Quantum Master Equation Approach to Singlet Fission Dynamics of

Subscriber access provided by Northern Illinois University

Article

Quantum Master Equation Approach to Singlet Fission Dynamics of Realistic/ Artificial Pentacene Dimer Models: Relative Relaxation Factor Analysis Masayoshi Nakano, Soichi Ito, Takanori Nagami, Yasutaka Kitagawa, and Takashi Kubo J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b06402 • Publication Date (Web): 08 Sep 2016 Downloaded from http://pubs.acs.org on September 16, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Quantum Master Equation Approach to Singlet Fission Dynamics of Realistic/Artificial Pentacene Dimer Models: Relative Relaxation Factor Analysis Masayoshi Nakano,*,†,¶ Soichi Ito,† Takanori Nagami,† Yasutaka Kitagawa,†,¶ Takashi Kubo§ †

Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University,

Toyonaka, Osaka 560-8531, Japan ¶

Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University,

Toyonaka, Osaka 560-8531, Japan §

Department of Chemistry, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT:

Page 2 of 58

The singlet fission (SF) dynamics of realistic/artificial pentacene dimer

models are investigated using the quantum master equation method in order to obtain new insight into the SF dynamics and its rational design guidelines.

We comprehensively clarify

the effects of the energy offsets of diabatic Frenkel exciton (FE) and charge transfer (CT) exciton states to the double-triplet (TT) exciton state, excitonic couplings, and state-dependent vibronic couplings on the exciton population dynamics using relative relaxation factors (RRFs) between the adiabatic exciton states.

As shown in previous studies,

efficient sequential/superexchange CT-mediated SF is observed in the energy level matching region (E(TT) – E(FE) < 0).

On the other hand, it is predicted that almost the perfect energy

level matching (E(TT) – E(FE) ~ 0) causes the significant reduction of TT yields though exhibits remarkably fast SF rates, when the corresponding adiabatic double-triplet ( TT′ ) and Frenkel exciton ( FE′ ) states are near-degenerate to each other with common diabatic configurations.

The excitonic coupling is also found to have a possibility of causing

significant change of SF dynamics when it has a large amplitude comparable to those of the other electronic coupling elements.

Furthermore, the large vibronic coupling of CT state

shows striking enhancement of SF rates with keeping high TT yields in the CT-mediated superexchange region, while the large vibronic couplings of FE and TT states do not show such striking enhancement.

These features are understood by analyzing their RRFs, which

are proportional to the product of the square of common diabatic exciton configuration coefficients in the concerned two adiabatic exciton states, multiplied by the spectral density (vibronic coupling).


Page 3 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1. INTRODUCTION Recently, singlet fission (SF), which is a photophysical process with splitting a singlet exciton generated by light irradiation into two triplet excitons, has attracted much attention both experimentally and theoretically due to its possibility of improving photoelectric conversion efficiency through multi-exciton generation in organic solar cells.

1, 2

In addition, the

generated triplet excitons tend to have longer lifetimes than singlet ones since the recombination to the singlet ground state is a spin-forbidden process.

This feature

contributes to further improving the photoelectric conversion efficiency since they have higher possibility to diffuse to the donor-acceptor interfaces producing free charges.

Thus,

lots of investigations have been conducted to reveal the detailed mechanism of SF as well as to design highly efficient SF molecular systems, together with measurements of the SF of novel molecules and crystals. 1-15 The investigation of SF is composed of three steps: (i) energy level matching at single molecular level,

1, 2

(ii) electronic coupling at molecular aggregate

1, 2, 16-19

or

covalently-linked multimer

20, 21

(vibronic) couplings. 22–38

The molecular design guideline in the first step (i), which was

proposed by Michl et al.,

1, 2

level, and (iii) exciton dynamics including exciton-phonon

is based on the energy level matching conditions for the

monomer: (a) 2E(T1 ) ~ E(S1 ) or 2E(T1 ) < E(S1 ) and (b) 2E(T1 ) < E(T2 ) .

Here, (a) is

required for splitting a singlet exciton (with energy E(S1)) into two triplet excitons (with energy E(T1)), while (b) is required for suppressing the recombination of the split two triplet excitons into another triplet exciton (with energy E(T2)) represented by T1 + T1 → T2 + S0. The molecular systems satisfying these conditions are found to be distinguished by using diradical (y0) and tetraradical (y1) characters (0 ≤ y0, y1 ≤ 1), which are concerned with conditions (a) and (b), respectively. 4, 5, 39

Namely, molecules with relatively small diradical

character (~0.1 < y0 < ~0.5) as well as with much smaller tetraradical character (y1/y0 < ~0.2) 3 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

tend to satisfy these two conditions. 4, 5

Page 4 of 58

Also, relatively small-size hydrocarbon systems with

relatively small y0 values mostly tend to give much smaller y1 values (y1 > Vll +Vhh

(= 24.3 meV), where Vll − Vhh

electronic coupling between respectively.

are found to originate in the relation Vll −Vhh (=

S+

and

and Vll +Vhh

CT+ , and that between

This also contributes to the larger split in

S−

S′+ – CT+′

indicate the and

CT− ,

gap than in

S′− – CT−′ gap, that is, E(S′– ) > E(S′+ ) and E(CT+′ ) > E(CT–′ ), for Vex = 0 meV (see Figure 2a).

On the other hand, the dominant configuration in TT′

(~91.7 %), which leads to the high TT yield. relatively large amounts of CA phase.

is shown to be TT

In addition, TT′

is found to include

and AC configurations with the mutually opposite

This is found to originate in the relation Vlh −Vhl (= 226.3 meV) >> Vhl +Vlh (= 13 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 58

17.1 meV), where ( 3 / 2) Vlh −Vhl and ( 3 / 2) Vhl + Vlh indicates the electronic coupling between TT and CT− , and that between TT and CT+ , respectively. although this also contributes to the enhancement of TT′ – CT−′ this enhancement is smaller than that of S′+ – CT+′ in TT – CT–

gap than in S+ – CT+

Note here that

gap like S′+ – CT+′

gap,

gap due to the larger energy difference

gap as well as to Vll −Vhh > ( 3 / 2) Vlh − Vhl ,

which holds the energy relationships, E (S′– ) > E (S′+ ) and E (CT+′ ) > E (CT–′ ) for Vex = 0 meV. As a result, since TT′ and S′+

include relatively large amounts of CA and AC

configurations in contrast to S′− , ∆Γ (S′+ → TT′) is found to be much larger than

∆Γ (S′− → TT′) . This implies that primary exciton relaxation occurs from S′+ to TT′ . Next, we clarify the effects of Vex on the RRFs of the relaxation pathways from

S′+ / S′−

to TT′ .

As compared to the case of Vex = 0 meV, the positive FE coupling (Vex

= 100 meV) is found to increase and decrease the energies of S′+ due to the difference in the relative phase between S1S0 involved in those states. and S′−

and S′− , respectively,

and S0S1

configurations

This behavior results in the inversion of the energy levels of S′+

in the present case (see Figure 2a and b).

Because the CA

and AC

configurations in those states are not shown to be so changed in this case, the ∆Γ (S′+ → TT′) ,

∆Γ (S′− → TT′) and the amount of TT configuration in TT′ are found to be similar to those in the case of Vex = 0 meV, respectively.

However, the much larger ∆Γ (S′+ → S′– )

than ∆Γ (S′+ → TT′) is predicted to reduce the possibility of taking the relaxation pathway

S′+ → TT′ , which is a primary relaxation pathway for Vex = 0 meV. This is the reason of the slower SF rate for Vex = 100 meV than for Vex = 0 meV though the TT yield is the same in both the cases.

Some additional increase of positive Vex is predicted to further slow SF rate 14 ACS Paragon Plus Environment

Page 15 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


until S′−

is close to TT′ since the ∆Γ (S′+ → S′– ) becomes larger due to the S′+ – S′−

gap being closer to the spectral density peak frequency Ω .

Indeed, for Vex = 200 meV, we

obtain a slower SF rate (τ = 1141 fs) and a high TT yield = 0.917.

On the other hand, the

negative excitonic coupling (Vex = –100 meV) is found to enhance the S′+ – S′−

gap as

compared to the case of Vex = 0 meV since the negative Vex is found to decrease and increase the S′+

and S′–

amounts of CA

energy levels, respectively. and AC

This feature leads to slight decrease in the

configurations in S′+

due to slight enhancement of the

S′+ – CT+′ gap. Also, the amount of each diabatic configuration in TT′ is shown to be almost unchanged as compared to the case of Vex = 0 meV.

Considering these facts, the

primary ∆Γ (S′+ → TT′) is found to slightly decrease as compared to the case of Vex = 0 meV. This leads to slightly lower SF rate for Vex = –100 meV than for Vex = 0 meV.

In contrast to

positive Vex case, some additional increase in the amplitude of negative Vex is found to slightly lower SF rate (τ = 252.2 fs and TT yield = 0.915 for Vex = -200 meV) until S′−

TT′ , since the inversion of the energy level between S′+ case.

and S′–

is close to

does not occur in this

These results indicate that the excitonic coupling has a possibility of causing

significant change of SF rate when it has a large amplitude comparable to those of the other coupling elements and thus causes the inversion of the energy level between S′+ only one of which is involved in a primary SF relaxation pathway. |Vex| makes the energy of either S′+

or S′−

such case, the amounts of S1S0 , S0S1 large in those near-degenerate states, S′+ couplings exist.

and S′− ,

If further enhancement of

closer to that of TT′ , what will happen?

In

and TT configurations are predicted to become or S′−

and TT′ , when the related electronic

Namely, it is predicted that the SF rate is significantly enhanced, while that

the TT yield is largely reduced due to the relatively large configurations of S1S0 15 ACS Paragon Plus Environment

and


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 58

S0S1 , that is, relatively small configuration of TT , in TT′ . Indeed, we obtain τ = 19.10 fs and TT yield = 0.525 for Vex = 450 meV ( E(S′– ) − E(TT′) = 15.95 meV), and τ = 81.45 fs and TT yield = 0.676 for Vex = –350 meV ( E(S′+ ) − E(TT′) = 28.53 meV). course, if either

S′+

or

S′−

became lower-lying than

TT′

Of

due to the further

enhancement of |Vex|, the TT yield would be further reduced since the final state in thermal equilibrium is dominantly composed of the lowest state S′+

or S′− .

Indeed, we obtain

TT yield = 0.155 for Vex = 500 meV ( E(S′– ) − E(TT′) = –44.26 meV) and TT yield = 0.015 for Vex = –500 meV ( E(S′+ ) − E(TT′) = –109.0 meV).

3.2. Effects of CT Energy Offset to TT State and Energy Level Matching on SF Dynamics.

First, we clarify the effect of E(CT) and E(FE) offsets to E(TT) on the SF

dynamics (SF rate (τ-1) and TT yield (a)).

To this end, we examine two artificial models,

which have E(CT) – E(TT) = 1000 meV (Figure 2d), and E(FE) – E(TT) = 20 meV (Figure 2e), respectively, as well as the realistic model (E(CT) – E(TT) = 600 meV and E(FE) – E(TT) = 400 meV (Figure 2a)).

Note here that the other parameters are the same as those in

Table 1 and that we adopt Vex = 0 meV, which is approximately satisfied in the realistic pentacene dimer as mentioned in Sec. 3.1.

As seen from Figure 2d, the relative increase of

E(CT) significantly slows the SF rate (τ = 1380 fs), while it slightly increases the TT yield (a = 0.965), cf. τ = 189.8 fs and a = 0.917 for the realistic pentacene dimer model (Figure 2a). This is understood by the relative decreases in the CT (CA and AC) mixings in S′+

and

TT′ , which reduce ∆Γ (S′+ → TT′) , due to the increase in the energy gap between CT and FE/TT states (Figure 2a and 2d).

Indeed, the ∆Γ (S′+ → TT′) is found to be much reduced

by a factor of ~7 in the artificial model with E(CT) – E(TT) = 1000 meV (Figure 2d) as compared to the realistic model (Figure 2a).

Concurrently, the relative decreases in the CT 16

ACS Paragon Plus Environment

Page 17 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


mixings in TT′ lead to the relative increase of TT configuration in TT′ . reason of the slight increase in the TT yield in this case.

This is the

These results are also predicted by

the perturbation analysis of superexchange effect of CT states on the SF rate. 1, 2

On the

other hand, as seen from Figure 2e, the decrease of |E(FE) – E(TT)| from 400 meV to 20 meV remarkably enhances the SF rate (decrease of τ = 189.8 fs to 18.71 fs), while it significantly reduces the TT yield from a = 0.917 to 0.511.

As seen from the adiabatic energy state

diagram, these features originate in the significant FE and TT mixings in S′+ , S′−

and

TT′ , which lead to the significant enhancement of ∆Γ (S′+ → TT′) and ∆Γ (S′− → TT′) . At the same time, such relative increases in FE mixings in TT′ are found to cause the relative decrease in the amount of TT configuration in TT′ , resulting in the significant reduction of the TT yield.

This result suggests that almost the perfect energy level matching

(E(TT) – E(FE) ~ 0) significantly reduces the TT yield though speeds up the SF dynamics, when S′+

or S′−

and TT′ are near-degenerate to each other with common diabatic

configurations. Next, in order to more comprehensively clarify the relationship between the energy level matching condition for monomer and SF dynamics based on the RRF analysis, we examine the artificial pentacene dimer models with varying E(FE) – E(TT) and E(CT) – E(TT). Figure 3.

The diabatic exciton state map with respect to these two energy offsets is shown in Apparently, the energy level matching condition corresponds to the right quadrant

plane (E(FE) – E(TT) ≥ 0).

As the CT-mediated SF process, which is found to be major in

most SF systems, 1, 2 there are known to be two cases, sequential (Seq) and superexchange (Spx) mechanisms. 1, 2, 32

In the former case, CT state lies between FE and TT states, while

in the latter case CT state lies above or below FE and TT states.

Considering the fact that SF

usually occurs from FE to TT state and that the thermodynamic equilibrium favors the lowest 17 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 58

energy state, high TT population is expected to appear in the first quadrant, Spx+(TT) and Seq+(TT) regions.

On the other hand, in the triplet-triplet annihilation (TTA), 56 which is an

opposite process to the SF process, the high FE yield is expected to appear in Spx+(FE) and Seq–(FE) regions.

The TT yields after several periods of time for the realistic pentacene

dimer model with Vex = 0 meV and ( λ , Ω ) = (50, 180) meV are shown in Figure 4.

The

features of these figures are in qualitative agreement with those obtained by Berkelbach et al. 32

Firstly, as expected, the non-zero P(TT) values at all times are found to be distributed

primarily in the first quadrant (Spx+(TT) and Seq+(TT)) as well as somewhat in Spx–(TT) region.

At 1 ps (Figure 4d), we found that the maximum TT yield area mainly appears in

Seq+(TT) region though it somewhat penetrates to Spx+(TT) region.

As shown in the

previous study, 32 at relatively early time region (≤ ~ 1ps), the increase of |E(CT) – E(TT)| is found to decrease TT yield as shown in Figure 2d, while in the region with small difference of |E(FE) – E(TT)|, relatively large TT yield is observed even in the high |E(CT) – E(TT)| region. This is predicted to stem from the same origin of the fast SF dynamics shown in Figure 2e. Also in Spx–(TT) region, somewhat TT yield is found to be obtained if |E(CT) – E(TT)| is small.

As seen from Figure 4a and b, fast SF occurs in two regions, that is, (i) Seq+(TT)

region with E(FE) – E(TT) < ~600 meV and E(CT) – E(TT) < 400 meV, and (ii) Spx+(TT) region with 0 < E(FE) – E(TT) < 100 meV. the previous result. 32

These results are in qualitative agreement with

Also, note here that at almost the perfect energy level matching region

(E(FE) – E(TT) ~ 0), only small TT yields are shown to be obtained at all the time regardless of E(CT) – E(TT) value (see Figure 4a-d and 2e).

The energy region of the realistic

herringbone pentacene dimer (indicated by a white dot) is found to be included in the maximum TT yield area in Spx+(TT) region.

This indicates that the real pentacene crystal is

suitable for efficient SF (with fast SF rate and high TT yield).

These results suggest that the

energy level matching condition is useful for predicting TT yield at long time region


Page 19 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(thermodynamic equilibrium region) except for almost the perfect matching region, while the relative energy level of CT state is also essential for determining the SF rate and TT yield at early time region, which is important for SF applications.

In addition, it is predicted that

there is a trade-off relationship between TT yield and SF rate within the identical spectral density (vibronic coupling) approximation for each diabatic exciton state.

This is

understood by the fact that the faster (slower) SF rate, that is, the larger (smaller)

∆Γ (S′– → TT′) or ∆Γ (S′+ → TT′) , originates in the larger (smaller) CT mixings in those adiabatic states.

Namely, such larger (smaller) CT mixing in TT′ corresponds to the

relative decrease (increase) of the amount of TT configuration in TT′ , resulting in smaller (larger) TT yield in the thermodynamic equilibrium.

Thus, optimal design for the SF

rate as well as for the TT yield is desired for the practical applications.

3.3. State-Dependent Vibronic Coupling Effect on SF Dynamics. As seen from the expression of RRF (eq 9), the exciton relaxation between the adiabatic exciton states also depends on the relaxation rate γ m (ω ) (eq 7), which is a function of the spectral density

J m (ω ) (eq 8) concerned with the exciton-phonon (vibronic) coupling.

Although this

spectral density depends on the diabatic exciton state in principle, such state-dependence has not been investigated in most previous studies. 32, 37

In this section, we examine the effects

of state-dependent J m (ω ) on the SF dynamics by using different spectral density ( λm , Ω m ) for each diabatic state { m }.

Here, we adopt a reasonable approximation that the same

parameters are employed for Frenkel exciton states FE ( = S1S0 charge transfer states CT

(= CA or AC ), respectively. 30

the feature of vibronic couplings (

or S0S1 ) and for

In our previous paper, 30

in eq 4) has been clarified for a tetracene dimer



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

model using vibronic coupling density analysis. 57

Page 20 of 58

It has been found (i) that FE state exhibits

large vibronic coupling peaks in the high frequency region (1200 – 1700 cm-1 (150 – 210 meV)) such as carbon(C)-carbon(C) stretching mode, (ii) that TT state exhibits almost the same spectral shape with that of FE state, while the peak amplitudes of TT state are twice as large as those of FE state, and (iii) that CT state exhibits larger peaks in the low frequency region (60 – 800 cm-1 (7 – 100 meV)) such as out-of-plane bending and ring breathing modes. Such larger spectral peaks for TT and CT states than for FE state are predicted to significantly affect the SF dynamics through eq 7.

In order to clarify the qualitative effects of

state-dependent spectral density on the SF dynamics, we here consider three model spectral densities reflecting the above features, ( λm , Ω m ) = (50, 180) meV for FE, (1290, 98) meV for CT, and (200, 180) meV for TT.

Here, the first one is the same as those used in the

previous study, 32 while the second and the third ones are qualitatively modeled after the results in our previous study. 30

As mentioned before, the spectral densities of FE and TT

states exhibit the peak at the same phonon energy (180 meV), and the peak intensity of TT state (200 meV) is four times that of FE state (50 meV) due to the reorganization energy .

In contrast, the spectral density of CT state has a much larger peak (1290

meV) at a lower phonon energy (98 meV).

Figure 5 shows the spectral densities for these

three cases: (λ, Ω) = (50, 180) meV, (1290, 98) meV, and (200, 180) meV.

The present

spectral density takes a peak (reorganization energy λ) at the cutoff frequency Ω.

Figure 6

shows the diabatic exciton population dynamics for the realistic pentacene dimer model with the state-dependent spectral densities described above.

Compared with the case of identical

spectral density ( λ , Ω ) = (50, 180) meV (Figure 2a), it is found that the SF time constant τ is remarkably reduced in the case of the state-dependent spectral density though the TT yield is unchanged in both the cases.

Note here that the absolute value of the unusually faster SF

rate for the state-dependent spectral density is predicted to be somewhat overestimated 20 ACS Paragon Plus Environment

Page 21 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


because the ( λ , Ω ) is obtained by the approximate vibronic coupling calculation in our previous study.

30

However, the feature of the faster SF rate for state-dependent spectral

density than for identical spectral density is predicted to be qualitatively correct because our previous study well reproduces the qualitative feature of the state dependence of vibronic couplings.

Thus, we here focus on the relative differences in the SF dynamics originating in

the state-dependent spectral densities.

As seen from the adiabatic state energy diagrams with

RRFs for these cases (Figure 6 and Figure 2a), the coefficients of all the configurations in each adiabatic state are the same in both the cases due to the same electronic coupling matrix. Thus, the differences in RRFs between them are predicted to originate in the state-dependent spectral densities, which are found to cause the enhancement of RRFs between the adiabatic states including large CT and/or TT mixings. In order to further elucidate the effect of the state-dependent spectral density on the SF dynamics in a wide range of the energy offsets, E(CT) – E(TT) and E(FE) – E(TT), we compare the TT yield maps at 50 fs, 100 fs, 300 fs and 1000 fs between the state-dependent (Figure 7) and the identical (Figure 4) spectral density cases.

As compared to the identical

spectral density case, the rapid evolution of large TT population is observed in the state-dependent spectral density case, in particular, in Spx+(TT) region with high |E(CT) – E(TT)| and |E(CT) – E(FE)| values. in detail.

To clarify the origin of this feature, we investigate eq 9

Equation 9 states that the RRF from adiabatic state α to β ( ∆Γ (α → β ) ) is given 2

by the sum of the product of Cmα Cmβ configurations {m}.

2

and γ m (ωα − ω β ) over all the diabatic state

From eq 7, γ m (ωα − ω β ) is a function of spectral density J m (ωα − ω β ) ,

where α = S′+ or S′– and β = TT′ in the SF case.

In the present systems, as shown in Sec.

3.1, the primary SF pathway is S′+ → TT′ , so that the SF rate depends on the amount of CT configurations in S′+ / TT′ and J m (ωS′+ − ω TT′ ) .

From these results, under constant CT



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 58

configurations in S′+ / TT′ , the ∆Γ (S′+ → TT′) becomes large when the spectral density of diabatic CT state has the cutoff frequency ΩCT near to ω S′+ − ω TT′ as well as the large reorganization energy λCT .

In contrast, large λm for the other diabatic states FE and TT

are not predicted to cause such significant enhancement of the SF rate.

In order to confirm

these predictions, we investigate the SF dynamics by using five different spectral density combinations for the same pentacene dimer model with E(CT) – E(TT) = 1000 meV (which belongs to Spx+(TT) region): (i) (λm, Ωm) = (50, 180) meV for all diabatic states (Figure 2d), (ii) (λm, Ωm) = (1000, 180) meV for CT states (Figure 8a), (iii) (λm, Ωm) = (1000, 90) meV for CT states (Figure 8b), (iv) (λm, Ωm) = (1000, 180) meV for FE states (Figure 8c), and (v) (λm, Ωm) = (1000, 180) meV for TT state (Figure 8d).

Here, for (ii)–(v), (λm, Ωm) = (50, 180)

meV is employed for all the diabatic states except for CT.

As mentioned in Sec. 3.1, (i)

indicates the significantly slower SF rate (τ = 1380 fs) but larger TT yield (a = 0.965) than the realistic pentacene dimer model (τ = 189.8 fs, a = 0.917, see Figure 2a).

(ii) (with only the

increase in the CT vibronic coupling (λ = 1000 meV) and the same Ω (= 180 meV)) is found to cause significantly faster SF rate (τ = 79.13 fs) with keeping the same TT yield, and (iii) (with lower Ω (= 90 meV)) is shown to somewhat decrease the SF rate (τ = 133.0 fs) due to the more off-resonance of Ω with ω S′+ − ω TT′ (= 310.7 meV).

In contrast, (iv) (with only

the increase in the FE vibronic coupling (λ = 1000 meV)) and (v) (with only the increase in the TT vibronic coupling (λ = 1000 meV)) do not show such remarkable enhancement of the SF rates observed in (ii) and (iii) (775.9 fs for (iv) and 544.2 fs for (v) vs. 79.13 fs for (ii) and 133.0 fs for (iii)) though they enhance the SF rates as compared to (i) (τ = 1380 fs).

Note

here that all the TT yields are the same higher value 0.965 than that of the original pentacene dimer model (0.917).

These results confirm our predictions.

Namely, in Spx+(TT) region

with relatively high |E(CT) – E(TT)| and |E(CT) – E(FE)| values (except for E(FE) – E(TT) ~ 0 region), we can realize the efficient SF dynamics with both large SF rates and high TT 22 ACS Paragon Plus Environment

Page 23 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


yields if the reorganization energy of CT state is significant even in the low frequency region. In summary, significant state dependence of spectral density has a possibility of affecting the RRF ∆Γ (α → β ) through the common diabatic exciton state mixings in the concerned adiabatic states (α and β), and thus is likely to change the SF rate with keeping the TT yield.

4. CONCLUSION We have investigated the singlet fission (SF) dynamics for realistic/artificial pentacene dimer models with a wide range of physical parameters using the quantum master equation method. In particular, we have aimed to obtain general conclusions for SF dynamics and its design guidelines beyond the specific pentacene dimer, regarding the origin of the effects of excitonic coupling, charge transfer (CT) energy offset to double-triplet (TT) state, energy level matching, and state-dependent exciton-phonon (vibronic) coupling on the SF rate and TT yield.

From the analysis of the quantum master equation, we have found that the feature

of SF dynamics is described by the relative relaxation factor (RRF), which is proportional to the product of the square of common diabatic exciton configuration coefficients included in the concerned two adiabatic exciton states, multiplied by the spectral density (vibronic coupling) (see eq 9).

On the basis of the RRF analysis of the pentacene dimer models, it is

found (i) that excitonic coupling between diabatic Frenkel exciton (FE) states can affect the SF rate by changing the relative energies of adiabatic exciton states ( FE′ = ( S′+ or S′– ),

CT′ = ( CA′ or AC′ ), TT′ ) and thus by changing the amounts of diabatic exciton configurations (FE (S1S0 or S0S1), CT = (CA or AC), TT) in those adiabatic exciton states, (ii) that fast SF, observed in the CT-mediated sequential (Seq+(TT)) and limited superexchnage (Spx+(TT)) regions, originates in the significant mixing of CT configurations in adiabatic



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 58

FE′ and TT′ states, and (iii) that almost the perfect energy level matching (E(TT) – E(FE) ~ 0) conversely reduces the TT yield though speeds up the SF dynamics, when S′+

or S′−

and TT′ are near-degenerate to each other with common diabatic configurations.

From

these results, the trade-off relationship between the SF rate and TT yield is predicted within the approximation of using the identical spectral density for each diabatic exciton state.

On

the other hand, in the case of state-dependent spectral density, we have found striking enhancement of the SF rates particularly in the Spx+(TT) region.

This is because the

vibronic coupling of CT state has a much larger intensity than that of FE state and thus significantly compensates the effect of the small amount of CT mixings in adiabatic TT′ and

FE′ states, which is caused by the large energy gaps |E(CT) – E(TT)| and |E(CT) – E(FE)|. Namely, in contrast to the identical spectral density case, both fast SF rates and high TT yields have a possibility to be concurrently realized in the Spx+(TT) region with large vibronic coupling of CT state.

In contrast, the large vibronic couplings of FE and TT states

are not found to cause such striking enhancement of SF rates.

Because such a large vibronic

coupling at a low frequency region of CT state is often observed for dimers of acenes and other conjugated organic molecules, 30 the optimal tuning of the energy level matching of the monomer, the CT energy offset and the electronic coupling is particularly desired for further improving the SF efficiency for such systems.

To this end, the relationships between the

crystal packing or covalently-linked manner and the electronic/vibronic couplings have to be clarified in detail for a variety of SF candidate monomers screened by the diradical character based design guideline. 4, 5, 39

Furthermore, the remaining issues include (i) CT offset energy

tuning, which is achieved, for example, by tuning the ionization potential and electron affinity of monomer and intermolecular interaction between positively- and negatively-charged monomers as well as by designing the crystalline environment of polarizable molecules, and (ii) clarification of the effect of Peierls coupling, for example, intermolecular vibronic 24 ACS Paragon Plus Environment

Page 25 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


couplings, which tend to have primary peaks in the low frequency region. 37

In the latter

issue, consideration of non-Markovian nature of the exciton relaxation dynamics and the design of spectral density distribution, 32, 36, 37 that is, phonon-bath engineering, would be also important for tuning the SF dynamics.

The present RRF analysis will be useful for such

investigations, which are in progress in our laboratory.

ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: ????. Cartesian coordinates for the herringbone pentacene dimer model

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Notes The authors declare no competing financial interests.

ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Number JP25248007 in Scientific Research (A), Grant Number JP24109002 in Scientific Research on Innovative Areas 25 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

“Stimuli-Responsive Chemical Species”, Grant Number JP15H00999 in Scientific Research on Innovative Areas “π-System Figuration”, Grant Numbers JP26107004 and JP15H01086 in Scientific Research on Innovative Areas “Photosynergetics”, and Grant Number JP2645050 in JSPS Research Fellowship for Young Scientists.


Page 26 of 58

Page 27 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


REFERENCES (1) Smith, M. B.; Michl, J. Singlet Fission. Chem. Rev. 2010, 110, 6891–6936. (2) Smith, M. B.; Michl, J. Recent Advances in Singlet Fission. Annu. Rev. Phys. Chem. 2013, 64, 361–386. (3) Schwerin, A. F.; Johnson, J. C.; Smith, M. B.; Sreearunothai, P.; Popović, D.; Černý, J.; Havlas, Z.; Paci, I.; Akdag, A.; MacLeod, M. K.; Chen, X.; David, D. E.; Ratner, M. A.; Miller, J. R.; Nozik, A. J.; Michl, J. Toward Designed Singlet Fission: Electronic States and Photophysics of 1,3-Diphenylisobenzofuran. J. Am. Chem. Soc. 2010, 114, 1457–1473. (4) Minami, T.; Nakano, M. Diradical Character View of Singlet Fission. J. Phys. Chem. Lett. 2012, 3, 145−150. (5) Minami, T.; Ito, S.; Nakano, M. Fundamental of Diradical Character Based Molecular Design for Singlet Fission. J. Phys. Chem. Lett. 2013, 4, 2133–2137. (6) Minami, T.; Ito, S.; Nakano, M. Theoretical Study of Singlet Fission in Oligorylenes. J. Phys. Chem. Lett. 2012, 3, 2719–2723. (7) Ito, S.; Minami, T.; Nakano, M. Diradical Character Based Design for Singlet Fission of Condensed-Ring Systems with 4nπ Electrons. J. Phys. Chem. C 2012, 116, 19729–19736. (8) Akdag, A. Havlas, Z. Michl, J. Search for a Small Chromophore with Efficient Singlet Fission: Biradicaloid Heterocycles. J. Am. Chem. Soc. 2012, 134, 14624–14631. (9) Zeng, T.; Ananth, N.; Hoffmann, R. Seeking Small Molecules for Singlet Fission: A Heteroatom Substitution Strategy. J. Am. Chem. Soc. 2014, 136, 12638–12647. (10) Chen, Y.; Shen, L.; Li, X. Effects of Heteroatoms of Tetracene and Pentacene Derivatives on Their Stability and Singlet Fission. J. Phys. Chem. A 2014, 118, 5700−5708. (11) Varnavski, O.; Abeyasinghe, N.; Arago, J.; Serrano-Peŕez, J. J.; Ortí, E.; Navarrete, J. T. L.; Takimiya, K.; Casanova, D.; Casado, J.; Goodson, III, T. High Yield Ultrafast Intramolecular Singlet Exciton Fission in a Quinoidal Bithiophene. J. Phys. Chem. Lett. 2015,



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

6, 1375−1384. (12) Ito, S. Minami, T. Nakano, M. Theoretical Molecular Design of Heteroacenes for Singlet Fission: Tuning the Diradical Character by Modifying π–Conjugation Length and Aromaticity. J. Phys. Chem. C 2015, 119, 148–157. (13) Eaton, S. W.; Miller, S. A.; Margulies, E. A.; Shoer, L. E.; Schaller, R. D.; Wasielewski, M. R. Singlet Exciton Fission in Thin Films of tert-Butyl-Substituted Terrylenes. J. Phys. Chem. A, 2015, 119, 4151−4161. (14) Wen, J.; Havlas, Z.; Michl, J. Captodatively Stabilized Biradicaloids as Chromophores for Singlet Fission. J. Am. Chem. Soc. 2015, 137, 165−172. (15) Zirzlmeier, J.; Lehnherr, D.; Coto, P. B.; Chernick, E. T.; Casillas, R.; Basel, B. S.; Thoss, M.; Tykwinski, R. R.; Guldi, D. M. Singlet Fission in Pentacene Dimers, Proc. Natl. Acad. Sci. USA 2015, 112, 5325–5330. (16) Johnson, J. C.; Nozik, A. J.; Michl, J. The Role of Chromophore Coupling in Singlet Fission. Acc. Chem. Res. 2013, 46, 1290−1299. (17) Wang, L.; Olivier, Y.; Prezhdo, O. V.; Beljonne, D. Maximizing Singlet Fission by Intermolecular Packing. J. Phys. Chem. Lett. 2014, 5, 3345−3353. (18) Zeng, T.; Hoffmann, R.; Ananth, N. The Low-Lying Electronic States of Pentacene and Their Roles in Singlet Fission. J. Am. Chem. Soc. 2014, 136, 5755−5764. (19) Monahan, N.; Zhu, X.-Y. Charge Transfer-Mediated Singlet Fission. Annu. Rev. Phys. Chem. 2015, 66, 601−618. (20) Zirzlmerier, J.; Lehnherr, D.; Coto, P. D.; Chernick, E. T.; Casillas, R.; Basel, B. S.; Thoss, M.; Tykwinski, R. R.; Guldi, D. M. Singlet Fission in Pentacene Dimers. Proc. Natl Acad. Sci. U. S. A. 2015, 112, 5325–5330. (21) Ito, S.; Nagami, T.; Nakano, M. Design Principles of Electronic Couplings for Intramolecular Singlet Fission in Covalently-Linked Systems. J. Phys. Chem. A 2016, 120, 28 ACS Paragon Plus Environment

Page 28 of 58

Page 29 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6236–6241. (22) Greyson, E. C.;Vura-Weis,J.; Michl, J.; Ratner, M. A. Maximizing Singlet Fission in Organic Dimers: Theoretical Investigation of Triplet Yield in the Regime of Localized Excitation and Fast Coherent Electron Transfer, J. Phys. Chem. B 2010, 114, 14618-14177. (23) Piland, G. B.; Burdett, J. J.; Dillon, R. J.; Bardeen, C. J. Singlet Fission: From Coherences to Kinetics. J. Phys. Chem. Lett. 2014, 5, 2312–2319. (24) Zimmerman, P. M.; Bell, F.; Casanova, D.; Head-Gordon, M. Mechanism for Singlet Fission in Pentacene and Tetracene: From Single Exciton to Two Triplets. J. Am. Chem. Soc. 2011, 133, 19944–19952. (25) Havenith, R. W. A.; de Gier, H. D.; Broer, R. Explorative Computational Study of the Singlet Fission Process. Mol. Phys. 2012, 110, 2445–2454. (26) Akimov, A. V.; Prezhdo, O. V. Nonadiabatic Dynamics of Charge Transfer and Singlet Fission at the Pentacene/C60 Interface. J. Am. Chem. Soc. 2014, 136, 1599−1608. (27) Musser, A. J.; Liebel, M.; Schnedermann, C.; Wende, T.; Kehoe, T. B.; Rao, A.; Kukura, P. Evidence for Conical Intersection Dynamics Mediating Ultrafast Singlet Exciton Fission. Nature Phys. 2015, 11, 352–357. (28) Tamura, H.; Huix-Rotllant, M.; Burghardt, I.; Olivier, Y.; Beljonne, D. First-Principles Quantum Dynamics of Singlet Fission: Coherent versus Thermally Activated Mechanisms Governed by Molecular π Stacking. Phys. Rev. Lett. 2015, 115, 107401–1–5. (29) Alguire, E. C.; Subotnik, J. E.; Damrauer, N. H. Exploring Non-Condon Effects in a Covalent Tetracene Dimer: How Important are Vibrations in Determining the Electronic Coupling for Singlet Fission? J. Phys. Chem. A 2015, 119, 299–311. (30) Ito, S.; Nagami, T.; Nakano, M. Density Analysis of Intra- and Intermolecular Vibronic Couplings toward Bath Engineering for Singlet Fission. J. Phys. Chem. Lett. 2015, 6, 4972−4977.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(31) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. Microscopic Theory of Singlet Exciton Fission. I. General Formulation. J. Chem. Phys. 2013, 138, 114102–1–16. (32) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. Microscopic Theory of Singlet Exciton Fission. II. Application to Pentacene Dimers and the Role of Superexchange. J. Chem. Phys. 2013, 138, 114103–1–12. (33) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. Microscopic Theory of Singlet Exciton Fission. III. Crystalline Pentacene. J. Chem. Phys. 2014, 141, 074705–1–12. (34) Mirjani, F.; Renaud, N.; Gorczak, N.; Grozema, F. C. Theoretical Investigation of Singlet Fission in Molecular Dimers: The Role of Charge Transfer States and Quantum Interference. J. Phys. Chem. C 2014, 118, 14192−14199 (35) Yost, S. R.; Lee, J.; Wilson, M. W. B.; Wu, T.; McMahon, D. P.; Parkhurst, R. P.; Thompson, N. J.; Congreve, D. N.; Rao, A.; Johnson, K.; Sfeir, M. Y.; Bawendi, M. G.; Swager, T. M.; Friend, R. H.; Baldo, M. A.; Vooorhis, T. V. A Transferable Model for Singlet-Fission Kinetics. Nature Chem. 2014, 6, 492–497. (36) Tao, G. Bath Effect in Singlet Fission Dynamics. J. Phys. Chem. C 2014, 118, 27258− 27264. (37) Renaud, N.; Grozema, F. C. Intermolecular Vibrational Modes Speed up Singlet Fission in Perylenediimide Crystals. J. Phys. Chem. Lett. 2015, 6, 360−165. (38) Zeng, T.; Goel, P. Design of Small Intramolecular Singlet Fission Chromophores: An Azaborine Candidate and General Small Size Effects. J. Phys. Chem. Lett. 2015, 7, 1351–1358. (39) Nakano, M.; Open-Shell-Character-Based Molecular Design Principles: Applications to Nonlinear Optics and Singlet Fission, Chem. Rec. 2016, in press. DOI: 10.1002/tcr.201600094

(40) Breuer, H.-P.; Petruccione, F. The Theory of Open Quantum Systems. Oxford University 30 ACS Paragon Plus Environment

Page 30 of 58

Page 31 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Press, Oxford, 2002. (41) Takahata, M.; Nakano, M.; Fujita, H.; Yamaguchi, K. Mechanism of Exciton Migration of Dendritic Molecular Aggregate: A Master Equation Approach Including Weak Exciton-Phonon Coupling. Chem. Phys. Lett. 2002, 363, 422–428. (42) Nakano, M.; Takahata, M.; Yamada, S.; Yamaguchi, K.; Kishi, R.; Nitta, T. Exciton Migration Dynamics in a Dendritic Molecule: Quantum Master Equation Approach Using Ab Initio Molecular Orbital Configuration Interaction Method. J. Chem. Phys. 2004, 120, 2359–2367. (43) Beck, M. H.; Jackle, A.; Worth, G. A.; Meyer, H. D. The Multiconfiguration Time-Dependent Hartree (MCTDH) Method: A Highly Efficient Algorithm for Propagating Wavepackets. Phys. Rep. 2000, 324, 1−105. (44)

Wang,

H.;

Thoss,

M.

Multilayer

Formulation

of

the

Multi-Configuration

Time-Dependent Hartree Theory. J. Chem. Phys. 2003, 119, 1289−1299. (45) Wesolowski, T. A.; Warshel, A. Frozen Density Functional Approach for Ab Initio Calculations of Solvated Molecules. J. Phys. Chem. 1993, 97, 8050−8053. (46) Swart, M.; van Duijnen, P. T. DRF90: a polarizable force field. Mol. Sim. 2006, 32, 471−484. (47) Holmes, D.; Kumaraswamy, S.; Matzger, A. J.; Vollhardt, K. P. C. On the Nature of NonPlanarity in the [N]Phenylenes. Chem.-Eur. J. 1999, 5, 3399–3412. (48) Biermann, D.; Schmidt, W. Diels-Alder Reactivity of Polycyclic Aromatic Hydrocarbons. 1. Acenes and Benzologs. J. Am. Chem. Soc. 1980, 102, 3163–3173. (49) Heinecke, E.; Hartmann, D.; Muller, R.; Hese, A. Laser Spectroscopy of Free Pentacene Molecules (I): The Rotational Structure of the Vibrationless S1 ← S0 Transition. J. Chem. Phys. 1998, 109, 906–911. (50) Yamagata, H.; Norton, J.; Hontz, E.; Olivier, Y.; Beljonne, D.; Brédas, J. L.; Silbey, R. J.;



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Spano, F. C. The Nature of Singlet Excitons in Oligoacene Molecular Crystals. J. Chem. Phys. 2011, 134, 204703/1–204703/11. (51) Sebastian, L.; Weiser, G.; Bässler, H. Charge Transfer Transitions in Solid Tetracene and Pentacene Studied by Electroabsorption. Chem. Phys. 1981, 61, 125–135. (52) Burgos, J.; Pope, M.; Swenberg, C. E.; Alfano, R. R. Heterofission in Pentacene-Doped Tetracene Crystals. Phys. Status Solidi B 1977, 83, 249–256. (53) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, Revision B.01; Gaussian, Inc.: Wallingford, CT, 2010. (54) Chan, W.-L.; Ligges, M.; Jailaubekov, A.; Kaake, L.; Miaja-Avila, L.; Zhu, X.-Y. Observing the Multiexciton State in Singlet Fission and Ensuing Ultrafast Multielectron Transfer. Science 2011, 334, 1541–1545. (55) Wilson, M. W. B.; Rao, A.; Clark, J.; Kumar, R. S. S.; Brida, D.; Cerullo, G.; Friend, R. H. Ultrafast Dynamics of Exciton Fission in Polycrystalline Pentacene. J. Am. Chem. Soc. 2011, 133, 11830–11833 (2011). (56) Zhao, J.; Ji, S.; Guo, H. Triplet–Triplet Annihilation Based Upconversion: From Triplet Sensitizers and Triplet Acceptors to Upconversion Quantum Yields. RSC Adv. 2011, 1, 937–950. (57) Sato, T.; Tokunaga, K.; Tanaka, K. Vibronic Coupling in Cyclopenatadienyl Radical: A Method for Calculation of Vibronic Coupling Constant and Vibronic Coupling Density Analysis. J. Chem. Phys. 2006, 124, 024314/1−024314/12.


Page 32 of 58

Page 33 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1.

Electronic Coupling Matrix Elements for the Herringbone Pentacene Dimer

Model (Figure 1b) Described in the Text Electronic coupling matrix element [meV] E(FE)

a

2120

E(CT)

b

2320

E(TT) c

1720

Vhh d

101.5

Vll d

-125.8

Vhl d

104.6

Vlh

d

-121.7

a

b

Estimated from refs. 48 and 49.

d

Obtained from RB3LYP/cc-pVDZ calculations.

Estimated from refs. 50 and 51.


c

Ref. 52.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 58

FIGURE CAPTIONS

Figure 1.

(a) Schematic diagram of five diabatic exciton states { S1S0 , S0S1 , CA ,

AC , TT } for singlet fission in a dimer. Direct and CT-mediated pathways as well as electronic couplings for CT-mediated pathway are also shown.

(b)

Molecular packing of a herringbone pentacene dimer model. 44

Figure 2.

Time evolution of diabatic exciton populations {P(FE), P(CT), P(TT)} and adiabatic state energy diagram of { S′+ , S′− , CT+′ , CT−′ , TT′ } for the pentacene dimer models.

Model (a)–(c) employ Vex = 0 meV (a), 100 meV (b)

and –100 meV (c), respectively, together with the other parameters (Table 1). Model (d) and (e) use E(CT) = 2720 meV (E(CT) – E(TT) = 1000 meV) and E(FE) = 1740 meV (E(FE) – E(TT) = 20 meV), respectively, while the other parameters are the same as those in Table 1. (τ) are also shown. in the inset.

TT yield (a) and SF time constant

For (e), the time evolution in the early time region is shown

In the adiabatic state energy diagram, the coefficients Cmα ({ m }

= { 1 , 2 , 3 , 4 , 5 } = { S1S0 , S0S1 , CA , AC , TT }) for each adiabatic state { α } = { S′+ , S′− , CT+′ , CT−′ , TT′ } are shown, and the direction of relaxation is shown by an arrow with a relative relaxation factor [meV].

The diabatic state energies, E(FE), E(CT) and E(TT), are indicated by

blue, red, and green bars on the energy axis, respectively.

Note that the identical

spectral density with ( λ , Ω ) = (50, 180) meV is used for each diabatic state.

Figure 3.

Diabatic exciton state map with respect to E(FE) – E(TT) and E(CT) – E(TT)


Page 35 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


offsets.

Superexchange (Spx) and sequential (Seq) regions are also shown with a

superscript (+ and – means E(CT) > E(TT) and E(CT) < E(TT), respectively) and (TT) or (FE), which indicates the lower state between FE and TT.

The dashed

line qualitatively separates the diabatic energy state configurations.

The colored

regions in the first (the right upper) and the third (the left lower) quadrants indicate the sequential regions, which are thermodynamically favorable for singlet fission and triplet-triplet annihilation, respectively.

Figure 4.

Double-triplet yield P(TT) contour map on the (E(FE) – E(TT))–(E(CT) – E(TT)) plane after 50 fs (a), 100 fs (b), 300 fs (c), and 1000 fs (d) for the realistic pentacene dimer model with Vex = 0 meV and ( λ , Ω ) = (50, 180) meV. Electronic coupling matrix elements except for E(FE) and E(CT) are given in Table 1. dimer.

Figure 5.

The white dot indicates the energy region for the realistic pentacene See the legend of Figure 3 for more details.

Spectral densities (eq 8) for three cases: (a) ( λ , Ω ) = (50, 180) meV, (b) (1290, 98) meV, and (c) (200, 180) meV.

The vertical black and red dashed lines

indicate the cutoff frequency Ω and ωS′+ − ω TT′ = 310.7 meV for the realistic pentacene dimer model, respectively.

Figure 6. Time evolution of diabatic exciton populations {P(FE), P(CT), P(TT)} and adiabatic state energy diagram of { S′+ , S′− , CT+′ , CT−′ , TT′ } for the realistic pentacene dimer model with state-dependent spectral densities (( λm ,

Ω m ) = (50, 180) meV for FE, (1290, 98) meV for CT, and (200, 180) meV for TT). The time evolution in the early time region is shown in the inset. 35 ACS Paragon Plus Environment

See the legend


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 58

of Figure 2 for more details.

Figure 7.

Double-triplet yield P(TT) contour map on the (E(FE) – E(TT))–(E(CT) – E(TT)) plane after 50 fs (a), 100 fs (b), 300 fs (c) and 1000 fs (d) for the realistic pentacene dimer model with state-dependent spectral densities (( λm , Ω m ) = (50, 180) meV for FE, (1290, 98) meV for CT, and (200, 180) meV for TT). while dot indicates the energy region for the realistic pentacene dimer.

The

See the

legend of Figure 4 for more details.

Figure 8.

Time evolution of diabatic exciton populations {P(FE), P(CT), P(TT)} and adiabatic state energy diagram of { S′+ , S′− , CT+′ , CT−′ , TT′ } for the pentacene dimer models (with E(CT) – E(TT) = 1000 meV) with state-dependent spectral densities: (ii) ( λm , Ω m ) = (1000, 180) meV for CT states (a), (iii) ( λm , Ω m ) = (1000, 90) meV for CT states (b), (iv) ( λm , Ω m ) = (1000, 180) meV for FE states (c), and (v) ( λm , Ω m ) = (1000, 180) meV for TT state (d). = (50, 180) meV are employed for the other diabatic states.

( λm , Ω m )

See Figure 2d for

the same pentacene dimer model using the identical spectral density with ( λ , Ω ) = (50, 180) meV.

See the legend of Figure 2 for more details.


Page 37 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 1.

(a) Schematic diagram of five diabatic exciton states { S1S0 , S0S1 , CA ,

AC , TT } for singlet fission in a dimer. Direct and CT-mediated pathways as well as electronic couplings for CT-mediated pathway are also shown. (b) Molecular packing of herringbone pentacene dimer model. 37 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 58

Figure 2 38 ACS Paragon Plus Environment

(Continued)

Page 39 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2. Time evolution of diabatic exciton populations {P(FE), P(CT), P(TT)} and adiabatic state energy diagram of { S′+ , S′− , CT+′ , CT−′ , TT′ } for pentacene dimer models. Model (a)–(c) employ Vex = 0 meV (a), 100 meV (b) and –100 meV (c), respectively, together with the other parameters (Table 1). Model (d) and (e) use E(CT) = 2720 meV (E(CT) – E(TT) = 1000 meV) and E(FE) = 1740 meV (E(FE) – E(TT) = 20 meV), respectively, while the other parameters are the same as those in Table 1. TT yield (a) and SF time constant (τ) are also shown. For (e), the time evolution in the early time region is shown in the inset. In the adiabatic state energy diagram, the coefficients Cmα ({ m } = { 1 , 2 , 3 , 4 , 5 } = { S1S0 , S0S1 , CA , AC , TT }) for each adiabatic state { α } = { S′+ , S′− , CT+′ , CT−′ , TT′ } are shown, and the direction of relaxation is shown by an arrow with a relative relaxation factor [meV]. The diabatic state energies, E(FE), E(CT) and E(TT), are indicated by blue, red, and green bars on the energy axis, respectively. Note that the identical spectral density with ( λ , Ω ) = (50, 180) meV is used for each diabatic state. 39 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 3. Diabatic exciton state map with respect to E(FE) – E(TT) and E(CT) – E(TT) offsets. Superexchange (Spx) and sequential (Seq) regions are also shown with a superscript (+ and – means E(CT) > E(TT) and E(CT) < E(TT), respectively) and (TT) or (FE), which indicates the lower state between FE and TT. The dashed line qualitatively separates the diabatic energy state configurations. The colored regions in the first (the right upper) and the third (the left lower) quadrants indicate the sequential regions, which are thermodynamically favorable for singlet fission and triplet-triplet annihilation, respectively.


Page 40 of 58

Page 41 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 4. Double-triplet yield P(TT) contour map on the (E(FE) – E(TT))–(E(CT) – E(TT)) plane after 50 fs (a), 100 fs (b), 300 fs (c), and 1000 fs (d) for the realistic pentacene dimer model with Vex = 0 meV and ( λ , Ω ) = (50, 180) meV. Electronic coupling matrix elements except for E(FE) and E(CT) are given in Table 1. The white dot indicates the energy region for the realistic pentacene dimer. See the legend of Figure 3 for more details.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 5. Spectral densities (eq 8) for three cases: (a) ( λ , Ω ) = (50, 180) meV, (b) (1290, 98) meV, and (c) (200, 180) meV. The vertical black and red dashed lines indicate the cutoff frequency Ω and ωS′+ − ω TT′ = 310.7 meV for the realistic pentacene dimer model, respectively.


Page 42 of 58

Page 43 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 6. Time evolution of diabatic exciton populations {P(FE), P(CT), P(TT)} and adiabatic state energy diagram of { S′+ , S′− , CT+′ , CT−′ , TT′ } for the realistic pentacene dimer model with state-dependent spectral densities (( λm , Ω m ) = (50, 180) meV for FE, (1290, 98) meV for CT, and (200, 180) meV for TT). The time evolution in the early time region is shown in the inset. See the legend of Figure 2 for more details.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 7. Double-triplet yield P(TT) contour map on the (E(FE) – E(TT))–(E(CT) – E(TT)) plane after 50 fs (a), 100 fs (b), 300 fs (c) and 1000 fs (d) for the realistic pentacene dimer model with state-dependent spectral densities (( λm , Ω m ) = (50, 180) meV for FE, (1290, 98) meV for CT, and (200, 180) meV for TT). The while dot indicates the energy region for the realistic pentacene dimer. See the legend of Figure 4 for more details.


Page 44 of 58

Page 45 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 8 (Continued)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 8. Time evolution of diabatic exciton populations {P(FE), P(CT), P(TT)} and adiabatic state energy diagram of { S′+ , S′− , CT+′ , CT−′ , TT′ } for the pentacene dimer models (with E(CT) – E(TT) = 1000 meV) with state-dependent spectral densities: (ii) ( λm , Ω m ) = (1000, 180) meV for CT states (a), (iii) ( λm , Ω m ) = (1000, 90) meV for CT states (b), (iv) ( λm , Ω m ) = (1000, 180) meV for FE states (c), and (v) ( λm , Ω m ) = (1000, 180) meV for TT state (d). ( λm , Ω m ) = (50, 180) meV are employed for the other diabatic states. See Figure 2d for the same pentacene dimer model using the identical spectral density with ( λ , Ω ) = (50, 180) meV. See the legend of Figure 2 for more details.


Page 46 of 58

Page 47 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC graphic



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22(a) 23 CT-mediated pathway 24 25 26 27 3 28 S1S0 H ex CA ≈ Vll CA H ex TT ≈ Vlh A B 2 29 CA 30 31 32 33 , Direct pathway 34 A B A B A B 35 S1S0 TT S0S1 36 Frenkel exciton (FE) state Double-triplet (TT) state 37 38 3 AC H ex TT ≈ Vhl S1S0 H ex AC ≈ −Vhh 39 2 40 41 A B 42 AC 43 44 Charge transfer (CT) state! 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Page 48 of 58

(b)

Top view

Front view

Herringbone pentacene dimer model

Vex = 0 meV (b)

Vex = 100 meV (c)

Vex = –100 meV

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(a)

Page 49 of 58



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(d)

(e)

E(CT) – E(TT) = 1000 meV E(FE) – E(TT) = 20 meV


Page 50 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


E(CT) – E(TT)

Page 51 of 58

CT

TT

FE CT TT FE

CT FE TT

Spx+(FE) Spx+(TT)

CT TT

Spx+(FE)

Seq+(TT)

Seq–(FE)

Spx–(TT) E(FE) – E(TT) FE

TT CT

FE

Spx–(FE) Spx–(TT)

TT

FE TT

CT

FE TT

FE CT

CT



(b) 100 fs 0.9 0.9

Spx+(TT)

1000 1000

0.7 0.7

Seq+(TT)

500 500

0.5 0.5

0.3 0.3

E(CT) – E(TT) [meV]

(a) 50 fs


1000 1000

500 500

00

00

Spx–(TT) -500 -500 -500 -500

00

500 500

0.1 0.1

-500 -500 -500 -500

1000 1000

0 0


1000 1000

(d) 1000 fs

(c) 300 fs 1000 1000

1000 1000

500 500

00

-500 -500 -500 -500

500 500

E(FE) – E(TT) [meV]



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 52 of 58

00

500 500

1000 1000

500 500

00

-500 -500 -500 -500



00

500 500

1000 1000


Page 53 of 58

(a) J(ω) [meV]

200 150 100 50

λ

00

100

ω Sʹ+ − ω TTʹ Ω 200 300 400

500

600

500

600

500

600

ω [meV]

(b) J(ω) [meV]

1200 λ 1000 800 600 400 200 00

Ω 100

ω Sʹ+ − ω TTʹ 200 300 400

ω [meV]

(c)

200 λ

J(ω) [meV]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


150 100 50 00

100

ω Sʹ+ − ω TTʹ Ω 200 300 400


ω [meV]


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Page 54 of 58

Page 55 of 58

(b) 100 fs 0.9 0.9

Spx+(TT)

1000 1000

0.7 0.7

Seq+(TT)

500 500

0.5 0.5

0.3 0.3

00

Spx–(TT) -500 -500 -500 -500

00

500 500



(a) 50 fs

1000 1000

500 500

00

0.1 0.1 -500 -500 -500 -500

1000 1000

0 0

1000 1000

(d) 1000 fs E(CT) – E(TT) [meV]

(c) 300 fs 1000 1000

1000 1000

500 500

00

-500 -500 -500 -500

500 500




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


500 500

00

00

500 500

1000 1000

-500 -500 -500 -500



00

500 500

1000 1000



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a)

Page 56 of 58

(b)

(λ,Ω) = # (1000,180) meV for CT# (50,180) meV for others

(λ,Ω) = # (1000,90) meV for CT# (50,180) meV for others


Page 57 of 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(d)

(c)

(λ,Ω) = # (1000,180) meV for FE# (50,180) meV for others

(λ,Ω) = # (1000,180) meV for TT# (50,180) meV for others



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Relative Relaxation Factor Analysis of Singlet Fission Dynamics

• CT Energy Offset and Energy Level Matching • Excitonic Coupling • State-Dependent Vibronic Coupling


Page 58 of 58

Quantum Master Equation Approach to Singlet Fission Dynamics of

Recommend Documents