are viewed end- on. Green spheres indicate d(H3···H8’) = 2.90 Å through the sheet. on. Green spheres indicate d(H3···H8’) = 2.90 Å through the sheet. An assembly of 2D-sheets into a 3D structure represents the ﬁnal step of crystal lattice An assembly of 2D-sheets into a 3D structure represents the final step of crystal lattice reconstruction.An assembly The multipliedof 2D-sheets sheets into area brought3D structure closer. represents Individual the sheets final mutually step of interlockcrystal lattice well reconstruction. The multiplied sheets are brought closer. Individual sheets mutually interlock well andreconstruction. minimize voids The multiplied in between. sheets The onlyare brought criterion closer. is the Individual possible collision sheets mutually of van der interlock Waals radii. well and minimize voids in between. The only criterion is the possible collision of van der Waals radii. Thereand minimize are no distance voids in restraints between. left, The which only mightcriterion be usedis the for possible the mutual collision positioning of van der of neighboring Waals radii. There are no distance restraints left, which might be used for the mutual positioning of neighboring 2D-sheets.There are no However, distance the restraints sheets should left, which be positioned might be soused that for the the volume mutual of positioning empty space of in neighboring the crystal is minimized (Figure 10).

Crystals 2019, 9, x FOR PEER REVIEW 8 of 11 Crystals 2019, 9, x FOR PEER REVIEW 8 of 11 2D-sheets. However, the sheets should be positioned so that the volume of empty space in the crystal Crystals2D-sheets.2019, 9 However,, 159 the sheets should be positioned so that the volume of empty space in the crystal8 of 11 is minimized (Figure 10). is minimized (Figure 10).

FigureFigureFigure 10. 10.An Anassembly assembly of ofof 2D-sheets 2D-sheets2D-sheets into into the the finalﬁnal final 3D 3D 3D structure. structure. structure. The TheThe coordinatecoordinate coordinate systemsystem system depicteddepicted depicted was was was chosenchosenchosen arbitrarily arbitrarily and andand has hashas only onlyonly illustrativeillustrative illustrative function. function.

Finally,Finally,Finally, the the approximate approximate positionposition position of of the the water water mo moleculemoleculelecule inin in the thethe crystal crystalcrystal packing packingpacking can cancan be bebe determined. determined. However,However,However, itit cannotcannot bebe locatedlocatedlocated precisely,precisely, asas its its hydrogen hydrogen atoms atomsatoms are areare not notnot spatially spatiallyspatially resolved resolvedresolved in in inthe thethe availableavailableavailable NMR NMR data.data. TheThe accurate accurateaccurate locationlocation and and orient orientationorientationation of ofof the thethe water waterwater molecule moleculemolecule has hashas to to tobe bebe examined examinedexamined exexex post postpost,,, taking takingtaking into into account accountaccount possible possiblepossible involvementinvolvement in inin hydrogen-bond hydrogen-bondhydrogen-bond networking. networking.networking. On OnOn the thethe other otherother hand, hand,hand, thethethe location locationlocation of of thethe chloride chloride anionanionanion cannotcannotcannot be be determined determined at atat all, all,all, as asas therethere there is is isno no no information information information of of ofchloride chloride chloride atomatomatom involvement involvementinvolvement inin thethe molecular molecularmolecular packing.packing. Therefore, Therefore,Therefore, the thethe remaining remainingremaining voids voidsvoids in inin the thethe crystal crystalcrystal structure structurestructure andandand possible possiblepossible hydrogen-bonding hydrogen-bonding interactions interactionsinteractions have have to to be be examined examined examined for for for its its its location. location. location. OnceOnceOnce thethe molecular packingpacking packing isis is established,established, established, it it is itis possible ispossible possible to to derive derive to derive crystal crystal crystal cell cell parameters. parameters. cell parameters. This This Thisinformationinformation information is hidden is hidden inin thethe in periodicityperiodicity the periodicity of of a a single single of ahistidine histidine singlehistidine molecule, molecule, molecule, of of each each of of ofits its eachatoms. atoms. of Selecting its Selecting atoms. Selectingoneone particularparticular one particular atom in atom thethe histidinehistidine in the histidine moleculemolecule molecule an andd deleting deleting and deleting the the rest rest the of restof them, them, of them,the the information theinformation information on on ontranslationtranslation translation becomes becomes obviousobvious obvious (Figure(Figure (Figure 11).11). 11 ).ThenThen Then cellcell cell parameters parameters parameters are are are easily easily determined, determined,determined, and andand the thethe structurestructurestructure can can bebe classiﬁedclassified intointointo thethethe corresponding correspondingcorresponding sy system,system,stem, in inin this thisthis case casecase an anan orthogonal orthogonal orthogonal crystal crystalcrystal system systemsystem identical with the system found in ref. [16]. identicalidentical with with the the system system found found in in ref. ref. [ 16[16].].

FigureFigure 11. 11.A A derivationderivation ofof thethe cellcell lattice parametersparameters from from the the molecular molecular framework; framework; (a ()a molecular) molecular packingFigurepacking in11. in stick Astick derivation presentation, presentation, of the (b) (cell projectionb) projectionlattice ofpara molecular ofmeters molecular from packing packingthe using molecular using only one framework;only atom one of atom each (a) molecule.molecularof each packingmolecule. in stick presentation, (b) projection of molecular packing using only one atom of each Formolecule. this particular solution, parameter a is found in the 2D-sheet between chain A and C (see FigureFor9). Parameterthis particularb is solution, identified parameter as the distance a is found between in the 2D-sheet 2D-sheet A between and B (see chain Figure A and 10). C Finally, (see parameterFigureFor 9). this cParameteremerges particular asb is the solution,identified distance parameter as between the distance a histidine is foundbetween molecules in 2Dthe-sheet 2D-sheet A A and and between C B within (see Figure chain a single 10).A and molecular Finally, C (see chainFigure (see 9). Parameter Figure5). Fourb is identified histidine as molecules the distance were between found 2D in-sheet the crystal A and unit B (see cell. Figure One 10). molecule Finally, is situated at the origin of the orthogonal system while the remaining positions correspond to the Crystals 2019, 9, 159 9 of 11 symmetry operations characteristic for the given space group. The deduction of possible space groups is then rather straightforward. There are 59 space groups available for the orthorhombic crystal system. The requirements for the space group are two: a primitive cell based on the resulting molecular packing (Figure 11b) and no mirror planes, as histidine is a chiral compound and only its L-form is present. Therefore, the number of possible space groups is reduced to 4, namely P222, P2221, P21212 and P212121. Based on symmetry operations, each space group generates different positions for four histidine molecules in the unit cell. The only match of positions generated by symmetry operations with the positions found in the reconstructed crystal structure was found for space group P212121 providing x, y, z, −x + 1/2, −y, z + 1/2, x + 1/2, −y + 1/2, −z and –x + 1/1, y + 1/2, −z + 1/2 sites. The found crystallographic parameters can then be confronted with X-ray powder diffraction data for verification of the crystal model proposed. When a significant match is found, the crystal structure can be further refined by Rietveld refinement. Alternatively, the crystal’s structure can be optimized by quantum-chemical methods and NMR parameters can be calculated, which may serve for comparison [21] with experimental data.

3. Conclusions A general procedure of crystal packing reconstruction using a certain number of intermolecular interactions was introduced and demonstrated on the crystal structure of L-histidine·HCl·H2O. The geometric restrictions are applied gradually within a supervised procedure, narrowing the scope of possible arrangement of two adjacent molecules. Using a list of available intermolecular interactions and implementing the additional condition that every molecule has the same surroundings in the crystal lattice, a basic repetitive motif of histidine molecules was derived. This motif was subsequently used for the construction of molecular chains containing information on translation in one dimension. The other two dimensions can then be obtained by the construction of 2D-sheets and their mutual assembly into the final 3D structure. However, the procedure for the whole 3D structure from molecular chains was not attempted; it was only briefly outlined here. Complete molecular packing would enable the derivation of standard crystallographic parameters that can be used for verification of the structural model. The procedure described can be employed especially when standard crystallographic parameters are not available and traditional methods based on X-ray diffraction fail. Similarly to other fields of NMR spectroscopy, hydrogen atoms show high relevance for direct structure solution using solid-state NMR data. Generally speaking, further development of methods for detection and quantification of intra- and intermolecular interactions is in high demand. Intramolecular distances can serve for construction of the initial molecule with a proper conformation, while the intermolecular distances are used for a molecular assembly within crystal packing and derivation of standard crystallographic parameters. More interactions mean lower degrees of freedom and a narrower scope of possible solutions. Regarding intermolecular interactions, only the shortest ones ranging from 2.5–4.0 Å are meaningful. The number of required distances can vary for different crystallographic systems. It seems that the construction of molecular frameworks of systems of low symmetry will be easier due to the low number of molecules in the unit cell, and therefore also in the basic repetitive motif. In some cases, a coarse crystal packing could be obtained using just a fragment of the molecule. Furthermore, the construction of crystal packing of achiral molecules crystallizing in centrosymmetric groups will probably be more difficult, as a mirror image has to be considered in every single step.

4. Materials and Methods

4.1. DFT Calculations The geometry of histidine hydrochloride monohydrate crystal structure was taken from the Cambridge Crystallographic Database (Refcode HISTCM12) [1,19]. The geometry optimization of the crystal structure was performed with the CASTEP program [20], which is a DFT-based code, Crystals 2019, 9, 159 10 of 11 using pseudopotentials to model the effects of core electrons and plane waves to describe the valence electrons. Electronic exchange and correlation effects were modelled using the PBE functional [22]. A plane wave basis set energy cutoff of 600 eV, default ‘on the ﬂy generation’ pseudopotentials, and a k-point spacing of 0.05 Å−1 over the Brillouin zone via a Monkhorst-Pack grid [23] was used.

4.2. Construction of Molecular Framework The search for the arrangement of two histidine molecules and the building of a molecular pair was done in MATLAB R2016b (MathWorks, USA). The inspection of the 3D space deﬁned by angles α, β and θ (Figure3) was done using a 5-degree step for each angle. The ﬁnal optimization of the mutual arrangement within a molecular chain in order to obtain a linear object was done using a 1-degree step for each angle. In this step, a dispersion of distances between equivalent atoms in molecules A and C was minimized.

4.3. Visualization of Generated Structures Individual steps of the creation of a basic repetitive motif and subsequent proposal of the construction of the complete molecular packing were visualized in Biovia Discovery Studio Visualizer (Accelrys Inc., USA).

Author Contributions: Conceptualization, J.S.; methodology, J.S. and V.H.; formal analysis, J.S. and M.D.; investigation, V.H., J.S. and M.D.; resources, J.S. and M.D.; writing—original draft preparation, J.S.; writing—review and editing, J.S. and M.D.; visualization, J.S. Funding: This work was supported by The Czech Science Foundation (grant No. 15-12719S and 18-11851S). Acknowledgments: The authors are grateful to Andrew Christensen for proofreading. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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