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M. Miyata and S. Tsuzuki
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 P2 1 /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.
7.4.4 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.
7.5 Hierarchical Structures and Interaction Energies
7.5.1 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].
M. Miyata and S. Tsuzuki
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 P2 1 /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.
7.4.4 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.
7.5 Hierarchical Structures and Interaction Energies
7.5.1 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].