7 Supramolecular, Hierarchical, and Energetical Interpretation …
117
The other subject deals with hierarchical structures of organic crystals, such as 1D
columns, 2D layers, 3D stacked layers [16–18]. 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.
7.2 Rectangular Triangle Unit with a Molecule
7.2.1 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.
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
117
The other subject deals with hierarchical structures of organic crystals, such as 1D
columns, 2D layers, 3D stacked layers [16–18]. 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.
7.2 Rectangular Triangle Unit with a Molecule
7.2.1 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.
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