7 Supramolecular, Hierarchical, and Energetical Interpretation …
127
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
(12, 30, 27) −2.31
(20, 22, 27) −2.41
(20, 30, 21) −1.99
(14, 30, 27) −2.34
(20, 24, 27) −2.44
(20, 30, 23) −2.15
(16, 30, 27) −2.35
(20, 26, 27) −2.44
(20, 30, 25) −2.25
(18, 30, 27) −2.35
(20, 28, 27) −2.40
(20, 30, 27) −2.32
(20, 30, 27) −2.32
(20, 30, 27) −2.32
(20, 30, 29) −2.33
(22, 30, 27) −2.22
(20, 32, 27) −2.19
(20, 30, 31) −2.30
constructed to increase the stabilization by interactions with many molecules in the
crystal.
7.4.3 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 P2 1 /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
1 2
2
4
4
3
3
a
1
2
3
4
(trala) (2 1 -helix)
(2 1 -helix)
(trala)
Distance: 0.54
0.46
0.60
0.54
Energy:
-2.0
-2.3
-1.0
-2.0
Distance: 0.59
0.50
0.78
0.86
Energy:
-3.5
-4.1
-1.5
-0.31
(Benzene)
(Naphthalene)
b
c
d
e
Fig. 7.15 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)