124
3 Fundamentals of the Analysis Tools
the total energy per unit cell E/N and the Mulliken populations, for instance, can be
obtained from the final total density matrix element. In the optimization process of the
1D polymer, this E/N value is minimized with respect to all the atomic nuclei in the
unit cell and the translation length. In the calculation method by the DFT framework,
the basic scheme to obtain the CO’s is essentially the same and the prescription of
solving the Fock equations is changed into the KS equations mentioned in Sect. 3.2.
It is further noted that the CO ψ s (k, x) is a complex function, in general, having
an imaginary part and, hence, that it is useless to draw their patterns unlike those of
MO’s being real function as seen, e.g., in Sect. 2.2. However, the highest occupied
CO (HOCO) and the lowest unoccupied CO (LUCO) appear mostly at k = 0 or ±
π
a
,
and any CO’s become real functions there, one can check the HOCO and the LUCO
patterns in most cases. In Fig. 3.11, those patterns of polyacetylene and polythiophene
are shown, for instance.
HOCO (π type)
LUCO (π type)
(a)
n
LUCO (π type)
HOCO (π type)
(b)
S
S
n
Fig. 3.11 The HOCO and the LUCO patterns of a polyacetylene and b polythiophene. They are
drawn only in the unit cells. All of these are of π types
3 Fundamentals of the Analysis Tools
the total energy per unit cell E/N and the Mulliken populations, for instance, can be
obtained from the final total density matrix element. In the optimization process of the
1D polymer, this E/N value is minimized with respect to all the atomic nuclei in the
unit cell and the translation length. In the calculation method by the DFT framework,
the basic scheme to obtain the CO’s is essentially the same and the prescription of
solving the Fock equations is changed into the KS equations mentioned in Sect. 3.2.
It is further noted that the CO ψ s (k, x) is a complex function, in general, having
an imaginary part and, hence, that it is useless to draw their patterns unlike those of
MO’s being real function as seen, e.g., in Sect. 2.2. However, the highest occupied
CO (HOCO) and the lowest unoccupied CO (LUCO) appear mostly at k = 0 or ±
π
a
,
and any CO’s become real functions there, one can check the HOCO and the LUCO
patterns in most cases. In Fig. 3.11, those patterns of polyacetylene and polythiophene
are shown, for instance.
HOCO (π type)
LUCO (π type)
(a)
n
LUCO (π type)
HOCO (π type)
(b)
S
S
n
Fig. 3.11 The HOCO and the LUCO patterns of a polyacetylene and b polythiophene. They are
drawn only in the unit cells. All of these are of π types