We theoretically investigated the effect of mixed Frenkel (F) and charge transfer (CT) states on the spectral properties of perylene bisimide (PBI) derivatives, focusing on the role of strong electron–phonon interactions. The model consists of a four-level system described by the Holstein Hamiltonian coupled to independent local heat-baths on each site, described by Brownian spectral distribution functions. We employ the reduced hierarchical equations of motion (HEOM) approach to calculate the time evolution of the system and compare it to the pure F exciton cases. We compute the absorption and time-gated fluorescence (TGF) spectra for different exciton transfer integrals and F-CT bandgap conditions. The coherence length of excitons (Ncoh) is evaluated employing two different definitions. We observe the presence of an excited hot state peak whose intensity is associated with the delocalization of the excited species and ultrafast dynamics that are solely dependent on the frequency of the local bath. The results indicate that the inclusion of CT states promotes localization of the excitons, which is manifested in a decrease in the intensity of the hot state peak and the 0–1 peak and an increase in the intensity of the 0–0 emission peak in the TGF spectrum, leading to a decrease of Ncoh.

1.
G.
Horowitz
, “
Organic field-effect transistors
,”
Adv. Mater.
10
,
365
377
(
1998
).
2.
C. D.
Dimitrakopoulos
and
P. R. L.
Malenfant
, “
Organic thin film transistors for large area electronics
,”
Adv. Mater.
14
,
99
117
(
2002
).
3.
J.
Zhang
,
H. S.
Tan
,
X.
Guo
,
A.
Facchetti
, and
H.
Yan
, “
Material insights and challenges for non-fullerene organic solar cells based on small molecular acceptors
,”
Nat. Energy
3
,
720
731
(
2018
).
4.
K.
Sato
,
J.
Mizuguchi
,
Y.
Sakai
, and
S.
Aramaki
, “
Crystal structure of parallel-stacked peryleneimides and their application to organic field-effect transistor devices
,”
J. Appl. Phys.
103
,
013702
(
2008
).
5.
C.
Li
,
J. H.
Yum
,
S. J.
Moon
,
A.
Herrmann
,
F.
Eickemeyer
,
N. G.
Pschirer
,
P.
Erk
,
J.
Schöneboom
,
K.
Müllen
,
M.
Grätzel
, and
M. K.
Nazeeruddin
, “
An improved perylene sensitizer for solar cell applications
,”
ChemSusChem
1
,
615
618
(
2008
).
6.
U.
Gómez
,
M.
Leonhardt
,
H.
Port
, and
H. C.
Wolf
, “
Optical properties of amorphous ultrathin films of perylene derivatives
,”
Chem. Phys. Lett.
268
,
1
6
(
1997
).
7.
J.
Mizuguchi
and
K.
Tojo
, “
Electronic structure of perylene pigments as viewed from the crystal structure and excitonic interactions
,”
J. Phys. Chem. B
106
,
767
772
(
2002
).
8.
Z. G.
Soos
,
M. H.
Hennessy
, and
G.
Wen
, “
Frenkel and charge transfer states of conjugated polymers and molecules
,”
Chem. Phys.
227
,
19
32
(
1998
).
9.
M. H.
Hennessy
,
Z. G.
Soos
,
R. A.
Pascal
, and
A.
Girlando
, “
Vibronic structure of PTCDA stacks: The exciton phonon-charge-transfer dimer
,”
Chem. Phys.
245
,
199
212
(
1999
).
10.
T.
Fujita
,
S.
Atahan-Evrenk
,
N. P. D.
Sawaya
, and
A.
Aspuru-Guzik
, “
Coherent dynamics of mixed Frenkel and charge-transfer excitons in dinaphtho[2,3-b:2′3′-f]thieno[3,2-b]-thiophene thin films: The importance of hole delocalization
,”
J. Phys. Chem. Lett.
7
,
1374
1380
(
2016
).
11.
W.
Popp
,
M.
Polkehn
,
K. H.
Hughes
,
R.
Martinazzo
, and
I.
Burghardt
, “
Vibronic coupling models for donor-acceptor aggregates using an effective-mode scheme: Application to mixed Frenkel and charge-transfer excitons in oligothiophene aggregates
,”
J. Chem. Phys.
150
,
244114
(
2019
).
12.
S.
Jiang
,
Y.
Xie
, and
Z.
Lan
, “
The role of the charge-transfer states in the ultrafast excitonic dynamics of the DTDCTB dimers embedded in a crystal environment
,”
Chem. Phys.
515
,
603
613
(
2018
).
13.
M.
Hoffmann
and
Z. G.
Soos
, “
Optical absorption spectra of the Holstein molecular crystal for weak and intermediate electronic coupling
,”
Phys. Rev. B
66
,
024305
(
2002
).
14.
M.
Hoffmann
,
K.
Schmidt
,
T.
Fritz
,
T.
Hasche
,
V. M.
Agranovich
, and
K.
Leo
, “
The lowest energy Frenkel and charge-transfer excitons in quasi-one-dimensional structures: Application to MePTCDI and PTCDA crystals
,”
Chem. Phys.
258
,
73
96
(
2000
).
15.
J.-L.
Brédas
,
J. E.
Norton
,
J.
Cornil
, and
V.
Coropceanu
, “
Molecular understanding of organic solar cells: The challenges
,”
Acc. Chem. Res.
42
,
1691
1699
(
2009
).
16.
S.
Kang
,
C.
Kaufmann
,
Y.
Hong
 et al, “
Ultrafast coherent exciton dynamics in size controlled perylene bisimide aggregates
,”
Struct. Dyn.
6
,
064501
(
2019
).
17.
C.
Kaufmann
,
W.
Kim
,
A.
Nowak-Król
,
Y.
Hong
,
D.
Kim
, and
F.
Würthner
, “
Ultrafast exciton delocalization, localization, and excimer formation dynamics in a highly defined perylene bisimide quadruple π-stack
,”
J. Am. Chem. Soc.
140
,
4253
4258
(
2018
).
18.
W.
Poppa
,
M.
Polkehn
,
R.
Binder
, and
I.
Burghardt
, “
Coherent charge transfer exciton formation in regioregular P3HT: A quantum dynamical study
,”
J. Phys. Chem. Lett.
10
,
3326
3332
(
2019
).
19.
N. J.
Hestand
and
F. C.
Spano
, “
Expanded theory of H- and J-molecular aggregates: The effects of vibronic coupling and intermolecular charge transfer
,”
Chem. Rev.
118
,
7069
7163
(
2018
).
20.
F. C.
Spano
, “
Excitons in conjugated oligomer aggregates, films, and crystals
,”
Annu. Rev. Phys. Chem.
57
,
217
243
(
2006
).
21.
K.
Sun
,
X.
Liu
,
W.
Hu
,
M.
Zhang
,
G.
Long
, and
Y.
Zhao
, “
Singlet fission dynamics and optical spectra of pentacene and its derivatives
,”
Phys. Chem. Chem. Phys.
23
,
12654
12667
(
2021
).
22.
K.
Sun
and
Y.
Zhao
, “
Beating maps of singlet fission: Simulation of coherent two-dimensional electronic spectroscopy by Davydov ansatz in organic molecules
,”
J. Chem. Phys.
147
,
224905
(
2017
).
23.
G.
Yang
,
N.
Wu
,
T.
Chen
,
K.
Sun
, and
Y.
Zhao
, “
Theoretical examination of long-range energy propagation in nano-engineered light-harvesting antenna arrays
,”
J. Phys. Chem. C
116
,
3747
3756
(
2012
).
24.
F.
Zheng
,
L.
Chen
,
J.
Gao
, and
Y.
Zhao
, “
Fully quantum modeling of exciton diffusion in mesoscale light harvesting systems
,”
Materials
14
,
3291
(
2021
).
25.
T.
Meier
,
Y.
Zhao
,
V.
Chernyak
, and
S.
Mukamel
, “
Polarons, localization, and excitonic coherence in superradiance of biological antenna complexes
,”
J. Chem. Phys.
107
,
3876
(
1997
).
26.
Y.
Zhao
,
T.
Meier
,
W. M.
Zhang
,
V.
Chernyak
, and
S.
Mukamel
, “
Superradiance coherence sizes in single-molecule spectroscopy of LH2 antenna complexes
,”
J. Phys. Chem. B
103
,
3954
3962
(
1999
).
27.
F. C.
Spano
,
J.
Clark
,
C.
Silva
 et al, “
Determining exciton coherence from the photoluminescence spectral line shape in poly(3-hexylthiophene) thin films
,”
J. Chem. Phys.
130
,
074904
(
2009
).
28.
F. C.
Spano
and
H.
Yamagata
, “
Vibronic coupling in J-aggregates and beyond: A direct means of determining the exciton coherence length from the photoluminescence spectrum
,”
J. Phys. Chem. B
115
,
5133
5143
(
2011
).
29.
J.
Sung
,
P.
Kim
,
B.
Fimmel
,
F.
Würthner
, and
D.
Kim
, “
Direct observation of ultrafast coherent exciton dynamics in helical π-stacks of self-assembled perylene bisimides
,”
Nat. Commun.
6
,
8646
(
2015
).
30.
R.
Tempelaar
,
F. C.
Spano
,
J.
Knoester
, and
T. L. C.
Jansen
, “
Mapping the evolution of spatial exciton coherence through time-resolved fluorescence
,”
J. Phys. Chem. Lett.
5
,
1505
1510
(
2014
).
31.
M. F.
Gelin
,
A. V.
Pisliakov
, and
W.
Domcke
, “
Time- and frequency-gated spontaneous emission as a tool for studying vibrational dynamics in the excited state
,”
Phys. Rev. A
65
,
062507
(
2002
).
32.
M. F.
Gelin
and
R.
Borrelli
, “
Simulation of nonlinear femtosecond signals at finite temperature via a thermo field dynamics-tensor train method: General theory and application to time- and frequency-resolved fluorescence of the Fenna–Matthews–Olson complex
,”
J. Chem. Theory Comput.
17
,
4316
(
2021
).
33.
M. F.
Gelin
,
D.
Egorova
, and
W.
Domcke
, “
Efficient calculation of time- and frequency-resolved four-wave-mixing signals
,”
Acc. Chem. Res.
42
,
1290
1298
(
2009
).
34.
Y.
Tanimura
and
S.
Mukamel
, “
Optical Stark spectroscopy of a Brownian oscillator in intense fields
,”
J. Phys. Soc. Jpn.
63
,
66
77
(
1994
).
35.
M.
Tanaka
and
Y.
Tanimura
, “
Quantum dissipative dynamics of electron transfer reaction system: Nonperturbative hierarchy equations approach
,”
J. Phys. Soc. Jpn.
78
,
073802
(
2009
).
36.
X.
Xie
,
A.
Santana-Bonilla
, and
A.
Troisi
, “
Nonlocal electron–phonon coupling in prototypical molecular semiconductors from first principles
,”
J. Chem. Theory Comput.
14
,
3752
3762
(
2018
).
37.
K.
Nakamura
and
Y.
Tanimura
, “
Optical response of laser-driven charge-transfer complex described by Holstein-Hubbard model coupled to heat baths: Hierarchical equations of motion approach
,”
J. Chem. Phys.
155
,
064106
(
2021
).
38.
A. A.
Bakulin
,
A.
Rao
,
V. G.
Pavelyev
,
P. H. M.
van Loosdrecht
,
M. S.
Pshenichnikov
,
D.
Niedzialek
,
J.
Cornil
,
D.
Beljonne
, and
R. H.
Friend
, “
The role of driving energy and delocalized states for charge separation in organic semiconductors
,”
Science
335
,
1340
1344
(
2012
).
39.
S.
Gélinas
,
A.
Rao
,
A.
Kumar
,
S. L.
Smith
,
A. W.
Chin
,
J.
Clark
,
T. S.
van der Poll
,
G. C.
Bazan
, and
R. H.
Friend
, “
Ultrafast long-range charge separation in organic semiconductor photovoltaic diodes
,”
Science
343
,
512
516
(
2014
).
40.
L.
Gisslén
and
R.
Scholz
, “
Crystallochromy of perylene pigments: Interference between Frenkel excitons and charge-transfer states
,”
Phys. Rev. B
80
,
115309
(
2009
).
41.
M.
Cainelli
and
Y.
Tanimura
, “
Exciton transfer in organic photovoltaic cells: A role of local and nonlocal electron–phonon interactions in a donor domain
,”
J. Chem. Phys.
154
,
034107
(
2021
).
42.
A.
Ishizaki
and
Y.
Tanimura
, “
Quantum dynamics of system strongly coupled to low-temperature colored noise bath: Reduced hierarchy equations approach
,”
J. Phys. Soc. Jpn.
74
,
3131
3134
(
2005
).
43.
Y.
Tanimura
, “
Stochastic Liouville, Langevin, Fokker–Planck, and master equation approaches to quantum dissipative systems
,”
J. Phys. Soc. Jpn.
75
,
082001
(
2006
).
44.
S.
Mukamel
,
Y.
Tanimura
, and
P.
Hamm
, “
Coherent multidimensional optical spectroscopy
,”
Acc. Chem. Res.
42
,
1207
1209
(
2009
).
45.
Y.
Tanimura
, “
Numerically ‘exact’ approach to open quantum dynamics: The hierarchical equations of motion (HEOM)
,”
J. Chem. Phys.
153
,
020901
(
2020
).
46.
Y.
Tanimura
and
A.
Ishizaki
, “
Modeling, calculating, and analyzing multidimensional vibrational spectroscopies
,”
Acc. Chem. Res.
42
,
1270
1279
(
2009
).
47.
L.
Chen
,
M. F.
Gelin
,
Y.
Zhao
, and
W.
Domcke
, “
Mapping of wave packet dynamics at conical intersections by time- and frequency-resolved fluorescence spectroscopy: A computational study
,”
J. Phys. Chem. Lett.
10
,
5873
5880
(
2019
).
48.
L.
Chen
,
R.
Borrelli
,
D. V.
Shalashilin
,
Y.
Zhao
, and
M. F.
Gelin
, “
Simulation of time- and frequency-resolved four-wave-mixing signals at finite temperatures: A thermo-field dynamics approach
,”
J. Chem. Theory Comput.
17
,
4359
(
2021
).
49.
O.
Kühn
and
V.
Sundström
, “
Pump–probe spectroscopy of dissipative energy transfer dynamics in photosynthetic antenna complexes: A density matrix approach
,”
J. Chem. Phys.
107
,
4154
(
1997
).
50.
T.
Meier
,
V.
Chernyak
, and
S.
Mukamel
, “
Multiple exciton coherence sizes in photosynthetic antenna complexes viewed by pump-probe spectroscopy
,”
J. Phys. Chem. B
101
,
7332
7342
(
1997
).
51.
C.
Smyth
,
F.
Fassioli
, and
G. D.
Scholes
, “
Measures and implications of electronic coherence in photosynthetic light-harvesting
,”
Philos. Trans. R. Soc., A
370
,
3728
3749
(
2012
).
52.
A. G.
Dijkstra
and
Y.
Tanimura
, “
Linear and third- and fifth-order nonlinear spectroscopies of a charge transfer system coupled to an underdamped vibration
,”
J. Chem. Phys.
142
,
212423
(
2015
).
53.
M. F.
Gelin
,
A.
Velardo
, and
R.
Borrelli
, “
Efficient quantum dynamics simulations of complex molecular systems: A unified treatment of dynamic and static disorder
,”
J. Chem. Phys.
155
,
134102
(
2021
).
54.
C.
Shao
,
M.
Grüne
,
M.
Stolte
, and
F.
Würthner
, “
Perylene bisimide dimer aggregates: Fundamental insights into self-assembly by NMR and UV/Vis spectroscopy
,”
Chem. Eur. J.
18
,
13665
13677
(
2012
).
55.
N. J.
Hestand
and
F. C.
Spano
, “
Molecular aggregate photophysics beyond the kasha model: Novel design principles for organic materials
,”
Acc. Chem. Res.
50
,
341
350
(
2017
).
56.
A.
Capobianco
,
R.
Borrelli
,
A.
Landi
,
A.
Velardo
, and
A.
Peluso
, “
Absorption band shapes of a push–pull dye approaching the cyanine limit: A challenging case for first principle calculations
,”
J. Phys. Chem. A
120
,
5581
5589
(
2016
).
57.
N. J.
Hestand
,
R. V.
Kazantsev
,
A. S.
Weingarten
,
L. C.
Palmer
,
S. I.
Stupp
, and
F. C.
Spano
, “
Extended-charge-transfer excitons in crystalline supramolecular photocatalytic scaffolds
,”
J. Am. Chem. Soc.
138
,
11762
11774
(
2016
).
58.
S.
Kang
,
T.
Kim
,
Y.
Hong
,
F.
Würthner
, and
D.
Kim
, “
Charge-delocalized state and coherent vibrational dynamics in rigid PBI H-aggregates
,”
J. Am. Chem. Soc.
143
,
9825
9833
(
2021
).
59.
R.
Borrelli
, “
Density matrix dynamics in twin-formulation: An efficient methodology based on tensor-train representation of reduced equations of motion
,”
J. Chem. Phys.
150
,
234102
(
2019
).
60.
R.
Borrelli
and
S.
Dolgov
, “
Expanding the range of hierarchical equations of motion by tensor-train implementation
,”
J. Phys. Chem. B
125
,
5397
5407
(
2021
).
61.
Y.
Ke
,
R.
Borrelli
, and
M.
Thoss
, “
Hierarchical equations of motion approach to hybrid fermionic and bosonic environments: Matrix product state formulation in twin space
,”
J. Chem. Phys.
156
,
194102
(
2022
).
62.
R.
Borrelli
and
M. F.
Gelin
, “
Finite temperature quantum dynamics of complex systems: Integrating thermo-field theories and tensor-train methods
,”
WIREs Comput. Mol. Sci.
11
,
e1539
(
2021
).
63.
Q.
Shi
,
Y.
Xu
,
Y.
Yan
, and
M.
Xu
, “
Efficient propagation of the hierarchical equations of motion using the matrix product state method
,”
J. Chem. Phys.
148
,
174102
(
2018
).
64.
Y.
Yan
,
T.
Xing
, and
Q.
Shi
, “
A new method to improve the numerical stability of the hierarchical equations of motion for discrete harmonic oscillator modes
,”
J. Chem. Phys.
153
,
204109
(
2020
).
65.
R.
Borrelli
and
M. F.
Gelin
, “
Quantum electron-vibrational dynamics at finite temperature: Thermo field dynamics approach
,”
J. Chem. Phys.
145
,
224101
(
2016
).
66.
R.
Borrelli
and
M. F.
Gelin
, “
Simulation of quantum dynamics of excitonic systems at finite temperature: An efficient method based on thermo field dynamics
,”
Sci. Rep.
7
,
9127
(
2017
).

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