First Principle Prediction of Intramolecular Singlet Fission and Triplet

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Quantum Electronic Structure

First Principle Prediction of Intramolecular Singlet Fission and Triplet Triplet Annihilation Rates Hung-Hsuan Lin, Karl Y. Kue, Gil C. Claudio, and Chao-Ping Hsu J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b01185 • Publication Date (Web): 12 Mar 2019 Downloaded from http://pubs.acs.org on March 19, 2019

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Journal of Chemical Theory and Computation

First Principle Prediction of Intramolecular Singlet Fission and Triplet Triplet Annihilation Rates Hung-Hsuan Lin,† Karl Y. Kue,‡,‡ Gil C. Claudio,‡ and Chao-Ping Hsu∗,† †Institute of Chemistry, Academia Sinica, 128 Section 2 Academia Road, Nankang, Taipei 115, Taiwan ‡Institute of Chemistry, University of the Philippines Diliman, Quezon City, 1101, Philippines E-mail: [email protected] Phone: +886 2 2789 8659. Fax: +886 2 2783 1237 Abstract Intramolecular singlet fission and triplet-triplet annihilation (TTA) has been experimentally observed and reported. However, problems remain in theoretically accounting for the corresponding intramolecular electronic couplings and their rates. We used the fragment excitation difference (FED) scheme to calculate the coupling with states from restricted active-space spin-flip configuration interaction. We investigated three covalently linked pentacene dimers via a phenyl group in an ortho-, meta- and para-arrangement. The singlet fission and TTA couplings were enhanced when two chromophores were covalently linked. With the Fermi golden rule, both the estimated singlet fission and TTA rates were in line with the experimental results. For systems with significant singlet-fission coupling, charge-transfer components were observed in the excited states involved, and charge-transfer states were also seen within 1 eV above

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the singlet excited states. Our approach allows for an analysis of through-bond versus through-space singlet fission in the full electronic wave functions. The FED scheme is useful for calculating intramolecular singlet-fission and TTA couplings.

Introduction During singlet fission, a singlet exciton splits into two triplet excitations 1,2 . By increasing the yield of triplet excitons 3–7 , singlet fission can increase the efficiency of solar cells. 7,8 . Because of its potential for solar energy conversion, singlet fission has been widely investigated. 4–7,9 As shown in eq. (1), singlet fission involves the conversion of a singlet excited state (S1 + S0 ) into a multiexcitonic state (ME, 1(TT), and the subsequent decoupling and separation of a correlated triplet pair (T1 ): 1,2 S1 + S0 →1 (TT) → T1 + T1 .

(1)

This process competes with others such as radiationless relaxation pathways. To achieve an efficient singlet fission, designing molecules with a high singlet fission rate has been much discussed 10–12 For this purpose, first-principle prediction of singlet-fission rates plays a crucial role 10,13,14 . The kinetics of singlet fission is primarily determined by the energetics of the processes involved 15,16 . In the weak-coupling limit, the rate for singlet fission can be estimated by the Fermi golden rule, kif =

2π |Vif |2 (FCWD), ~

(2)

where Vif is the electronic coupling between the initial and final diabatic states, and FCWD is the Franck-Condon weighted density of states, which arises from the energy conservation 17,18 . For efficient singlet fission, it is important to have a large electronic coupling factor. Because singlet fission involves changes of two electron occupation, the coupling can be approximated

2


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as a standard two-electron exchange integral; this result implies that the coupling strength would be too small to produce picosecond singlet-fission rates. However, further studies have shown that the strength of singlet fission couplings could be mediated by charge-transfer (CT) states, 19–27 , which requires a careful consideration of the electronic structure. Lowlevel and easily accessible models for excited states such as configuration interaction singles (CIS) and Time-dependent density functional theory (TDDFT) have various of problems in treating the CT states 28,29 , so the CT mediation effect cannot be properly predicted with such models. It is, therefore, important to test singlet-fission calculation with other excited state models. Electronic coupling is sensitive to the intermolecular configuration, as shown in studies with crystals. 21,26,30–33 However, controlling the arrangements of chromophores in crystals or other amorphous states is difficult. Covalently linked dimers, whereby the two chromophores involved in the singlet fission are linked with well-defined structures, were developed and studied. 11,34–39 For example, the ortho, para or meta linkage in covalent tetracenes with their singlet fission rates was reported. 34,40,41 Similar approaches were also reported for pentacene. 10,35,36,42,43 In these works, the effects of through-bond coupling, as well as the spatial arrangements for singlet fission were investigated. The electronic effects of linkage position in a phenyl linker were also studied and reported, with picosecond time scales typically observed. Theoretical computation for covalently linked systems, such as those studied experimentally in Ref. 11,36 is also seen. Characterization of diabatic coupling for covalently linked systems requires forming diabatic states for the overall systems. The commonly used approaches with separately calculated wavefunctions of the fragments risk a chance of underestimating the through-bond effects of the linker, and they require inclusion of an explicit set of CT states 36,44 . Another approach is to reduce the problem to 1-electron molecular orbital representation with a Green’s function formalism 10 where only 1-electron Fock matrix elements were considered. Since a direct singlet fission itself involves an exchange of

3



two electrons, neglecting many small, two-electron coupling terms may be over-simplifying. Nonadiabatic coupling (NAC) as estimated by the norm of transition density matrices for the adiabatic states (eigenstates) were reported 45,46 . The norm is proportional to the intermolecular NACs and it exhibits small variation with large change in intramolecular NACs. 46 There is also no intuitive way to estimate the corresponding rate directly from the norm of transition density. To the best of our knowledge, a proper quantitative tool of describing intramolecular SF process remains a challenge. Therefore, it is desirable to explore the computational methodology and the underlying physics for such systems. Another factor that affects the triplet yields of singlet fission is triplet-triplet annihilation (TTA), which takes two triplet excitations and annihilates back to one singlet excited state. 34,47–49 . Theoretical characterization of TTA rates is not common 50,51 . TTA is the reverse process of singlet fission, and the two processes share essentially the same theoretical grounds in electronic coupling, hS1 S0 | H0 |1 (T1 T1 )i. Therefore, it is necessary to extend the electronic coupling calculation and rate estimation for singlet fission to those of TTA and report the feasibility of such extensions. Fragment spin or excitation difference (FSD/FED) schemes are useful for calculating the singlet-fission coupling. 26 , and an extension of FED to wavefunctions with multiple excitation was developed 52 . Restricted-active-space spin-flip (RAS-SF) has been shown a useful method to obtain the S1 S0 and T1 T1 states simultaneously. 16,20,53–56 In the present work we used the FED scheme with RAS-SF method to obtain singlet-fission/TTA coupling, for the ortho-, meta- and para-linked pentacene dimers studied and reported recently 35 . We found that the FED scheme combined with the Fermi Golden Rule can be useful for estimating the rate for singlet fission as well as TTA. We also found that the CT state plays a role in singlet fission by quantifying the contribution of CT configuration. The phenyl linker offers a channel for CT for the two pentacene fragments, and enhance the singlet-fission couplings.

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Methods The systems studied in this work are as shown in Figure 1. The molecules ortho, meta and para were used to model the molecules o-2, m-2 and p-2 studied in Ref. 35, with the triisobutylsilyl group replaced by a methyl group to reduce computational complexity. We also developed through-space models by removing the linking phenyl groups and capping the broken bonds with a hydrogen atom, forming ortho-1, meta-1 and para-1 systems. To mimic the structure in the geometry of the S1 S0 state, the initial state of singlet fission, the molecular structure was first fully optimized at the ground state, and one of the 6,13-bis(ethenyl)pentacene moieties was replaced by its S1 -optimized structure, followed by another optimization but with the S1 -optimized fragment fixed. All the structural optimizations were done with TDDFT and DFT with the B3LYP density functional and DZ* basis sets. Subsequently, RAS-SF was calculated with the 6-31G* basis sets. RAS-SF requires a high spin reference and the triplet restricted open-shell Hartree-Fock was used here. We used 4 active electrons and 4 active orbitals in RAS-SF calculations because singlet fission involves the highest occupied and lowest unoccupied molecular orbitals of the donor and acceptor. The setting was verified by using different numbers of active orbitals and electrons, and insignificant changes in excitation energies for the relevant states were seen. (Table S1 in the supplementary information accompanying this work). As an intermediate excited state model, we have also tested similar calculation with spin-flip extended configuration interaction singles (SF-XCIS) 57,58 . We included such results in the supplementary information accompanying this work, for the interested readers. All calculations reported in the present work were performed with the a developmental version of Q-Chem. 59

The FED scheme In an earlier work, the FSD scheme was used to characterize singlet fission coupling 26 . However, since RAS-SF is a spin-restricted model, without spin polarization for the singlet states

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* * *

*

*

*

* *

*

(a) ortho

(b) meta

(c) para

(d) ortho-1

(e) meta-1

(f) para-1

Figure 1: Model molecules calculated in the present work, where two pentacenes are covalently linked with a phenyl group in an (a) ortho-, (b) meta-, and (c) para- arrangement. (d)-(f), the corresponding truncated models. involved in singlet fission, it is impossible to utilize FSD. In this work, we generalized the FED scheme for calculating singlet-fission/TTA coupling. In its original form, FED is used to form two excitation-localized diabatic states, |ψi i and |ψj i, from the linear combination of two eigenstates,|ψ1 i and |ψ2 i

|ψi i = cos φ |ψ1 i + sin φ |ψ2 i ,

(3)

|ψf i = − sin φ |ψ1 i + cos φ |ψ2 i ,

(4)

where θ is the transformation parameter. To obtain the excitation-localized states, the excitation density is defined as the sum of attachment and detachment density 60 :

ρex (r) ≡ ρdetach (r) + ρattach (r)

(5)

where ρdetach (r) and ρattach (r) are the negative-definite and positive-definite part of the difference density between the ground and excited state. The excitation difference between the

6


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donor and acceptor can be calculated by Z

ρ(mn) ex (r)dr

∆xmn =

Z

ρ(mn) ex (r)dr,

−

r∈D

(6)

r∈A

where D and A represent the donor and acceptor regions, which is now listed for diagonal (where n = m) or off-diagonal (with n 6= m) matrix elements. The excitation-localized state can be obtained by diagonalizing the difference matrix ∆x. We note that ∆x is generally not symmetric. The symmetrized ∆x was often used, 61,62 :

∆¯ x=

1 ∆x + ∆x† . 2

(7)

FED relies on the partition of a difference density into an attachment and detachment density. The corresponding off-diagonal ∆¯ x requires a similar treatment for the transition density between the two states involved. However, the definition of attachment and detachment densities (Eq. (5)) is unclear when more than single excitation is considered. Here, we used a newly developed method, the angle-scan scheme, to find the angle with which the Hamiltonian can be transformed on an excitation-localized basis. 52 . With this method, the extent of the excitation localization is calculated by

∆X(θ) = ∆¯ x1 (θ) − ∆¯ x2 (θ).

(8)

The excitation-localized states are obtained by finding the largest ∆X(θ) and its corresponding angle θ. The FED coupling is then obtained from the off-diagonal element of the Hamiltonian transformed into the excitation-localized basis VFED =

E1 − E2 2

sin 2θ,

where E1 and E2 are the excitation energies for the two excited states.

7


(9)


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FED scheme is a general method to calculate electronic coupling. In applying FED to singlet fission problems, we note that it is helpful to start from an asymmetric structure with a localized S1 S0 state. In this way, there are distinct values in ∆x for the two states, allowing FED to find the linear combination for the diabatic state by the best separation in ∆X. Treating highly symmetric systems for singlet fission would require a further improvement in FED, and this is a problem we will aim to solve in the near future.

Important configurations and their weightings The states involved in singlet fission/TTA couplings have the configurations listed in FigA ure 2. 63,64 Panel (a) describes the localized excitation (LE) on the donor |SD 1 S0 i (configuraA tions 1 and 2 in Figure 2) and the acceptor |SD 0 S1 i (configurations 3 and 4). Collectively,

the localized excitation wavefunction can be denoted as

|ΨLE i = c1 |Ψ1 i + c2 |Ψ2 i + c3 |Ψ3 i + c4 |Ψ4 i .

(10)

The final state is mainly a multiexcitonic state |1 (T1 T1 )i (configurations 5-10), which can be denoted as ME

|Ψ

i=

10 X

ci |Ψi i

(11)

i=5

The CT state |D− A+ i (configurations 11 and 12) and |D− A+ i (configurations 13 and 14) may be mixed with the initial state, influencing the singlet couplings. In the bottom of Figure 2 the CT configurations were included,

|ΨCT i = |D− A+ i + |D− A+ i = c11 |Ψ11 i + c12 |Ψ12 i + c13 |Ψ13 i + c14 |Ψ14 i .

8


(12) (13)

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Finally, the total electronic wave function is generally composed of these components 65,66 :

|Ψi = |ΨLE i + |ΨCT i + |ΨM E i + |ΨRes i ,

(14)

where |ΨRes i includes the residual configuration. To quantify the contribution of each character, we defined the following weighting factor wX :

wX =

X

|ci |2 ,

(15)

i∈X

where X is one of the characters (LE, ME, or CT) and ci is the coefficient of configurations with character X. For example, the weight of excitation localized in the donor is given by w(LE) = |c1 |2 + |c2 |2 .

LE

ME

CT

L H L H L H

D A 1 5 11

2

9

10

13

8

7

4

6

12

3

14

Figure 2: Relevant configurations of the donor and acceptor in terms of molecular orbitals. L and H represent LUMO and HOMO and D and A represent the donor and acceptor. LE A D A denotes localized excitation configurations, SD 1 S0 and S0 S1 ; ME is for multiexcition configu1 rations, (T1 T1 ); and CT is for charge-transfer configurations, D− A+ and D+ A− .

The Franck-Condon weighted density of states and the rate estimation The Fermi golden rule is a practical approach to estimate the energy transfer rates 18,21 . To include a dense collection of states in the condensed phases, the delta function in the

9



original form is replaced with the Franck-Condon weighted density of states (FCWD). In the singlet excitation energy transfer, the Förster theory evaluates the FCWD by the overlap of the acceptor absorption and donor emission spectra. In our previous work, we showed that this approach can be generalized to estimate triplet excitation energy transfer rates. 18 Here we followed the same principle and obtained the required parameters from experimental data (for 0-0 transition, vibronic frequencies and spectral line width) 67–69 and DFT/TDDFT (for reorganization energy). With other details included in the supporting information, the FCWD for singlet fission obtained was 2.097 eV−1 , and for TTA, was 0.005 eV – 1 .

Results and discussions Estimating singlet fission and TTA rates The calculated electronic couplings of meta, ortho and para are as listed in Table 1. The A 1 FED couplings between the SD 1 S0 and (T1 T1 ) states are denoted as VFED . The largest

coupling is in ortho, whereas for para, it is of the same order of magnitude, with the weakest coupling in meta. The predicted singlet fission rates are also listed, with the fastest being that of ortho, followed by para and then meta. With RAS-SF, our calculation yields results that are largely similar to those derived from transient spectroscopy experiments 35 . To study the effect of the linkers, results for truncated systems, ortho-1, meta-1 and para-1, are also included in Table 1. The couplings of meta-1 and para-1 nearly disappeared (at the order of 10−7 eV), while a through-space component is seen in ortho-1. We have also estimated the corresponding TTA rates as summarized in Table 1, by assuming that the electronic coupling hS1 S0 | H |1 (T1 T1 )i were the same but with a different FCWD value (0.005 eV−1 ). Singlet fission and TTA share the same set of initial and final states, and thus they share the same electronic coupling. The main difference in the rates are in the FCWD, which comes from the lower energy for T1 T1 states as compared to the S1 S0 state. We have also test our calculation with both 6,13-bis(ethenyl)pentacene moieties 10


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Table 1: Singlet-fission and corresponding triplet-triplet annihilation (TTA) coupling rates.

Molecule ortho meta para ortho-1 meta-1 para-1 a b c d

Singlet Fission RAS-SF Exp. a b |VFED | kSF kSF c 16.4 0.563 13.9 1.57 3.36×10−4 3.15×10−4

5.4×1012 6.3×109 3.9×1012

2.0×1012 1.6×1010 3.7×1011

TTA Calc. kTTA

Exp. kT d

2.1×1010 1.5×107 9.2×109

1.3×1010 4.6×108 5.8×1010

calculated with the basis set 6-31G*, in meV units. in the units of 1/sec. Ref. 35 The decay rate of T1 reported in Ref. 35.

optimized at the T1 state under a ground-state DFT calculation. The corresponding coupling values are similar: 20.9, 0.83 and 21.6 meV for ortho, meta and para, respectively, yielding rates in the same order of magnitudes. These rates are generally smaller, in a similar order of magnitudes, as compared to experimentally observed triplet decay rates 35 . Since the experimental triplet decay rates can be regarded as an upper bound for the TTA rates, we believe our results are consistent with these experimental observations, and they could offer useful insights for understanding the triplet decay channels. We also note that the largest difference in the rates of singlet fission and TTA is in their FCWDs. FCWD is essentially the probability for an initial state finding a final state that is in resonance, with thermal population in the vibronic levels considered. It is small if the 0-0 transition energy for a process is nearly zero or endothermic. On the other hand, FCWD is the largest when the process is exothermic, and the energy drop is roughly the reorganization energy, leading to a good overlap in the “donor emission” and “acceptor absorption” spectra (Figure S2 in supporting information 18 ). With TIPS-pentacene as the model monomer, the excess energy in S1 allows for a good overlap in probability (i.e. weighted state density) for initial-final state resonance in the direction of singlet fission. While for TTA such a chance

11



is minimal: only a small amount of thermally populated higher vibronic states can proceed with TTA. For other systems, such as tetracenes where the triplet pair and the singlet states are almost the same in energy or slightly endothermic 34 , their FCWD would basically be just an overlap of the zeroth vibronic peaks, which leads to a fraction of the current pentacene’s case (such as (90 ps)−1 for crystalline tetracene 70 ).

Effects of CT configurations Previously, Ito et al. reported singlet-fission coupling with the assumption that CT mediation dominates singlet-fission coupling and used only one-electron coupling terms, the off-diagonal Fock matrix elements 71 ; their coupling value for ortho was about one order of magnitude larger than that of para, and it is also about 10 times larger than our corresponding results. Such a discrepancy and the differences in our coupling values imply the difficulty in predicting coupling values for singlet fission: it is essentially a two-electron integral, which is typically small and sensitive to the composition of the wavefunctions, and the small CT component could contribute significantly. Therefore, reducing the interaction of two full electronic wavefunctions to that of two one-electronic molecular orbitals, and skipping many two-electron terms may be problematic. Other contributions arising from overlaps (if the two fragments were calculate separately) as well as the orbital relaxation effects are supposed to be minor, but nevertheless may play an important role in determining the small singlet fission coupling. With the calculated electronic couplings obtained from the FED scheme combined with RAS-SF, the theoretically estimated rates are summarized in Table 1. As we can see, the experimental results can be successfully reproduced with the FED scheme combined with the Fermi golden rule. A The composition of the RAS-SF for the SD 1 S0 state were analyzed with Eqn. 15 and the

results are included in Figure 3. The contributions of CT configuration for ortho and para were 9.7% and 4.5%, respectively, but less than 1% for meta. 12


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1

1

1 (b) meta

0.6 0.4 0.2

0.8

(c) para

Composition

0.8

Composition

0.6 0.4 0.2

0

0 − + + − A D A SD 1 S0 S0 S1 D A D A

0.8 0.6 0.4 0.2 0

− + + − A D A SD 1 S0 S0 S1 D A D A

1

− + + − A D A SD 1 S0 S0 S1 D A D A

1 (d) ortho-1

1 (e) meta-1

Composition

0.8 0.6 0.4 0.2

0.8

Composition

Composition

(a) ortho

Composition

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


0.6 0.4 0.2

0

0.6 0.4 0.2

0 − + + − A D A SD 1 S0 S0 S1 D A D A

(f) para-1

0.8

0 − + + − A D A SD 1 S0 S0 S1 D A D A

− + + − A D A SD 1 S0 S0 S1 D A D A

A Figure 3: Composition of the SD 1 S0 states.

We note that the the S1 S0 state of meta has insignificant CT components. This can be understood by the quantum interference, with a node in the HOMO at the meta position prevents the couplings between two chromophores. Therefore, the destructive interference leads to poor charge transfer coupling. 71,72 The CT components for truncated systems, ortho-1, meta-1 and para-1, are also included in Figure 3. The S1 S0 state of ortho-1, the case with a small through-space coupling, had some contribution from CT configurations (> 1.2%). The CT configurations for meta-1 and para-1 were not seen. We observed a tight correlation in the appearance of CT configurations and the singlet-fission coupling magnitude in the present work. In Table 2, we list the excitation energies of the two CT states and the S0 S1 excitonic states relative to the S1 S0 state, where singlet fission occurs. The CT state energies are mostly within 0.5 to 1.5 eV, with that of ortho being the lowest. Both the D− A+ and D+ A− states for meta were raised by about 0.4 eV in relative to ortho, whereas for para, they were 0.2 to 0.3 eV higher. In the truncated system, the CT states for the well-separated para-1 were out of the scope of calculation, which was at least 2 eV above the S1 S0 state.

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The relative position of CT states of ortho-1 were similar to those of ortho. For meta-1 the CT state were about 0.3 eV higher as compared to the linked meta. The large energy gap between the CT states and the S1 S0 state in meta and meta-1, as well as para-1 were also in line with the small CT components in the S1 S0 state, and the small singlet fission coupling. Table 2: Energies of charge-transfer states relative to locally excited S1 S0 states Molecule ortho meta para ortho-1 meta-1 para-1 a b c d e

A b Couplingc Diabatic energy gapd ∆E[D− A+ ]a ∆E[D+ A− ]a ∆E[SD 0 S1 ]

0.598 1.075 0.821 0.559 1.359 -e

0.715 1.145 0.982 0.849 1.421 -e

0.298 0.150 0.218 0.322 0.231 0.202

141.8 51.3 100.0 90.8 29.1 21.7

0.0913 0.110 0.0858 0.265 0.223 0.197

A Relative to the SD 1 S0 state, in the units of eV. D A D A S1 S0 (S0 S1 ) represents the donor (acceptor) being locally excited. A D A Coupling between |SD 1 S0 i and |S0 S1 i, calculated with FED, in the units of meV. Derived from a simple two-state model, in the units of eV. See text for details. A CT states are not seen within 2 eV above SD 1 S0 .

Excitonic energy gaps and couplings A D A In Table 2, the energy gaps and the FED coupling values between the SD 1 S0 and S0 S1 states A D A are included. The coupling of the SD 1 S0 and S0 S1 states contains the Förster’s coupling, or

more specifically, the Coulomb coupling between the transitions, and the short-range effects including Dexter’s exchange coupling and overlap effects 73–75 . It is seen that linked molecules have larger coupling values The linking phenyl group can enhance the coupling through an dielectric mediating effect 76 . It is also interesting to see that the coupling values follows the order of ortho > para > meta, but in the unlinked model, it is ortho-1 > meta-1 > para1. We can see that in the truncated models the distance between the two fragments plays an important role. The π-orbitals of two 6,13-bis(ethenyl)pentacene fragments in both ortho and ortho-1 slightly overlap but not in others, allowing for possible short-range coupling, 14


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contributing to the much larger coupling values in these two systems. We also note that the low value of the linked meta indicates that the electronic resonance effect is important. D A A The energy gap between the two excitonic states SD 1 S0 and S0 S1 is essentially a combined

result of the local S1 energy difference of the two fragments, and the coupling. If we employ a simple two-state model, with eigenvalue difference being the energy gap listed (∆E), and with the coupling value for the diabatic state coupling (V ), then the diabatic state energy gap p can be modeled as 2 ∆E 2 /4 − V 2 , as listed in the last column of Table 2. The resonance structure of the linking phenyl provides a change in the molecular orbitals and the subsequent excited state energy. A resonance allows a smaller energy gaps among the π and π ∗ orbitals, which can contribute to lower energy gaps in excited states. Indeed the diabatic energy gap for the non-resonant meta was slightly higher as compared to that ortho and para. Another consistent observation is that the diabatic energy gaps for the truncated models are also much higher than those for the linked molecules. Therefore, the energy gap in the linked molecules was largely determined by their excitonic coupling, while that for the truncated model, it follows that of the “intrinsic” diabatic state energies.

Other considerations RAS-SF is a model that includes non-dynamical correlation, and its predictive power is limited by the dynamical correlation. The dynamical correlation can basically change two different aspects of the present work: the state energy and the coupling strength. In the former case, the state energy influences E0 in the FCWD, but the E0 values employed in the present work was from experiments. In the latter case, we note that the dynamical correlation effects were discussed for electron transfer coupling 77 . For symmetric cases where energy gap equals twice coupling, it was found that the dynamical correlation mainly shifts both states together without changing much the energy gap (coupling). With the importance CT configurations involved in singlet fission coupling, it is possible that dynamical correlation may play a minor role in determining the quality of coupling. 15



Conclusion In the present work, we used an improved FED scheme to calculate the electronic coupling of singlet fission and TTA in covalently linked systems, and the rates were estimated with the Fermi golden rule. With RAS-SF, our approach successfully produced the rates for singlet fission and TTA. The analysis of configurations revealed that the CT components play a crucial role in mediating singlet-fission couplings. Linkers lead to larger singletfission couplings by offering the through-bond CT. The calculation results demonstrate the applicability of the FED scheme for intramolecular systems, for estimating the rates for singlet fission and TTA in photovoltaics.

Acknowledgement We acknowledge support from Academia Sinica Investigator Award (AS-IA-106-M01) and the Ministry of Science and Technology of Taiwan (project 105-2113-M- 001-009-MY4). This work also benefited from events organized by the National Center for Theoretical Sciences.

Supporting Information Available The structure of the molecules studied, and the detailed scheme together with the parameters for FCWD in determining the rates, are included in the supporting information for this work.

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Graphical TOC Entry

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* *

* * *

*

*

* *

(a) ortho

(b) meta

(c) para

(d) ortho-1

(e) meta-1

(f) para-1

Figure 1, Lin, et. al.



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ME

CT

L H L H L H

D A 1 5 11

2

13

8

7

4

6

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3

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14



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1

1

0.6 0.4 0.2

(c) para

0.8

Composition

0.8

1 (b) meta

Composition

0.6 0.4 0.2

0

0.8 0.6 0.4 0.2

0 − + + − A D A SD 1 S0 S0 S1 D A D A

0 − + + − A D A SD 1 S0 S0 S1 D A D A

1

− + + − A D A SD 1 S0 S0 S1 D A D A

1 (d) ortho-1

Composition

0.8 0.6 0.4 0.2

1 (e) meta-1

0.8

Composition

Composition

(a) ortho

Composition

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


0.6 0.4 0.2

0

0.6 0.4 0.2

0 − + + − A D A SD 1 S0 S0 S1 D A D A

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0.8

0 − + + − A D A SD 1 S0 S0 S1 D A D A



− + + − A D A SD 1 S0 S0 S1 D A D A

First Principle Prediction of Intramolecular Singlet Fission and Triplet

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