7 Supramolecular, Hierarchical, and Energetical Interpretation …
125
Translation
(R,r)-1 to (R,r)-2
X7 1 Bc 2 90 3 Af1
X8 7 Ba 1 90 2 0
X9 8 Bb 7 90 1 -Af2
X10 9 1.0 8 φ 7 -90
X11 9 1.0 10 ψ 8 90
X12 9 1.0 11 90 10 -ω
Two-fold helix
(R,r)-1 to (R,r)-2
X7 2 Bc 1 90 3 Af1
X8 7 0.5Ba 2 90 1 180
X9 8 Bb 7 90 2 -Af1
X10 9 1.0 8 φ 7 -90
X11 9 1.0 10 ψ 8 90
X12 9 1.0 11 90 10 -ω
1
2
1
2
a
b
Fig. 7.12 Connection of two triangular triangles with symmetry operations: translation (a) and
two-fold helix (b) through the separating axes
(R,r)(R,r)-Dimer
through translation
X1
X2 1 Ba
X3 2 Bb 1 90
X4 3 1.0 2 Ax 1 -90
X5 3 1.0 4 Ay 2 90
X6 3 1.0 5 90 4 Dzr
C1 3 R1 5 90 6 0 0 1
C2 3 R1 5 90 6 60 0 1
C3 3 R1 5 90 6 120 0 1
C4 3 R1 5 90 6 180 0 1
C5 3 R1 5 90 6 240 0 1
C6 3 R1 5 90 6 300 0 1
H1 3 R2 5 90 6 0 0 1
H2 3 R2 5 90 6 60 0 1
H3 3 R2 5 90 6 120 0 1
H4 3 R2 5 90 6 180 0 1
H5 3 R2 5 90 6 240 0 1
H6 3 R2 5 90 6 300 0 1
X7 3 Bb 4 Ax 2 0
X8 2 Ba 3 90 1 180
X9 20 Bb 19 90 3 0
X10 21 1.0 20 Ax 19 -90
X11 21 1.0 22 Ay 20 90
X12 21 1.0 23 90 22 Dzr
C7 21 R1 23 90 24 0 0 2
C8 21 R1 23 90 24 60 0 2
(Abbreviated)
R1 1.394
R2 2.475
Ba 5.376
Bb 1.883
Ax 20
Ay 30
Dzr -27
(R,r)(R,r)-Dimer
through two-fold helix
X1
X2 1 Ba
X3 2 Bb 1 90
X4 3 1.0 2 Ax 1 -90
X5 3 1.0 4 Ay 2 90
X6 3 1.0 5 90 4 Dzr
C1 3 R1 5 90.0 6 0 0 1
C2 3 R1 5 90.0 6 60 0 1
C3 3 R1 5 90.0 6 120 0 1
C4 3 R1 5 90.0 6 180 0 1
C5 3 R1 5 90.0 6 240 0 1
C6 3 R1 5 90.0 6 300 0 1
H1 3 R2 5 90.0 6 0 0 1
H2 3 R2 5 90.0 6 60 0 1
H3 3 R2 5 90.0 6 120 0 1
H4 3 R2 5 90.0 6 180 0 1
H5 3 R2 5 90.0 6 240 0 1
H6 3 R2 5 90.0 6 300 0 1
X7 2 Ba2 3 90 1 0
X8 2 Ba2 3 90 1 180
X9 20 Bb 19 90 3 180
X10 21 1.0 20 Ax 19 -90
X11 21 1.0 22 Ay 20 90
X12 21 1.0 23 90 22 Dzr
C7 21 R1 23 90 24 0 0 2
C8 21 R1 23 90 24 60 0 2
(Abbreviated)
R1 1.394
R2 2.475
Ba 5.376
Ba2 2.688
Bb 1.883
Ax 20
Ay 30
Dzr -27
(R,r)(S,s)-Dimer
through inversion
X1
X2 1 Ba
X3 2 Bb 1 90.0
X4 3 1.0 2 Ax 1 -90
X5 3 1.0 4 Ay 2 90
X6 3 1.0 5 90 4 Dzr
C1 3 R1 5 90 6 0 0 1
C2 3 R1 5 90 7 60 0 1
C3 3 R1 5 90 8 60 0 1
C4 3 R1 5 90 9 60 0 1
C5 3 R1 5 90 10 60 0 1
C6 3 R1 5 90 11 60 0 1
H1 3 R2 5 90 7 0 0 1
H2 3 R2 5 90 8 0 0 1
H3 3 R2 5 90 9 0 0 1
H4 3 R2 5 90 10 0 0 1
H5 3 R2 5 90 11 0 0 1
H6 3 R2 5 90 12 0 0 1
X7 3 Ba 2 90 1 180
X8 19 Bb 3 90 2 180
X9 20 Ba 19 90 3 180
X10 19 1.0 20 Ax 21 90
X11 19 1.0 22 Ay 20 -90
X12 19 1.0 23 90 22 Dzs
C7 19 R1 23 90 24 0 0 2
C8 19 R1 23 90 25 60 0 2
(Abbreviated)
R1 1.394
R2 2.475
Ba 5.376
Bb 1.883
Ax 20
Ay 30
Dzr -27
Dzs 27
a
b
c
Fig. 7.13 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)
125
Translation
(R,r)-1 to (R,r)-2
X7 1 Bc 2 90 3 Af1
X8 7 Ba 1 90 2 0
X9 8 Bb 7 90 1 -Af2
X10 9 1.0 8 φ 7 -90
X11 9 1.0 10 ψ 8 90
X12 9 1.0 11 90 10 -ω
Two-fold helix
(R,r)-1 to (R,r)-2
X7 2 Bc 1 90 3 Af1
X8 7 0.5Ba 2 90 1 180
X9 8 Bb 7 90 2 -Af1
X10 9 1.0 8 φ 7 -90
X11 9 1.0 10 ψ 8 90
X12 9 1.0 11 90 10 -ω
1
2
1
2
a
b
Fig. 7.12 Connection of two triangular triangles with symmetry operations: translation (a) and
two-fold helix (b) through the separating axes
(R,r)(R,r)-Dimer
through translation
X1
X2 1 Ba
X3 2 Bb 1 90
X4 3 1.0 2 Ax 1 -90
X5 3 1.0 4 Ay 2 90
X6 3 1.0 5 90 4 Dzr
C1 3 R1 5 90 6 0 0 1
C2 3 R1 5 90 6 60 0 1
C3 3 R1 5 90 6 120 0 1
C4 3 R1 5 90 6 180 0 1
C5 3 R1 5 90 6 240 0 1
C6 3 R1 5 90 6 300 0 1
H1 3 R2 5 90 6 0 0 1
H2 3 R2 5 90 6 60 0 1
H3 3 R2 5 90 6 120 0 1
H4 3 R2 5 90 6 180 0 1
H5 3 R2 5 90 6 240 0 1
H6 3 R2 5 90 6 300 0 1
X7 3 Bb 4 Ax 2 0
X8 2 Ba 3 90 1 180
X9 20 Bb 19 90 3 0
X10 21 1.0 20 Ax 19 -90
X11 21 1.0 22 Ay 20 90
X12 21 1.0 23 90 22 Dzr
C7 21 R1 23 90 24 0 0 2
C8 21 R1 23 90 24 60 0 2
(Abbreviated)
R1 1.394
R2 2.475
Ba 5.376
Bb 1.883
Ax 20
Ay 30
Dzr -27
(R,r)(R,r)-Dimer
through two-fold helix
X1
X2 1 Ba
X3 2 Bb 1 90
X4 3 1.0 2 Ax 1 -90
X5 3 1.0 4 Ay 2 90
X6 3 1.0 5 90 4 Dzr
C1 3 R1 5 90.0 6 0 0 1
C2 3 R1 5 90.0 6 60 0 1
C3 3 R1 5 90.0 6 120 0 1
C4 3 R1 5 90.0 6 180 0 1
C5 3 R1 5 90.0 6 240 0 1
C6 3 R1 5 90.0 6 300 0 1
H1 3 R2 5 90.0 6 0 0 1
H2 3 R2 5 90.0 6 60 0 1
H3 3 R2 5 90.0 6 120 0 1
H4 3 R2 5 90.0 6 180 0 1
H5 3 R2 5 90.0 6 240 0 1
H6 3 R2 5 90.0 6 300 0 1
X7 2 Ba2 3 90 1 0
X8 2 Ba2 3 90 1 180
X9 20 Bb 19 90 3 180
X10 21 1.0 20 Ax 19 -90
X11 21 1.0 22 Ay 20 90
X12 21 1.0 23 90 22 Dzr
C7 21 R1 23 90 24 0 0 2
C8 21 R1 23 90 24 60 0 2
(Abbreviated)
R1 1.394
R2 2.475
Ba 5.376
Ba2 2.688
Bb 1.883
Ax 20
Ay 30
Dzr -27
(R,r)(S,s)-Dimer
through inversion
X1
X2 1 Ba
X3 2 Bb 1 90.0
X4 3 1.0 2 Ax 1 -90
X5 3 1.0 4 Ay 2 90
X6 3 1.0 5 90 4 Dzr
C1 3 R1 5 90 6 0 0 1
C2 3 R1 5 90 7 60 0 1
C3 3 R1 5 90 8 60 0 1
C4 3 R1 5 90 9 60 0 1
C5 3 R1 5 90 10 60 0 1
C6 3 R1 5 90 11 60 0 1
H1 3 R2 5 90 7 0 0 1
H2 3 R2 5 90 8 0 0 1
H3 3 R2 5 90 9 0 0 1
H4 3 R2 5 90 10 0 0 1
H5 3 R2 5 90 11 0 0 1
H6 3 R2 5 90 12 0 0 1
X7 3 Ba 2 90 1 180
X8 19 Bb 3 90 2 180
X9 20 Ba 19 90 3 180
X10 19 1.0 20 Ax 21 90
X11 19 1.0 22 Ay 20 -90
X12 19 1.0 23 90 22 Dzs
C7 19 R1 23 90 24 0 0 2
C8 19 R1 23 90 25 60 0 2
(Abbreviated)
R1 1.394
R2 2.475
Ba 5.376
Bb 1.883
Ax 20
Ay 30
Dzr -27
Dzs 27
a
b
c
Fig. 7.13 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)