Keywords

1 Introduction

Organic crystals are analyzed by using a large amount of reflection data according to X-ray crystallography which bases on symmetry theory in mathematics [1]. This indicates that the analysis has no relation to interaction energies among molecules. Substantially, the crystals are formed due to intermolecular interactions among organic molecules [2]. Conventionally, such interaction energies have been calculated with powerful supercomputers using high-level ab initio calculations by specialists in quantum chemistry. This mainly comes from the fact that a large basis set and electron correlation correction are necessary for an accurate evaluation of the dispersion interactions. Therefore, an accurate evaluation of the dispersion energies demands a vast amount of computational resources, introducing much difficulty to understand energetic aspect of organic crystals [3,4,5].

However, recent progresses in personal computers and computational methods give us a challenging time to overcome this difficulty as for the evaluation of the dispersion energies. Namely, new personal computers with multicore CPU perform rapid calculations for intermolecular interactions of organic molecules with more than a hundred carbons using dispersion-corrected DFT method [6]. Therefore, the interaction energies between two neighbored molecules in crystals can be briefly evaluated. Gaussian is one of the well-known ab initio molecular orbital and DFT calculation program [7]. GaussView is graphical user interface (GUI) program for Gaussian [8]. We can easily evaluate intermolecular interaction energies between molecules in organic crystals using Gaussian and GaussView programs.

Such a change would introduce the third stage of our studies on organic crystal chemistry (Fig. 7.1). In the first stage, on the basis of space group [1], we classified crystal structures (Fig. 7.1a) [9, 10]. In the second stage, according to the Cambridge Structural Database and its graphics Mercury [11], we devoted to two subjects about organic crystals (Fig. 7.1b) [12, 13]. The one is supramolecular chirality of two-fold helical (or screw) molecular assemblies, and the other is hierarchical structures through bundles of the assemblies. The former indicates a difference between space geometry and space group [14,15,16]. For example, two-fold helical assemblies are discriminable in handedness from an anisotropic view. In contrast, from an isotropic view of space group, two-fold helical operations form identical assemblies by both clockwise and anticlockwise rotations.

Fig. 7.1
figure 1

Three stages for understanding generation of supramolecular chirality in crystal structures from viewpoints of symmetry theory (a), molecular graphics (b), and molecular calculations (c)

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The other subject deals with hierarchical structures of organic crystals, such as 1D columns, 2D layers, 3D stacked layers [16,17,18]. For example, the preferred two-fold helical columns with three-axial chirality construct chiral layers, which are stacked to give chiral crystals. According to such a bundling model, chiral crystals are briefly defined as follows. Only right- or left-handed two-fold helices are bundled to form the corresponding right- or left-handed crystals, respectively.

It is considered that these two subjects demand any quantitative expressions in mathematics (Fig. 7.1c). It is noteworthy that the above-mentioned Gaussian program uses Z-matrix and permits dummy atoms for defining geometries of organic molecules [5]. Accordingly, positions of any atoms of a molecule can be defined by three parameters which determine distances between atoms, angles formed by three atoms, and dihedral angles. Significantly, the last dihedral angles are suitable for determining chirality, since the four points for defining the dihedral angles enable us to determine handedness of chirality. Likewise, the above-mentioned hierarchical structures demand any quantitative explanation from an energetical viewpoint. The Gaussian program serves as an excellent tool for providing interaction energies between molecules.

We aim to solve two challenging problems, generation of supramolecular chirality and hierarchical structures in organic crystals. This chapter deals with (i) a rectangular triangle model by using Z-matrix and dummy atoms for positioning molecules, (ii) generation of supramolecular chirality in molecular assemblies on the basis of three kinds of rotations along an axis, and (iii) generation of hierarchical structures on the basis of interaction energies between neighbored molecules in organic crystals. As described below, the solution has come from an elucidation of three kinds of rotations of a molecule at the fixed point along an axis.

2 Rectangular Triangle Unit with a Molecule

2.1 Connection Model of a Rectangular Triangle and a Molecule

Z-matrix defines the position of an atom (C4) by using the positions of preceding three atoms (C3, C2, C1), as shown in Fig. 7.2a. Firstly, the parameter B3 determines the distance between C4 and C3; secondly, the parameter A2 determines the angle formed by C4, C3, and C2; and thirdly, the parameter D1 determines the dihedral (torsion) angle formed by C4, C3, C2, and C1 with the direction from C4 to C1 (C4 → C3 → C2 → C1). Such a dihedral angle is possible to take a sign of either plus (right) or minus (left), prompting us to discriminate the chirality and its handedness. Figure 7.2b shows a Z-matrix description for the Gaussian program.

Fig. 7.2
figure 2

Definition of the position of atom (C4) based on the distance (B3) from C3, the angle (A2) formed by C4, C3 and C2, as well as a dihedral (torsion) angle (D1) defined by C4, C3, C2 and C1 (a), Z-matrix for an assembly of four carbon atoms (b), a rectangular triangle as an assembly of three dummy atoms and a carbon atom connected to the third dummy atom (c), and its Z-matrix (d)

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In addition to real atoms, we may employ dummy atoms that enable us to freely design assemblies of points, including lines, polygons, polyhedrons, and so on. Figure 7.2c shows an example of a rectangular triangle with three dummy atoms (X1, X2, X3). This model suggests that the fourth real atom is connected to the third dummy atom X3 which serves as a root for connecting any molecule (Fig. 7.2d).

Z-matrix provides multiple ways to define the structure of organic molecule and any others. Figure 7.3 shows an example of a benzene molecule. Figure 7.3a depicts the molecule with a dummy atom which was put on the center of benzene. Figure 7.3b displays a rectangular triangle composed of three dummy atoms. The dummy atom at the center of benzene and the X3 are overlapped (Fig. 7.3c).

Fig. 7.3
figure 3

A benzene molecule with a dummy atom (a), a rectangular triangle composed of three dummy atoms (b), the dummy atom of benzene and the dummy atom X3 are overlapped (c), and any movements of the molecule around the center (d)

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2.2 Relation to a Unit Cell of a Crystal

The above-mentioned connection induced us to design a triangle model which simulates a steric situation of a molecule in organic crystals. Thus, a perpendicular line (X1–X2) in Fig. 7.3c functions as a unit axis for a two-fold rotation or helix in a unit cell of a crystal. Another horizontal line (X2–X3) corresponds to a distance between the axis and the center of a molecule. Namely, X3 serves as a root for positioning the molecule.

It is necessary to use real parameters for understanding the above-mentioned triangle. One takes an example of benzene. The main lattice parameters for its crystal with space group P21/c are summarized in Fig. 7.4. The distance between X1 and X2 (Ba) is 5.376 Å which corresponds to a unit length of b-axis, while one between X2 and X3 (Bb) is 1.883 Å which does to the one-fourth length of c-axis. These Ba and Bb are available for Z-matrix in Fig. 7.2d.

Fig. 7.4
figure 4

Relation between lattice parameters and Z-matrix of a benzene crystal. Parameters of distances (a) and angles (b) for a unit cell and a triangle. Three kinds of rotations along b-axis (c). Bond and dihedral angles for Z-matrix (d)

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In this way, the triangle model is related to the crystal structure, but one problem remains. In spite of a fixed orientation to the axis in Fig. 7.3c, molecular graphics mostly displays diverse orientations in crystals. At last, we reached an idea that such a difference derives from any rotations of the molecule toward the axis (Fig. 7.3d). The next section describes how to simulate the rotations.

2.3 Positioning for Three Kinds of Rotations

The earth rotates around the sun (Fig. 7.5a). In daily life, we sit on a rotating chair at a desk and rotate in various ways (Fig. 7.5b). This movement can be simplified as rotations around three axes of a facial plate (Fig. 7.5c).

Fig. 7.5
figure 5

Rotations of a material around other material or axes, the earth around the sun (a), a person on a rotating chair at a desk (b), and a facial plate towards three axes (c)

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Mathematically, the rotations of the plate can be combined with the triangle. First of all, a point of a plate (o) is overlapped with the X3 of the rectangular triangle. Next, like our body on a chair, the plate can rotate at the X3 around three axes (x, y, z). In order to define each rotation angle (φ, ψ, ω), we employ the corresponding unit vectors (x, y, z) (Fig. 7.6a), which are expressed by three additional dummy atoms (X4, X5, X6) (Fig. 7.6b–d). These dummy atoms are connected to the above-mentioned triangle (X1, X2, X3) (Fig. 7.6e). The X3 serves as a root of the vectors, and the X4, X5, and X6 function for determining bond angles and dihedral angles of Z-matrix in the following way.

Fig. 7.6
figure 6

Three kinds of rotations of a facial plate with the corresponding three unit vectors (a), a unit vector x as an over-view (b), y as a side-view (c), z as a front-view (d) for denoting three rotation angles, and dummy atoms (X4, X5, X6) connected with the dummy atom X3 of the rectangular triangle (X1, X2, X3) (e)

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The first vector x, which is perpendicular to the plate, rotates around the x-axis by an angle (φ) and is denoted as the X4 (Fig. 7.6b). The rotational direction is right (clockwise) or left (anticlockwise) to the axis X3–X2 of the triangle. The second vector y, which is perpendicular to both x-axis and y-axis, rotates around the y-axis by an angle (ψ) and is denoted as the X5 (Fig. 7.6c). This corresponds to a tilt forward to the axis X1–X2. The third vector z, which is perpendicular to the vector y, indicates a rotation around the z-axis by an angle (ω) and is denoted as the X6 (Fig. 7.6d). This angle corresponds to the right (clockwise) or left (anticlockwise) rotation from the x-axis and is fully described as a dihedral angle in the next section.

In organic crystals, the amount of each angle is restricted much smaller than 90°, since the molecule touches to the axis X1–X2 on both sides. Their angles (φ, ψ, ω) are roughly measured by using a semicircular protractor on display for molecular graphics. Figure 7.4 includes their values for benzene crystal.

2.4 Position-Dependent Chirality

The additional dummy atoms (X4, X5, X6) require bond angles (Ax, Ay, Az) and dihedral angles (Dx, Dy, Dz) for Z-matrix. The results are summarized in Fig. 7.7, where you see four sets of three bond angles and three dihedral angles.

Fig. 7.7
figure 7

Combination of angles formed by three atoms and dihedral angles for three kinds of rotations. Right-slide (R) or left-one (S) from the axis X2–X3 (X4, Ax) (a), tilt to the axis X2–X3–X4 (X5, Ay) (b), right-rotation (r) or left-one (s) from X3–X4 (X6, Az) (c), and four combinations of the angles generated from three kinds of rotations on X3 (d)

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Figure 7.7a shows a clockwise or right-handed rotation (R) of X4 with a bond angle of Ax(φ). Such a horizontal rotation yields a constant dihedral angle Dx(X4 → X3 → X2 → X1) with −90°. The subsequent tilt rotations of X5 with a bond angle of Ay(ψ) (0 < ψ < 90) yield a constant dihedral angle Dy(X5 → X3 → X4 → X2) with +90° (Fig. 7.7b). So, chiral isomers with (−90, +90) are recognized as (R)-isomers.

In contrast, there exists an anticlockwise or left-handed rotation (S) of X4 with a bond angle of Ax(φ). Such rotations yield a dihedral angle Dx(X4 → X3 → X2 → X1) with +90°, followed by the subsequent tilt rotations of X5 having a constant dihedral angle Dy(X5 → X3 → X4 → X2) with −90°. So, chiral isomers (+90, −90) are recognized as (S)-isomers.

Furthermore, the third rotation of X6, which has a defined angle Az(X6–X3–X5) with +90°, yields a dihedral angle Dz(X6 → X3 → X5 → X4) with a value of either −ω or +ω (Fig. 7.7c), indicating generation of chiral (r)- or (s)-isomers, respectively. In this way, we discriminate four sets of bond angles and dihedral angles as (R, r), (R, s), (S, r), and (S, s) (Fig. 7.7d).

At last, we reached the novel idea that the triangle rotation model makes clear the existence of four stereoisomers. We term this chirality as position-dependent chirality, since the chirality depends on a position of a molecule toward an axis. This finding leads us to specify their Z-matrix by use of six dummy atoms involving (X1, X2, X3) for the triangle and (X4, X5, X6) for the three kinds of rotations (Fig. 7.8).

Fig. 7.8
figure 8

Z-matrix expression of position-dependent chiral isomers (R, r) (a), (R, s) (b), (S, r) (c) and (S, s) (d) by using the rectangular triangle and rotation model with an organic molecule. Three rotation angles (φ, ψ, ω) can be measured on molecular graphics

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2.5 Z-Matrix of a Molecule with Position-Dependent Chirality in Crystals

Now, one can write down the Z-matrix for any organic molecules in crystals by using the triangle and three kinds of rotations. The necessary parameters (distances, angles, and dihedral angles) are acquired from the known crystal parameters as well as the measured rotation angles on graphics. As described in the next section, the latter angles are determined by comparison of bond distances due to Mercury [11] and GaussView [8] more exactly than by a semicircular protractor on display. Figure 7.9 denotes an example of a benzene molecule which belongs to space group P21/c. The measured angles (φ, ψ, ω) are available in Fig. 7.4.

Fig. 7.9
figure 9

Z-matrix of a benzene molecule in its crystal on the basis of the rectangular triangle and three kinds of rotations. Four position-dependent isomers; (R, r)-isomer (a), (R, s)-isomer (b), (S, r)-isomer (c), and (S, s)-isomer (d)

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3 Bimolecular Assembly with Connection of the Triangle Units

3.1 Connection of Two Rectangular Triangles Through Symmetry Operations

Two rectangular triangular units are connected by using dummy atoms in various ways. Regular structures of dimers in crystals owe to symmetry operations. It is noteworthy that translation operation derives from translation of a cell unit of crystals, resulting in 1D columnar assemblies which serve as fundamental architecture in organic crystals. Figure 7.10 exemplifies various overlaps of points and lines. The one is an overlap of two points. Figure 7.10a–c depict translation, inversion, and reflection, respectively. The other is an overlap of two lines composed of two points, yielding two-fold rotation (Fig. 7.10d) and two-fold helix (Fig. 7.10e). Figure 7.10f illustrates an overlap on a glide plane.

Fig. 7.10
figure 10

Various overlaps of two triangular triangles: two points (a–c), two points and two lines (d, e), and a point and a line onto a glide plane (f)

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3.2 Connection of Molecules with Position-Dependent Chirality

We consider a triangle with a molecule involving three kinds of rotations. The resulting four stereoisomers of (R, r), (R, s), (S, r), and (S, s) are possible to combine together for dimers. Among them, the same stereoisomers of (R, r) or (S, s) are assembled by symmetry operations, such as translation, two-fold rotation, and two-fold helix along an axis to yield a chiral (R, r)(R, r)- or (S, s)(S, s)-dimer (and vice versa).

On the other hand, an enantiomer of (R, r) and (S, s) is assembled by symmetry operations, such as inversion at one point, reflection on a mirror plane, reflection on a mirror with one-half translation to yield an achiral (R, r)(S, s)-dimer. The other enantiomer of (R, s) and (S, r) is possibly combined to yield an achiral (R, s)(S, r)-dimer.

In principle, another diastereomeric (R, r)(R, s)-dimer, (S, s)(S, r)-dimer, (R, r)(S, r)-dimer as well as (S, s)(R, s)-dimer are possible. Symmetry operations do not express these diastereomeric dimers, and probably we can observe these dimers as two independent molecules in crystals. More detailed research is necessary for discussing these dimers.

3.3 Connection of Molecules Along Identical Axes

These assemble methods have the corresponding Z-matrix. The first triangle has the same six dummy atoms (X1 to X6) as the ones in Fig. 7.8, while the second triangle has additional six dummy atoms (X7 to X12). The representative four kinds of the dimers (only X7 to X12) are illustrated in Fig. 7.11. Z-matrix has a rule to specify each atom or dummy atom through a completely arranged number from the first one. These expressions by Z-matrix in Fig. 7.11 start from the number 1 for the first dummy atom X1 and go to the number 12 for the last dummy atom X12.

Fig. 7.11
figure 11

Connection of two triangular triangles with symmetry operations: translation (a), two-fold rotation (b), two-fold helix (c) along an identical axis, and inversion (d) through a point

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3.4 Connection of Molecules Along Separated Axes

Next, we consider other cases that the lines (X1–X2) of the triangles are separated from each other. The above-mentioned examples in Fig. 7.11 belong to the identical cases, while those in Fig. 7.12 to the separated ones. The latter is observed in stacking of layers as mentioned later.

Fig. 7.12
figure 12

Connection of two triangular triangles with symmetry operations: translation (a) and two-fold helix (b) through the separating axes

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The other dimers through reflection on a mirror as well as glide plane can be expressed in a similar way to give us achiral dimers, leading us to the world of achiral assemblies in comparison with one of chiral ones.

At last, we obtain whole Z-matrix of four kinds of benzene dimers. Three examples of them are shown in Fig. 7.13.

Fig. 7.13
figure 13

Z-matrix of benzene dimers together with a rectangular triangle and three kinds of rotations: dimers with symmetry operation, such as translation (a), two-fold helix (b), inversion (c)

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4 Intermolecular Interaction Energy Calculations by Gaussian Program

4.1 Methods of DFT Calculations

Intermolecular interaction energies were calculated using Gaussian 16 W [7]. The B3LYP functional and 6-311G** basis set were used for the DFT calculations with Grimme’s D3 dispersion correction [19]. The basis set superposition error (BSSE) [20] was corrected for all the interaction energy calculations using the counterpoise method [21]. Figure 7.14 shows an example of input file for the calculation of the interaction energy of an (R, r)(R, r)-dimer of benzene with translation (trala) for symmetry operation. The contents of Figs. 7.13 and 7.14 are merged according to the defined procedure to yield the final input file of the Gaussian program for calculating the intermolecular interaction energies of neighbored molecules in organic crystals.

Fig. 7.14
figure 14

Example of input file for Gaussian calculations

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4.2 Dependence of the Interaction Energies on Rotation Parameters

In order to evaluate effects of various parameters, we checked two distances of triangles as well as three kinds of rotations in the case of benzene crystal with space group P21/c (Fig. 7.4). The former is exactly obtained by crystal lattice parameters, while the latter angles of rotations are roughly measured on display by GaussView [8] and Mercury [11]. Namely, we check the distances between atoms in the designed structures by GaussView and those in crystal structures by Mercury with changes of the rotation values. Among many values, we employ the most suitable ones.

The interaction energies may naturally depend on the rotation angles. For an example, Table 7.1 displays the energy changes of an (R, r)(R, r)-dimer of benzene obtained through two-fold helix symmetry operation. The angles of Ax(φ), Ay(ψ), and Dzr(−ω) were changed with a range of 10°. Each minimum value approximately lies in Ax(φ) = 18, Ay(ψ) = 26, Dzr(−ω) = −29, which are somewhat different from the set of values (20, 30, −27) in Fig. 7.4. This suggests that crystal structures are constructed to increase the stabilization by interactions with many molecules in the crystal.

Table 7.1 Interaction energies (kcal/mol) of (R, r)(R, r)-dimer of benzene through two-fold helix symmetry operation on dependence of a set of rotation angles (φ, ψ, ω) for Ax(φ), Ay(ψ), and Dzr(−ω) in Z-matrix of Fig. 7.13b
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4.3 One Molecule Surrounded by Twelve Neighbored Molecules

A spherical material is generally surrounded by other twelve ones, termed as the closest packing structure [22, 23]. Organic crystals may keep the same principle in spite of non-spherical shape of molecules. Namely, one organic molecule is surrounded by twelve neighbored organic molecules. Figure 7.15a shows an example of benzene crystal which belongs to monoclinic, space group P21/c [39]. Figure 7.15b depicts centroids of the molecules, indicating that a center molecule (0) is surrounded by twelve neighbored molecules through symmetry operations as follows (Fig. 7.15c). The translation operations transfer two equal molecules to the directions of up-and-down (1) and front-and-back (4), yielding totally four neighbored molecules. On the other hand, each of four two-fold helix operations (2, 2, 3, 3) transfer two equal molecules per a helical axis, yielding totally eight ones. In case of this space group P21/c, translation and reversion provide the same neighbored molecules. Of course, case by case, we need to check distances and contacts between the neighbored molecules, since organic molecules possess diverse structures.

Fig. 7.15
figure 15

A center molecule (hidden 0) surrounded by twelve neighbored molecules (a) and their centroids (b) through symmetry operations such as translation (trala) (1), two-fold helix (2) and (3), translation (4). Distances (nm) between centroids of molecules and interaction energies (kcal/mol) for benzene (d) and naphthalene (e)

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4.4 Intermolecular Distances and Interaction Energies

It is known that the dispersion force is mainly responsible for the attraction between aromatic compounds without substituents [24, 25]. Therefore, the interaction energies approximately depend on distances between neighbored molecules. Figure 7.15d contains the distances between centroids of the neighbored molecules as well as their interaction energies by using (φ, ψ, ω) = (20, 30, 27). It is reasonable that the operations (1, 2) along the identical axes afford more energies than those (3, 4) along the separated axes. Further separations of the molecules lead to the energies less than −0.5 kcal/mole.

This relation is observed in naphthalene [39] more clearly than in benzene [39] (Fig. 7.15e). The interaction energies for naphthalene is calculated by using (φ, ψ, ω) = (33, 21, 12). It is notable that the energies by operations (1, 2) in naphthalene are about two times more as compared to those in benzene. This means that the energies of dispersion force increase with increasing carbon numbers of organic molecules, when the intermolecular distances are almost equal.

It is generally considered that organic molecules are possible to have more dispersion energies with increasing carbon atoms. Accordingly, when neighbored molecules are closely located through the operations of translation (1) and two-fold helix (2) along the identical axis, they acquire maximal energies. In the case of pentacene, the energies amount to more or less −10 kcal/mol in the translation and two-fold helix along the axes, whose values are almost comparable or more to those of hydrogen bonds.

5 Hierarchical Structures and Interaction Energies

5.1 1D Column Surrounded by Six Columns

It is noteworthy that the conventional symmetry operations do not include translation itself, but the crystal units repeatedly transfer a molecule to yield translation assemblies. Such assemblies along an identical axis can exhibit short intermolecular distances, and obtain a large amount of interaction energies. Therefore, the following three ideas may be introduced, as illustrated in Fig. 7.16 regarding anthracene [39]. Firstly, the translation operations yield a one-dimensional (1D) column along an identical axis with a preferential amount of energies (Fig. 7.16a). Secondly, combinations of the translation with two-fold helix operations produce the most amount of interaction energies among various combinations to give the strongest 1D columns (Fig. 7.16b). Thirdly, one of the columns (C) is surrounded by six neighbored columns (I to VI) (Fig. 7.16c). Such a columnar model plays a key role for understanding organic crystals in comparison with the known spherical model of inorganic crystals.

Fig. 7.16
figure 16

Anthracene crystal (monoclinic, P21/a) consisting of 1D columnar assemblies. Column (C) by translation (trala) along crystallographic b-axis (a), column (C-I) by two-fold helix along b-axis (b), central column (C) surrounded by six neighbored columns (I to VI) as viewed down b-axis (c), another translation column as viewed down c-axis (d), Distances (nm) between centroids of the molecules and interaction energies (kcal/mol) by using (φ, ψ, ω) = (28, 23, 6) (e)

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5.2 2D Layers: Alignment of Columns by Maximal Interaction Energies

The resulting columns are bundled with other columns in parallel or anti-parallel through symmetry operations such as translation, two-fold helix, inversion and so on, to afford two-dimensional (2D) layers. In the case of anthracene (Fig. 7.16c), the centered column (C) forms three kinds of 2D layered alignments (I-C-IV, II-C-V, III-C-VI). Among them, the layer I-C-IV is the strongest, because it has a minimal intermolecular distances (0.60 nm (C), 0.52 nm (C-I or C-IV)) and maximal interaction energies (−5.5 kcal/mol (C), −7.2 kcal/mol (C-I or C-IV)) by using (φ, ψ, ω) = (28, 23, 6) (Fig. 7.16e).

It is generally considered that a combination of the columns with maximal interaction energies makes the most preferential layer among the above-mentioned three kinds of layers.

5.3 3D Stacked Layers Related to Space Group of Crystals

The layers are also stacked in various ways. The simplest way is observed by translation of the 2D layers in the monoclinic P21/c crystals of benzene, naphthalene and anthracene. This is exemplified as the side-view of the stacked layer in anthracene (Fig. 7.16d). It should be mentioned that parallel or perpendicular two-fold helix operation between two layers forms stacked layers with the same or reverse direction with space group P21/c (Fig. 7.17a) or P212121 (Fig. 7.17b), respectively. In contrast, inversion operation between two layers induces the racemic stacked layers in anti-parallel (Fig. 7.17c).

Fig. 7.17
figure 17

Three kinds of stacking for 2D layers consisting of two-fold helix 1D columns through symmetry operations of translation (a), two-fold helix (b), inversion (c), as their side-views of the 2D layers by using the rectangular triangle models. Broken arrows depict two-fold helix axes, while circles do inversion centers between layers

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In addition, these layers can slide each other between the layers, explaining the angles of lattice parameters which are apart from 90° between crystallographic axes. Namely, one directional sliding corresponds to monoclinic crystals, while two directional one does to triclinic crystals.

6 Supramolecular Chirality and Handedness in Crystals

6.1 Chiral Assemblies Composed of Identical Isomers

Supramolecular chirality [26, 27] is an important subject for understanding chiral materials in universe [28, 29]. Position-dependent chirality clearly indicates that identical molecules yield chiral assemblies through symmetry operations, including translation, two-, three-, four-, or six-fold rotation and helix. For example, (R, r)-isomers produce a 1D chiral column through translation. Such columns are bundled to a 2D chiral layer, and further the layers are stacked to a 3D chiral crystal through the operations. Simply, they are termed as right-handed (R, r)-column, (R, r)-layer, (R, r)-crystal (and vice versa).

In 2007, we reported that a two-fold helical molecular assembly of benzene exhibits right- or left-handedness [30, 31]. This was responsible for 3D geometry on display by molecular graphics Mercury. Thus, a side-view of a face affords a line. So, two-fold helical assembly of the faces composes an assembly of the lines. When the lines in front exhibit right-tilt alignment along the two-fold helical axis, the helical assembly is right-handed (and vice versa). Now, such a visual and qualitative description has developed to a quantitative one according to three kinds of rotations along a two-fold helical axis. These rotations bring about position-dependent chirality, explaining the well-known fact that supramolecular chirality generates in molecular assemblies composed of achiral molecules [32]. We can say that the conventional idea focused on molecular chirality, but did not on supramolecular chirality in crystals.

6.2 Achiral Assemblies of Enantiomeric and Diastereomeric Isomers

A combination of racemic (R, r)- and (S, s)-isomers forms an achiral (R, r)(S, s)-dimer through inversion, reflection or glide operation. The operations are repeated to form achiral 1D columns, which align toward achiral 2D layers, and further achiral 3D crystals.

The diastereomeric (R, r)(R, s)- or (S, s)(S, r)-dimer may be observed in organic crystals. For example, pentacene [39] form crystals which belong to space group P-1. It can be seen on display by Mercury that their dimers have parallel and T-type arrangements, indicating the same ω values of the rotations around z-axis in Fig. 7.5(c). As a result, the two-fold helix operations disappear and only inversion operations remain. Namely, pentacene employs (φ, ψ, ω) = (16, 28, 22) for three rotation angles and forms four kinds of position-dependent chiral isomers with (Dx, Dy, Dz) as follows; (−90, 90, −22) for (R, r), (−90, 90, 22) for (R, s), (90, −90, −22) for (S, r) and (90, −90, 22) for (S, s) (see Fig. 7.7d).

6.3 Diverse Diastereomers Regarding Position-Dependent Chirality

In principle, it is possible that diastereomeric isomers are combined to yield other achiral dimers, including an enantiomeric (R, s)(S, r)-dimer, a diastereomeric (R, r)(S, r)- or (S, s)(R, s)-dimer. Symmetry operations do not describe these dimers, and probably one might observe these dimers as two independent molecules in crystals.

It should be mentioned once more that the tilt rotation along y-axis in Fig. 7.5c is limited to forward tilt (0° < ψ < 90°) in this article. If necessary, this can be extended to backward tilt or reclination. This extension would make the discussion more complex than that mentioned above.

7 Chiral Crystallization of Achiral Molecules

7.1 New Understanding for Chiral Crystallization

From an isotropic viewpoint of symmetry theory, symmetry operations such as translation, two-fold rotation and two-fold helix do not enable us to discriminate handedness of the resultant assemblies. Nevertheless, from an anisotropic viewpoint, such assemblies exhibit chirality and handedness according to 3D space geometry by molecular graphics [14,15,16]. Furthermore, the latter anisotropic insight led us to the hierarchical structures in organic crystals [16,17,18] as well as the linkage between molecular and supramolecular chirality [33, 34].

The present article describes a new understanding for chiral crystallization of achiral molecules, telling that the triangle method reasonably explains generation and handedness of supramolecular chirality of the assemblies on the basis of position-dependent chirality. It still remains unclear to elucidate a relationship between molecular structures and space group of their crystals. Hereafter one can find the relation throughout a series of relative compounds, including polycyclic aromatic compounds, chalcones, and so on.

It is noteworthy that chiral crystallization forms an equimolar amount of (R)- and (S)-crystals. Their separation needs another insight. Recent crystal engineering presents a great possibility for their separation [35, 36].

7.2 Connection of Organic Molecules via Their Centroids

Benzene, naphthalene and anthracene are achiral organic molecules with intramolecular inversion points (Fig. 7.18a), producing no chiral crystals. However, achiral molecules without such inversion points have a great potential to undergo the chiral crystallization. Instead of the points, centroids of organic molecules are briefly determined by the graphics Mercury, and useful for connecting the X3 of the triangles.

Fig. 7.18
figure 18

Achiral molecules with an inner inversion center; anthracene (a), instead with a centroid; phenanthrene (b), picene (c), triphenylene (d), and benzo[c]phenanthrene (e)

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7.3 Examples of Chiral Crystallization of Achiral Molecules

We continue a research for resolving a relation between molecular and crystal structures. Among them, three examples of polycyclic aromatic compounds are described below.

The first example focuses on chiral crystals which belong to monoclinic, P21. Phenanthrene [39] has a centroid in the middle ring (Fig. 7.18b), where the dummy atom X3 is connected like anthracene. Figure 7.19a shows its (R, r)-crystal. The centered translation (R, r)-column (C) is surrounded by six (R, r)-columns (I to VI). The columnar alignment (I-C-IV) constitutes a preferential 2D (R, r)-layer, which is stacked by translation to yield a 3D (R, r)-crystal. This hierarchical structure is the same as that of anthracene. The preferential layer comes from different intermolecular distances (0.68, 0.51, 0.53 nm for C, C-I, C-IV compared to 0.95, 1.03 nm for C-II, C-III). Figure 7.19b involves another example of the achiral molecule without the inversion center, called picene (Fig. 7.18c) [39]. One can see the same hierarchical structure as phenanthrene.

Fig. 7.19
figure 19

Chiral crystals with space group P21; phenanthrene (a) and picene (b), with space group P212121; triphenylene (c), benzo(c)phenanthrene (d), phenanthridine (e) and 1,5-diiodonaphthalene (f). The central column (C) is surrounded by six columns (I to VI). Symmetry operation; two-fold helix ((●) and (–)), translation (⋯)

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The second examples deal with crystals which belong to orthorhombic, P212121. Triphenylene (Fig. 7.18d) [39] is illustrated in Fig. 7.19c. The centered translation column (C) is surrounded by six columns (I to VI), and the main layer is (I-C-V) instead of (I-C-IV). The difference lies in two kinds of two-fold helical axes between the layers, causing a rotation by 180° between the layers. The axis perpendicular to the layers ascribes to steric and electronic complementarity between the neighbored molecules. Figure 7.19d involves another example of the achiral molecule, called benzo(c)phenanthrene (Fig. 7.18e) [39], which displays the same hierarchical structure.

Thirdly, substituted aromatic compounds (orthorhombic, P212121) are employed. Phenanthridine [39] involves nitrogen at 6-position. The nitrogen functions for a connection of 1D translation column through hydrogen bonds to form a preferential 2D layer, which is stacked with two-fold helix operation (Fig. 7.19e). In addition, 1,5-diiodonaphthalene [39] has iodine for halogen bonds (Fig. 7.19f). The translation columns are combined by halogen bonds.

8 Conclusions and Perspectives

It has been found that supramolecular chirality of molecular assemblies in organic crystals is attributable for position-dependent chirality. We designed a rectangular triangle model attached with a molecule, enabling us to evaluate a position of organic molecules through three kinds of rotations toward an axis. The combination of the rotations leads us to four kinds of isomers with position-dependent chirality. The isomers are combined through symmetry operations, including translation, two-fold rotation, two-fold helix, and so on, to construct the corresponding molecular assemblies with supramolecular chirality.

Z -matrix, which is used for the Gaussian program, enables us to describe such a triangle model, leading to the evaluation of intermolecular interaction energies between neighbored molecules in organic crystals. The resulting interaction energies explain the hierarchical structures involving 1D columns, 2D layers, and 3D layer-stacked crystals, prompting us to understand space group of crystals as well as chiral crystallization of achiral molecules.

Dispersion energies contribute to such hierarchical structures. Basically, short distances between neighbored molecules play a key role in determining the columns and the layers. Hopefully, this research prompts us to elucidate the relation between molecular structures and space group of crystals, to dissolve various hidden chirality [37, 38], and to overcome diversity of organic compounds.