7 First-Principles Investigations of Electronically …
175
large HOMO-LUMO gap reduction from the gas to the condensed phases is consistent
with the previous findings [68].
As discussed in Sect. 7.2.1, the MO energy change can be understood in terms of
the polarization energy. Here, we introduce the polarization energies as:
(N )
LUMO =
(N =1)
LUMO − P
−
,
(7.30)
(N )
HOMO =
(N =1)
HOMO + P
+
.
(7.31)
where
N denotes the average HOMO or LUMO energy of the three PEN molecules in
the (PEN) N cluster. As explained in Sect. 1.2.1 the ES and IP effects contribute to the
total polarization energy. Thus, the calculated polarization energies were decomposed
into the ES and IP contributions as follows:
P
+/−
= P
+/−
ES + P
+/−
IP .
(7.32)
Here, P ES represents the ES contribution of the polarization energies and is approximately described as the interaction between positive (or negative) charge and the
quadrupole moments of surrounding molecules. On the other hand, P IP indicates
the interaction between the charge and the induced dipole moments of surrounding
molecules. The calculated ES and IP polarization energies are shown Fig. 7.6c, d,
respectively. Here, the plus signs on P
+ /P
− indicate that the HOMO/LUMO energies are increased/decreased compared to the gas-phase values. The ES contributions
tend to increase both the HOMO and LUMO levels. By contrast, the IP contribution increases and decreases the HOMO and LUMO levels, respectively. Thus, the
IP effect is predominantly responsible for the HOMO–LUMO gap reduction in the
PEN clusters. This finding is consistent with the earlier experimental suggestions
that the IP terms are primarily responsible for the HOMO–LUMO gap reduction,
whereas the ES terms play a decisive role in the orientation dependence [118, 123].
We found a difference between the anion and cation polarization energies. Yamada
et al. [118] attempted to decompose experimental polarization energies into ES and
IP terms by assuming that magnitudes of the cation and anion energies are identical,
P
+
ES = −P
−
ES and P
−
IP = P
+
IP . By contrast, according to our calculations, the polarization energy of the anion state is larger than that of the cation state. The absolute
values of P
−
ES and P
−
IP are larger than P
+
ES and P
+
IP by 137 and 72 meV, respectively,
in the N = 33 cluster. This discrepancy may be attributed to the larger spatial extent
of the LUMO compared to the HOMO.
Next, we turn to the polarization energies of localized excited states. We consider
the S 1 excited state of a PEN molecule in the cluster as an LE state. The CT states
were defined as the intermolecular HOMO → LUMO transition. The computed
average energies of the LE and CT states as a function of cluster size are shown in
Fig. 7.7. Here, the LE energy was obtained as the average of the three PEN molecules,
whereas the CT energies were obtained as the average of six intermolecular CT states
within the three PEN molecules. The excitation energies of the LE states increase
175
large HOMO-LUMO gap reduction from the gas to the condensed phases is consistent
with the previous findings [68].
As discussed in Sect. 7.2.1, the MO energy change can be understood in terms of
the polarization energy. Here, we introduce the polarization energies as:
(N )
LUMO =
(N =1)
LUMO − P
−
,
(7.30)
(N )
HOMO =
(N =1)
HOMO + P
+
.
(7.31)
where
N denotes the average HOMO or LUMO energy of the three PEN molecules in
the (PEN) N cluster. As explained in Sect. 1.2.1 the ES and IP effects contribute to the
total polarization energy. Thus, the calculated polarization energies were decomposed
into the ES and IP contributions as follows:
P
+/−
= P
+/−
ES + P
+/−
IP .
(7.32)
Here, P ES represents the ES contribution of the polarization energies and is approximately described as the interaction between positive (or negative) charge and the
quadrupole moments of surrounding molecules. On the other hand, P IP indicates
the interaction between the charge and the induced dipole moments of surrounding
molecules. The calculated ES and IP polarization energies are shown Fig. 7.6c, d,
respectively. Here, the plus signs on P
+ /P
− indicate that the HOMO/LUMO energies are increased/decreased compared to the gas-phase values. The ES contributions
tend to increase both the HOMO and LUMO levels. By contrast, the IP contribution increases and decreases the HOMO and LUMO levels, respectively. Thus, the
IP effect is predominantly responsible for the HOMO–LUMO gap reduction in the
PEN clusters. This finding is consistent with the earlier experimental suggestions
that the IP terms are primarily responsible for the HOMO–LUMO gap reduction,
whereas the ES terms play a decisive role in the orientation dependence [118, 123].
We found a difference between the anion and cation polarization energies. Yamada
et al. [118] attempted to decompose experimental polarization energies into ES and
IP terms by assuming that magnitudes of the cation and anion energies are identical,
P
+
ES = −P
−
ES and P
−
IP = P
+
IP . By contrast, according to our calculations, the polarization energy of the anion state is larger than that of the cation state. The absolute
values of P
−
ES and P
−
IP are larger than P
+
ES and P
+
IP by 137 and 72 meV, respectively,
in the N = 33 cluster. This discrepancy may be attributed to the larger spatial extent
of the LUMO compared to the HOMO.
Next, we turn to the polarization energies of localized excited states. We consider
the S 1 excited state of a PEN molecule in the cluster as an LE state. The CT states
were defined as the intermolecular HOMO → LUMO transition. The computed
average energies of the LE and CT states as a function of cluster size are shown in
Fig. 7.7. Here, the LE energy was obtained as the average of the three PEN molecules,
whereas the CT energies were obtained as the average of six intermolecular CT states
within the three PEN molecules. The excitation energies of the LE states increase