7 First-Principles Investigations of Electronically …
163
Here, the excitation energy of the Ith molecule E I is given as, E I =
I
L −
I
H +
2V HLHL − U HHLL , where V HLHL and U HHLL are the exchange and attractive e–h
interactions, corresponding the third and fourth terms in Eq. 7.9, respectively. An
off-diagonal element of the FE Hamiltonian, e I h I |H |e J h J , is known as an excitonic
coupling and is responsible for excitation energy transfer and dispersion of the FE
state. The excitonic coupling can be divided into the long-range Coulomb and shortrange exchange interactions as e I h I |H |e J h J = V
C
+ V
Ex [36, 58]. For example,
the long-range Coulomb coupling is given by
V
C
= ∫ d r 1 ∫ d r 2
ρ
(eg)∗
I
(r 1 , r 1 )ρ
(eg)
J (r 2 , r 2 )
|r 1 − r 2 |
,
(7.12)
Here, ρ
(eg)
I
denotes the transition density between the ground and excited states,
ρ
(eg)
I
= ψ
I
H (r)ψ
I ∗
L (r). V
C can be well approximated as a dipole-dipole interaction, V
C
= (µ I · µ J − 3(µ I · e R )(µ J · e R ))/R
3
IJ , in long molecular distances,
where μ I is the transition dipole moment between the ground and excited states,
µ I = − ∫ rρ
(eg)
I
(r, r). By contrast, V
Ex describes the short-range exchange interaction. In most organic systems, the long-range interaction is much stronger than
the exchange interaction. V
C vanishes for triplet excited states, with only V
Ex
determining the triplet exciton diffusion.
Although the FE state can be regarded as being delocalized over multiple
molecules, the mechanism of the delocalization should be distinguished from that
of Bloch orbitals. The FE state is formed by the long-range Coulomb interactions
between transition densities of organic molecules, without the wave function overlaps. The FE state can also be denoted as a collective excited state, in which multiple
e–h configurations are involved in the electronic excitation. In J-aggregates [117],
for example, FE states emerge as the J-band characterized by red-shifted absorption
spectrum, larger oscillator strength, narrow bandwidth, and small stokes shifts [54,
102]. An FE state can also show super radiant decay, a coherence enhancement of
the radiative decay rate, which was reported by Dicke [27].
Another limit of delocalized excited states is the Wannier-Mott (WM) exciton
state (Fig. 7.3d). The WM exciton states can be derived when the e–h interaction
terms are much smaller than the transfer integrals. If the e–h interaction terms can be
treated perturbatively, the excited-state wave function is written as a direct product
of the eigenfunctions of the HOMO-derived state and the LUMO-derived state as
Ψ W M = |ψ LDS ⊗ |ψ HDS ,
(7.13)
where |ψ LDS or |ψ HDS is an eigenfunction of H
e or H
h . In a periodic system, the
WM Hamiltonian can be rewritten in the same form as that of a hydrogen atom. The
diagonalization of the one-body Hamiltonian under periodic boundary conditions
provides the energy-band dispersion, e.g., H
h
ψ k h = E(k h )
ψ k h . Using the onebody energies (e.g., E(k h ) k
2
h /2m h ) and neglecting the e–h exchange term, we can
reduce the Hamiltonian to:
163
Here, the excitation energy of the Ith molecule E I is given as, E I =
I
L −
I
H +
2V HLHL − U HHLL , where V HLHL and U HHLL are the exchange and attractive e–h
interactions, corresponding the third and fourth terms in Eq. 7.9, respectively. An
off-diagonal element of the FE Hamiltonian, e I h I |H |e J h J , is known as an excitonic
coupling and is responsible for excitation energy transfer and dispersion of the FE
state. The excitonic coupling can be divided into the long-range Coulomb and shortrange exchange interactions as e I h I |H |e J h J = V
C
+ V
Ex [36, 58]. For example,
the long-range Coulomb coupling is given by
V
C
= ∫ d r 1 ∫ d r 2
ρ
(eg)∗
I
(r 1 , r 1 )ρ
(eg)
J (r 2 , r 2 )
|r 1 − r 2 |
,
(7.12)
Here, ρ
(eg)
I
denotes the transition density between the ground and excited states,
ρ
(eg)
I
= ψ
I
H (r)ψ
I ∗
L (r). V
C can be well approximated as a dipole-dipole interaction, V
C
= (µ I · µ J − 3(µ I · e R )(µ J · e R ))/R
3
IJ , in long molecular distances,
where μ I is the transition dipole moment between the ground and excited states,
µ I = − ∫ rρ
(eg)
I
(r, r). By contrast, V
Ex describes the short-range exchange interaction. In most organic systems, the long-range interaction is much stronger than
the exchange interaction. V
C vanishes for triplet excited states, with only V
Ex
determining the triplet exciton diffusion.
Although the FE state can be regarded as being delocalized over multiple
molecules, the mechanism of the delocalization should be distinguished from that
of Bloch orbitals. The FE state is formed by the long-range Coulomb interactions
between transition densities of organic molecules, without the wave function overlaps. The FE state can also be denoted as a collective excited state, in which multiple
e–h configurations are involved in the electronic excitation. In J-aggregates [117],
for example, FE states emerge as the J-band characterized by red-shifted absorption
spectrum, larger oscillator strength, narrow bandwidth, and small stokes shifts [54,
102]. An FE state can also show super radiant decay, a coherence enhancement of
the radiative decay rate, which was reported by Dicke [27].
Another limit of delocalized excited states is the Wannier-Mott (WM) exciton
state (Fig. 7.3d). The WM exciton states can be derived when the e–h interaction
terms are much smaller than the transfer integrals. If the e–h interaction terms can be
treated perturbatively, the excited-state wave function is written as a direct product
of the eigenfunctions of the HOMO-derived state and the LUMO-derived state as
Ψ W M = |ψ LDS ⊗ |ψ HDS ,
(7.13)
where |ψ LDS or |ψ HDS is an eigenfunction of H
e or H
h . In a periodic system, the
WM Hamiltonian can be rewritten in the same form as that of a hydrogen atom. The
diagonalization of the one-body Hamiltonian under periodic boundary conditions
provides the energy-band dispersion, e.g., H
h
ψ k h = E(k h )
ψ k h . Using the onebody energies (e.g., E(k h ) k
2
h /2m h ) and neglecting the e–h exchange term, we can
reduce the Hamiltonian to: