2.1 Molecular Structure
9
Table 2.1 Bond lengths and angles of C 60 and POSS
Bond lengths and
angles a
X-ray diffraction b
Neutron diffraction c
Theoretical
calculation d
C 60
C–C
1.455
1.46
1.453
C=C
1.391
1.39
1.396
POSS
Si–O
1.619
1.623–1.626
1.640
Si–H
1.450
1.459–1.463
1.460
Si–O–Si
147.5
147.25–147.45
148.2
O–Si–O
109.6
109.14–109.53
109.6
O–Si–H
109.5
109.07–109.77
109.3
a Bond lengths in Å and angles in degrees
b C 60 by David et al. (1991) and POSS by Larsson (1960)
c C 60 by Li et al. (1991) and POSS by Törnroos (1994)
d C 60 by the author and POSS by Mattori et al. (2000). Both the calculations by DFT/B3LYP/6-31G**
to the Si atoms (Zhen et al. 2009). Detailed structural data for POSS is also listed in
Table 2.1.
The above optimization procedure of molecular structures is further applicable
to the one-dimensional (1D) polymers as to their primary structures as illustrated in
Fig. 2.7. This system is peculiar in that it is constructed with imposing the periodic
boundary condition for the whole alignment of the unit cells providing crystal orbitals
(CO’s) instead of MO’s. Details of the CO calculations for 1D polymer established
on this condition are to be explained in Sect. 3.3. The calculation method is based
on the CO concept being somewhat different from that of the usual MO. Note that
in the structural optimization process the 1D polymer has one more freedom, that
is, the translation length of the unit cell in addition to the numbers of the coordinate
variables 3N, where N signifies the number of atoms involved in the unit cell. Note
that the CO calculation is also possible for the two- and three-dimensional (2D, 3D)
polymers, such as graphene and graphite crystals, respectively.
H
H
H
H
H
H
S
S
H
H
H
H
(a)
(b)
1.091 Å
1.368 Å
1.428 Å
Translation length: 2.474 Å
Translation length: 7.847 Å
1.413 Å
1.757 Å
1.443 Å
1.382 Å
1.085 Å
Fig. 2.7 Polymer skeletons of a polyacetylene and b polythiophene optimized by DFTCO/B3LYP/6-31G**. The bracketed indicate the unit cells with the translation lengths
9
Table 2.1 Bond lengths and angles of C 60 and POSS
Bond lengths and
angles a
X-ray diffraction b
Neutron diffraction c
Theoretical
calculation d
C 60
C–C
1.455
1.46
1.453
C=C
1.391
1.39
1.396
POSS
Si–O
1.619
1.623–1.626
1.640
Si–H
1.450
1.459–1.463
1.460
Si–O–Si
147.5
147.25–147.45
148.2
O–Si–O
109.6
109.14–109.53
109.6
O–Si–H
109.5
109.07–109.77
109.3
a Bond lengths in Å and angles in degrees
b C 60 by David et al. (1991) and POSS by Larsson (1960)
c C 60 by Li et al. (1991) and POSS by Törnroos (1994)
d C 60 by the author and POSS by Mattori et al. (2000). Both the calculations by DFT/B3LYP/6-31G**
to the Si atoms (Zhen et al. 2009). Detailed structural data for POSS is also listed in
Table 2.1.
The above optimization procedure of molecular structures is further applicable
to the one-dimensional (1D) polymers as to their primary structures as illustrated in
Fig. 2.7. This system is peculiar in that it is constructed with imposing the periodic
boundary condition for the whole alignment of the unit cells providing crystal orbitals
(CO’s) instead of MO’s. Details of the CO calculations for 1D polymer established
on this condition are to be explained in Sect. 3.3. The calculation method is based
on the CO concept being somewhat different from that of the usual MO. Note that
in the structural optimization process the 1D polymer has one more freedom, that
is, the translation length of the unit cell in addition to the numbers of the coordinate
variables 3N, where N signifies the number of atoms involved in the unit cell. Note
that the CO calculation is also possible for the two- and three-dimensional (2D, 3D)
polymers, such as graphene and graphite crystals, respectively.
H
H
H
H
H
H
S
S
H
H
H
H
(a)
(b)
1.091 Å
1.368 Å
1.428 Å
Translation length: 2.474 Å
Translation length: 7.847 Å
1.413 Å
1.757 Å
1.443 Å
1.382 Å
1.085 Å
Fig. 2.7 Polymer skeletons of a polyacetylene and b polythiophene optimized by DFTCO/B3LYP/6-31G**. The bracketed indicate the unit cells with the translation lengths