Effects of Intermolecular Interactions on the SingletâTriplet Energy

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Effects of Intermolecular Interactions on the Singlet-Triplet Energy Difference: A Theoretical Study of the Formation of Excimers in Acene Molecules Dongwook Kim J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b02021 • Publication Date (Web): 11 May 2015 Downloaded from http://pubs.acs.org on May 15, 2015

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The Journal of Physical Chemistry

Effects of Intermolecular Interactions on the Singlet-Triplet Energy Difference: A Theoretical Study of the Formation of Excimers in Acene Molecules

Dongwook Kim

Department of Chemistry Kyonggi University 154-42 Gwanggyosan-ro, Yeongtong-gu, Suwon, 440-760, Korea

E-mail: [email protected]

ACS Paragon Plus Environment


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Abstract The effects of intermolecular interactions in the excited state of acene molecules on the singlet-triplet energy difference (∆EST) were investigated by carrying out ab initio calculations at the SOS-CIS(D0)/aug-cc-pVDZ level. Benzene, naphthalene, and anthracene molecules were employed, and their ∆EST values were compared with those of their respective cofacial excimers. Our theoretical results demonstrate that, upon the formation of excimer, the ∆EST values decrease significantly. By carrying out an excitation energy decomposition, we found that ∆EST, albeit also modulated by the changes in orbital energy difference and coulomb energy, is dominated by the difference in exchange energy between the singlet and triplet states, with the exchange energy decreasing as the intermolecular interactions become stronger. The natural transition orbital analysis suggests that the decrease in the exchange energy may be caused by the different nature of the hole and electron wavefunctions of the excimers (bonding vs anti-bonding), which gives rise to their spatial separation. Furthermore, it was found that the geometry relaxation effects depend on the spin state, thus leading to a further reduction of ∆EST.


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I.

Introduction

The energy difference between the lowest-lying singlet and triplet excited states (S1/T1), ∆EST, represents a critical factor for improving the performance of organic electronics. The use of the singlet-fission phenomenon has been proposed to obtain high-efficiency organic photovoltaics (OPVs);1-3 to this end, molecules are required to have a ∆EST value that is at least equal to or larger than their triplet energy.4 The singlet-triplet energy gap of a molecule is considered to be a tentative indicator of its performance as a host in the emissive layer of phosphorescent organic light-emitting diodes (OLEDs).5-7 In addition, recent studies have reported that the molecules with small ∆EST, so-called thermally activated delayed fluorescent (TADF) molecules, can facilitate intersystem crossing at room temperature, and hence highly efficient electroluminescence can be achieved without the assistance of heavy metal atoms such as Ir or Pt.8-23 The energy difference between singlet and triplet states is usually considered to be twice the exchange energy (K), i.e., 2K.24-26 This holds true only if both singlet and triplet states are of exactly the same nature, and the relaxations of their wavefunctions are neglected.7, 24 Nonetheless, this provides important inklings on how to manipulate the singlet-triplet energy difference; K must be reduced to decrease the singlet-triplet energy gap. This can be done by placing the hole and electron wavefunctions in separate regions. Even though their triplet excitons are typically localized, for example, host molecules for OLEDs may possess a smaller energy gap between the singlet and triplet states than their respective subunits by placing the HOMO and LUMO in different subunits.7 This strategy has also been



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successfully applied to the design and synthesis of TADF molecules.8-23 It is well known that intermolecular interactions often exert a significant influence on the intrinsic optical properties of molecules. For example, some Pt-containing blue phosphors, which were reported to emit orange light when forming excimers, were employed for single-dopant white OLEDs.27-32 In addition, despite having an intrinsic triplet energy as high as ca. 3 eV, carbazole can have a significantly lower triplet energy upon the formation of excimer, thereby undermining the performance of blue OLEDs.33-34 In the case of acene molecules, Casanova pointed out the exciton energies of tetracene derivatives decrease as the size of their aggregates increases;35 Coto et al. also carried out CASPT2 calculation on several configurations of pentacene dimers and trimers to explore the effects of intermolecular interactions on the singlet fission process.36 To the best of our knowledge, however, the effects of intermolecular interactions on the singlet-triplet energy gap have rarely been studied; in our very recent study about the benzene excimer, it was pointed out that, in the case of the cofacial configuration, the benzene dimer binds more strongly in the S1 state than in the T1 states, thereby reducing the energy gap between these states.37 It was tentatively concluded that the exchange energy might be involved in the reduction of ∆EST. A more comprehensive study was, however, then warranted. To address this critical point in more detail, both the singlet and triplet excimers of acene molecules were theoretically investigated in this study. In addition to benzene, naphthalene and anthracene were also taken into account; only the face-to-face eclipsed configuration was considered, because, in this configuration, acene molecules are known to be the most


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stable in the excited states.38-39 The effect of intermolecular interactions on the singlettriplet energy difference was also discussed and important conclusions were drawn.

II. Computational Details and Theory Time dependent Density Functional Theory (TD-DFT) has been widely and successfully employed to investigate the electronic structures of various organic semicondotors. In particular, recent studies suggested that certain exchange-correlation functionals can properly reproduce experimental ∆EST of various TADF molecules.40-41 However, it has not been fully explored that the proposed DFT functionals are also capable of appropriately describing intermolecular interactions among π-conjugated systems where dynamic electron correlations play a pivotal role. On the other hand, scaled opposite spin (SOS)MP242-43 and SOS-CIS(D0)44-46 methods can deal with dynamic electron correlations adequately and were successfully employed in the previous study of benzene excimer.37 Thus, they, albeit more time-consuming, are chosen again in this study In order to evaluate the excimer binding energies for the acene dimers, the potential energy curves (PECs) for both S1- and T1-state dimers were computed at the SOS-CIS(D0) level of theory with the aug-cc-pVDZ basis sets using the ground-state monomer geometries which were optimized at the SOS-MP2/aug-cc-pVDZ level. In addition, geometry relaxation effects, which turned out to depend on the spin states, were further investigated by deriving the excited-state optimal geometries for both monomers and dimers at the SOS-CIS(D0)



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level. The natural transition orbital (NTO) analysis47 was performed to elucidate the nature of the excited states of the acene molecules and their dimers.

Within the CIS approximation, excitation energies are obtained from the solution of the following eigenvalue problem48 AX = ΩCIS X

(1)

where X is the matrix of excitation amplitudes, and A is the Hamiltonian matrix for the singly excited determinants with its elements being Aar ,bs = ( ε r − ε a ) δ abδ rs + 2 ( ar bs ) − ( ab rs )

(2)

The indices a, b and r, s label occupied and virtual orbitals, respectively; ε a and ε r denote the eigenvalues of the respective corresponding canonical orbitals, φa and φr . In this equation, the first term corresponds to the energy difference between orbitals associated with the electronic transition; the second term,

( ar bs ) ,

and the third one,

( ab rs ) , can be interpreted as exchange- and coulomb-type integrals of hole-electron pair, respectively (K vs J).49 Considering the spatial symmetries of singlet and triplet excited determinants following Pauli exclusion principle, and ignoring orbital relaxation, the corresponding single excitation energies, 1 ∆Ear and 1

3

∆Ear , are derived as follows:50

∆Ear = ε r − ε a + 2 K ra − J ra


(3)

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3

∆Ear = ε r − ε a − J ra

where J ra and K ra correspond to

( rr aa )

and

( ra ar ) ,

(4) respectively. Hence, ∆EST is

calculated to be 2Kra when both singlet and triplet excited states correspond to the transition between φa and φr . Herein, one should note that, according to HF theory, ε r contains 2 J ra and − K ra , i.e.,51

ε a = ( a h a ) + ∑  2 ( aa bb ) − ( ab ba ) 

(5)

b≠ a

ε r = ( r h r ) + ∑  2 ( rr bb ) − ( rb br )  b

(

)

= r h r + ∑  2 ( rr bb ) − ( rb br )  + 2 ( rr aa ) − ( ra ar )

(6)

b≠ a

where Z 1 h (1) = − ∇12 − ∑ A 2 A r1 A

(7)

As a result, Kra does take part in both singlet and triplet excitation energy, but in the opposite ways, i.e., K ra for the former and − K ra for the latter. Therefore, in order to better understand the role of K, an excitation energy decomposition was carried out at the CIS level.52

All calculations described in this study were performed using the Q-Chem program



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package (version 4.0).53-54

III. Results and Discussion 1) ∆EST of acene monomers Table 1 lists the calculated vertical and adiabatic transition energies for the lowest-lying singlet and triplet states of benzene, naphthalene, and anthracene. A comparison of the theoretical results with the experimental values55 demonstrates that the latter, albeit consistently overestimated, are well reproduced, i.e., the energy differences range between ca. 0.12 and ca. 0.32 eV. Notably, the computational results associated with the ∆EST values are more accurate, i.e., the difference between the calculated and experimental values is only ca. 0.05–0.16 eV. This demonstrates that the ab initio calculations performed at the SOS-CIS(D0) level of theory can provide a sufficiently reliable prediction of the ∆EST values of the acene dimers.56

As discussed above, K is the important energy component that makes the singlet and triplet excited states different. In the excited state, K for hole and electron pair can be expressed as follows:

K = ψ h (1)ψ e ( 2 )

1 ψ e (1)ψ h ( 2 ) r12


(8)

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where r12

is inter-charge distance, and ψ h

and ψ e

denote hole and electron

wavefunctions, respectively. As indicated, K can qualitatively be assessed by the size of wavefunction overlap, ψ hψ e , at small inter-charge distance r12 , which helps understand the role of K in determining the natures of singlet and triplet states. Figure 1 presents natural transition orbital pairs for the lowest lying singlet and triplet excited states of benzene at the ground-state geometry. In the ground state, benzene has doubly degenerate HOMOs/LUMOs and thus the orbital energy differences for various π-π* transitions among these orbitals are expected to be the same. However, S1 and T1 states of benzene are, as clearly discerned, of different characteristics. In the case of S1 state, the hole wavefunction has maximum amplitudes where the electron wavefunction has nodes and vice versa, leading to a reduction in wavefunction overlap. On the other hand, T1 state is depicted by the pairs of hole and electron wavefunctions both of which have maximum amplitudes at the same positions. Therefore, S1 state corresponds to the linear combination of CSFs with small K while T1 state is characterized by those with significant K; K values for S1 and T1 states are calculated to be ca. 0.18 eV and ca. 0.79 eV, respectively (Table S1 in supporting information).57

2) ∆EST of acene dimers

K as a function of the intermolecular interactions In order to explore the effects of the intermolecular interactions on ∆EST, we firstly



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calculated the PEC as a function of the intermolecular distance using the frozen-monomer structures (Figure S1 in the supporting information); the ∆EST values are calculated on the basis of the minima of the PECs in the respective states (Table 2). Our results clearly show that the ∆EST values of the acene molecules significantly decrease upon the formation of excimer. ∆EST changes from ca. 0.92 eV to ca. 0.62 eV for benzene, and that of naphthalene is reduced from ca. 1.16 eV to ca. 0.96 eV. In particular, ∆EST of anthracene decreases from ca. 1.58 eV to ca. 0.95 eV, which corresponds to ca. 40% reduction. In addition, the ∆EST reduction exactly matches the binding energy difference between the singlet and triplet excimer. In the previous report, it was shown that the intermolecular interactions, which stabilize the binding of the benzene dimer in the excited states, depend on the spin state, i.e., the singlet excimer binds more strongly than its triplet counterpart.37 Therefore, in order to provide a deeper understanding of the effects of excimer formation on ∆EST, a detailed study on the excimer binding-energy difference was carried out.

As mentioned earlier, the excitation energy for a given excited state consists of the orbital energy difference, and J and K between hole and electron.48 Among these terms, only K depends on the spin state of the system. Thus, to clarify the reasons why ∆EST decreases upon the formation of excimer of the acene molecules, we explored the K dependence on the intermolecular distance. In particular, K and J were calculated at various intermolecular distances; data of the anthracene dimer in the S1 state are displayed in Figure 2, those of the benzene and naphthalene dimer in their S1 states are shown in Figures S2 and S3,


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respectively. As clearly shown in Figure 2, K decreases gradually with the intermolecular distance in the range of 5–10 Å, and then drops sharply as the molecules get closer; in the cases of the benzene and naphthalene dimers, a similar trend was observed (Figures S2 and S3). As a result, K in the S1 state of the anthracene dimer at the equilibrium distance of ca. 3.13 Å is calculated to be ca. 0.26 eV, while that of the monomer were computed to be ca. 0.47 eV. Unlike the benzene and naphthalene excimers (Figures 3 and S5), the triplet excimers of anthracene have a similar nature to that of its singlet counterpart (Figure 4). Therefore, the K values of both the singlet and triplet excimers are expected to be comparable. Based on the equilibrium distance of the triplet excimer, which was calculated to be 3.25 Å, the difference in the K values between the singlet and triplet excimers was estimated to be at least ca. 0.54 eV. As discussed above, however, K tends to contribute to the stabilization of triplet state,24, 37 and therefore, the K value for the triplet excimer is expected to be larger than that estimated from the singlet excimer;57 thus, the contribution of K to the total value of ∆EST is expected to be even larger. In summary thus far, ∆EST and K diminish along with the intermolecular distance and the latter takes the most part in the former.

As the acene molecules approach each other, the enhanced molecular orbital interactions decrease the HOMO-LUMO energy gap, facilitating the electronic transition between these orbitals. Because the HOMO of a dimer, in general, stems from the anti-bonding-type interactions between the HOMOs of each monomer, hole is driven out towards the



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outermolecular domain. On the other hand, owing to the bonding nature of the dimer’s LUMO, the electron density is concentrated in the intermolecular region; see Figures 3, 4, and S6 for benzene, anthracene, and naphthalene dimers, respectively. Therefore, the different nature of the hole and electron wavefunctions gives rise to their spatial separation, which entails a smaller overlap, ψ hψ e , at the closer inter-charge distance,

, and

consequently, a smaller K. In addition, this also leads to a net increase in the distance between hole and electron, and hence the binding energy between them, i.e., J is expected to be reduced. Figure 2 shows that the magnitude of J is indeed reduced as the molecular orbital interactions are strengthened, and its evolution is parallel to that of K, suggesting the same origin for their behaviors.58

K: singlet vs triplet states As stated above, the triplet excimer is expected to have a larger K value than its singlet counterpart; the NTO pairs for singlet and triplet excimer of anthracene shown in Figure 3 provide another important inkling for this. In the case of anthracene, the major and dominant contribution to both the S1 and T1 states arises from the same NTO pair, i.e., the combination of the anti-bonding type of ψ h

and the bonding type of ψ e . However, its

triplet excimer has an additional contribution from the transition between bonding nature and

of the

of the anti-bonding nature, which accounts for more than 7% of the

total characteristics of the T1 state. This transition clearly behaves against the major one,


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accumulating hole density and attenuating the electron density in the intermolecular region. Hence, both J and K are expected to increase. For example, the J value for the anthracene dimer was calculated to be ca. -4.65 eV and ca. -6.42 eV in the S1 state and in T1 state at the intermolecular distance of 3.25 Å, respectively. This type of NTO pair is also observed in the triplet excimer of benzene and naphthalene (Figures 3 and S5). Therefore, a larger K value in the T1 state can be considered to be a general tendency.

In addition, this picture is consistent with the fact that the triplet wavefunctions tend to be rather contracted compared to the singlet ones. The strong molecular orbital interactions of the dimer in close proximity lead to more diffuse singlet and triplet wavefunctions. The additional contribution to the T1-state wavefunction, however, somewhat retreats the charge density back to each monomer and hence shrinks the wavefunction accordingly.

The results presented so far allow us to provide a plausible explanation for the dependence of the excimer binding energy on the spin state. K is known to destabilize the singlet state, but to stabilize the triplet state.18 Thus, as a result of the reduction in K, the binding energy of the singlet excimer is further enhanced, while that of the triplet excimer is reduced. This is also consistent with the aforementioned fact that the triplet state has an additional contribution that restores the reduced values of K.



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It is also noteworthy that the effect of the intermolecular interactions on ∆EST is significantly larger for the anthracene excimer than that of the benzene or naphthalene excimers. This is likely related to the fact that the singlet and triplet anthracene excimers have similar characteristics, while those of benzene and naphthalene significantly differ; both benzene and naphthalene exhibit only a small K in the S1 state, and thus the effects of the intermolecular interactions for the singlet excimers are marginal. For example, from the calculation of the T1-like singlet state, K of T1-state naphthalene was estimated to be at least ca. 0.56 eV and is expected to drop by ca. 0.20 eV when the excimer forms (Table 3 and Figure S4). In contrast, K in the S1 state is computed to be ca. 0.13 eV, and the intermolecular interactions reduce it by only ca. 30 meV (Table 3). The reduction in K of singlet anthracene, however, was computed to be ca. 0.21 eV, which is comparable to that of the triplet state. Therefore, a state with a larger value of K tends to be more susceptible to intermolecular interactions.

Geometry relaxation effects In order to explore the geometry relaxation effects on ∆EST, excimer structures were fully optimized; Table 4 collects the calculated interplanar distances of benzene, naphthalene, and anthracene dimers. As these dimer structures relax in the excited states, monomers get closer to each other, and in the cases of naphthalene and anthracene, each monomer becomes bent in the middle as pointed out in the literature.32, 59 What is more intriguing, however, is that, as Tables 1 and 2 demonstrate, the reductions of ∆EST values upon excimer


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formation are even larger when the geometry relaxation is taken into account; compared with the values of the respective relaxed monomers, ∆EST of benzene, naphthalene, and anthracene dimers are reduced by ca. 0.37 eV, ca. 0.30 eV, and ca. 0.76 eV, respectively. Compared to the reduction of ∆EST, i.e., ∆(∆EST), which was computed using the frozen structure, these values are increased by ca. 2.9–6.2 %. As pointed out previously,37 this is due to different geometry relaxation effects of different spin states. Because the geometric changes for cofacial excimers are evenly shared by each monomer, the relaxation energies are expected to be reduced, leading to a reduction of the excimer binding energies; results in Table 5 demonstrate that reorganization energies, λex, for each molecule in the acene excimers are significantly smaller than those for corresponding monomers. For instance, λex values for an isolated naphthalene molecule are computed to be ca. 0.43 and ca. 0.93 eV for S1 and T1 states, respectively, and these values decrease to ca. 0.22 for S1 excimer and ca. 0.47 eV for T1 excimer. In addition, as λex. values in Table 1 suggest, the geometry relaxation for a monomer is more significant in the T1 state than S1 state; the reduction of the excimer binding energies follows a similar trend. For example, the relaxation energy of the anthracene molecule in the T1 state is larger by ca. 0.19 eV than that of the S1 state, and the reduction of the excimer binding energy upon geometry relaxation are calculated to be ca. 10 meV and ca. 0.14 eV for the singlet and triplet excimers, respectively. As a result, upon the formation of excimer, ∆EST further decreases (by ca. 0.13 eV) when the geometric relaxation effects are included.



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Excitation energies vs ∆EST From the perspective of interconversion processes between states with different multiplicities, e.g., singlet fission or triplet-triplet annihilation, relative ∆EST with respect to actual excitation energies is also noteworthy. As π-conjugation length of acene molecules increases, orbital energy difference is expected to decrease. In addition, the expansions of hole and electron wavefunctions likely reduce both J and K; results in Table 3 indeed demonstrate the reduction in K values for triplet states. As a result, actual excitation energies for both singlet and triplet states decrease as acene molecules become larger; T1 excitation energies, for example, continue to decrease from be ca. 3.87 eV for benzene via ca. 2.78 eV for naphthalene to ca. 1.94 eV for anthracene (Table 1). As indicated by the evolution of K value, ∆EST values are also expected to decrease as the molecule become larger. However, they are calculated to be otherwise; ca. 1.04 eV, ca. 1.38 eV, and ca. 1.65 eV for benzene, naphthalene, and anthracene, respectively (Table 2). This apparently surprising result is ascribed to the inconsistent nature of their singlet states. Unlike that for anthracene, S1 states for benzene and naphthalene is characterized by small K value (less than 0.2 eV; see Table 3), and hence different in nature from their triplet counterparts. When tetracene and pentacene are also considered, however, the reduction in ∆EST becomes more evident. ∆EST were experimentally measured to be ca. 1.49 eV for anthracene, ca. 1.07 eV for tetracene, and ca. 1.02 eV for pentacene.4 One should note that the size of K value itself is rather small (Table S1), and hence the effects of enlarged πconjugation length on K remain marginal. In the case of T1 state, the reduction in the


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excitation energies is computed to be ca. 1.81 eV while that in K is calculated to be ca. 0.32 eV. Therefore, the reduction in ∆EST is not as significant as that in excitation energy. Consequently, the ratio of ∆EST to actual excitation energy is expected to increase along with the size of molecules as long as S1 and T1 states are of similar nature, which is consistent with the fact that singlet fission process is endoergic for anthracene, and less endoergic for tetracene, but exoergic for pentacene.

IV. Synopsis In an attempt to elucidate the effects of intermolecular interactions on ∆EST, we carried out ab initio calculations on the formation of the excimers in acene molecules. As the intermolecular interactions get into action and become stronger, the K value of the holeelectron pair is reduced, leading to a reduction of ∆EST. This is mainly due to the spatial separation between the hole and electron wavefunctions induced by the strong molecular orbital interactions. In addition, the decrease in ∆EST is more significant in the case of anthracene excimers, where the S1 and T1 states are of similar nature. We have also shown that the effect of the geometric relaxation depends on the spin state, which leads to a further reduction of ∆EST. Given the increasing importance of the TADF phenomenon in enhancing OLED performance, this work provides a fundamental understanding of ∆EST, which may support the design and development of new TADF materials. Finally, our results offer a



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strong indication that any phenomenon that depends on the spin state, particularly in solidstate organic electronics such as OLEDs and OPVs, may be influenced by intermolecular interactions.

Supporting Information Calculated excitation energy components of acene monomers; cartesian coordinates of the optimized geometries of the acene monomers and dimers in the ground and excited states; the comparison of excitation energies of acene molecules calculated using aug-cc-pVDZ and aug-cc-pVTZ basis sets; potential energy curves (PECs) of the S1 and T1 excimers of the acene molecules studied in this work; pictorial representations of the variation of the exchange and coulomb energies as a function of the intermolecular distance for the benzene and naphthalene dimers; natural transition orbital (NTO) pairs for the S1 and T1 excimer of naphthalene dimers. This material is available free of charge via the Internet at http://pubs.acs.org

Acknowledgement The author would like to thank Dr. Yihan Shao for implementation of the code for the excitation energy decomposition into Q-Chem program. This work was supported by the Basis Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0025653).


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Table 1. Electronic transition energies (eV) of the benzene, naphthalene, and anthracene monomers. Naphthalene

Benzene

a

Anthracene

S1

T1

S1

T1

S1

T1

∆EVA.a

5.09

4.17

4.40

3.24

3.93

2.36

∆EVE.a

4.73

3.35

3.97

2.31

3.51

1.51

∆EAd.b

4.91

3.87

4.16

2.78

3.59

1.94

λex.c

0.36

0.82

0.43

0.93

0.42

0.85

Expt.

4.78,d 4.79e

3.67d

3.97f

2.64f

3.31f

1.82f

4.59f

3.66f

∆EVA. and ∆EVE. denote the vertical transition energies at the ground-state and the

respective excited-state geometries, respectively. b∆EAd. corresponds to the adiabatic transition energy based on the optimized geometries in the ground and respective excited states.

c

λex. represents relaxation energy in the excited state; this is calculated by the

difference between ∆EVA. and ∆EVE. dRef. 60, eRef. 61, fRef. 55.


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Table 2. Calculated singlet-triplet energy differences (∆EST) and excimer binding energies (-∆E) for the excimers of benzene, naphthalene, and anthracene.a Benzene Frozen

Naphthalene Relaxed

Frozen

Anthracene Relaxed

Frozen

Relaxed

1.38

1.58

1.65

Monomer ∆EST

0.92

1.04

1.16

Dimer ∆EST

0.62

0.67

0.96

1.08

0.95

0.89

∆(∆EST)b

0.30

0.37

0.20

0.30

0.63

0.76

-∆E(S1)b

0.83

0.74

0.99

0.90

1.72

1.71

-∆E(T1)b

0.53

0.37

0.79

0.60

1.09

0.95

Expt.c a

0.34,d >0.36e

0.73, 0.76f

All values (in eV) are computed at the SOS-CIS(D0)/aug-cc-pVDZ level of theory. b-

∆E(S1) and -∆E(T1) are the binding energies of the singlet and triplet excimers, respectively; ∆(∆EST) denotes the reduction of ∆EST upon excimer formation, which also corresponds to the binding energy difference between the singlet and triplet excimers. c

Experimental values correspond to the binding energies of singlet excimers. dRef. 62, eRef.

63. fAdjusted data from the experimental binding energy obtained from Ref. 64; for further details, see Ref. 65.



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Table 3. The difference in exchange energies (K) between hole and electron for monomers and excimers of benzene, naphthalene, and anthracene calculated via the excited energy decomposition at the CIS/aug-cc-pVDZ level.a Benzene

Naphthalene

Anthracene

S1

T1b

S1

T1b

S1

T1b

Monomer

0.18

0.79

0.13

0.56

0.47

0.47

Excimer

0.13

0.50

0.10

0.36

0.26

0.28

∆Kc

0.05

0.29

0.03

0.20

0.21

0.19

a

All values are in eV and obtained without geometry relaxation in the respective excited

states. bExchange energies, K, for the triplet states are approximately estimated via the calculation of the T1-like singlet state; these are expected to be underestimated. See text for further details. c∆K denotes the exchange energy difference between monomer and excimer.


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Table 4. Calculated interplanar distances, Re, of benzene, naphthalene, and anthracene dimers in the ground and excited states.a Benzene Frozen

Naphthalene Relaxed

Frozen

Anthracene Relaxed

Frozen

Relaxed

Re: This work (Å) S0

3.71b

3.69b

3.63

3.59 (3.57)

3.50

3.41 (3.36)

T1

3.03b

2.94b

3.25

3.14 (3.12)

3.25

3.20 (2.99)

S1

2.93b

2.93b

3.13

3.10 (3.05)

3.13

3.13 (2.81)

Re: Other works (Å) 3.08g

T1 3.15,c 3.05,e S1 a

3.0,d

3.00-3.08f

2.91-3.01,f

All the values in this work are computed at either SOS-MP2 (S0) or SOS-CIS(D0) (S1 and

T1) level of theory using aug-cc-pVDZ basis sets. In the cases of naphthalene and anthracene, each molecule in dimer configuration is bent in the middle, and thus both the inter-centroid distances and the closest distances between the C atoms of each molecule (parenthesized) are provided. b The values are taken from Ref. 37. c Ref. 38. d Ref. 66. e Ref. 39. f Ref. 65. g Ref. 59.



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Page 24 of 39

Table 5. Calculated reorganization energies, λex, of benzene, naphthalene, and anthracene excimers.a Benzene

a

Naphthalene

Anthracene

monomer

excimer

monomer

excimer

monomer

excimer

S1

0.36

0.17

0.43

0.22

0.66

0.42

T1

0.82

0.38

0.93

0.47

0.85

0.45

All the values were derived at SOS-CIS(D0)/aug-cc-pVDZ level and are in eV. In the cases

of excimers, λex corresponds to that of each monomer.


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Figure 1. Natural transition orbital (NTO) pairs for S1 (left) and T1 (right) states of benzene at its ground-state geometry. Hole and particle wavefunctions are placed below and above the arrows, respectively. The square of singular value, λ, are on the bottom.



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. Variations in the exchange energies (K) and coulomb energies (J) of the anthracene dimer in the S1 state as a function of the intermolecular distance.


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Figure 3. Natural transition orbital (NTO) pairs for singlet (S1) and triplet (T1) excimers of benzene at the optimized geometries in the respective excited states. Electron wavefunctions are placed on top and hole wavefunctions are below them. Square of singular values, λ, are also listed below the images.



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. Natural transition orbital (NTO) pairs for the singlet (S1) and triplet (T1) excimers of anthracene at the optimized geometries in the respective excited states. Electron wavefunctions are placed on top and hole wavefunctions are below them. Square of singular values, λ, are also listed below the images.


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In fact, coulomb and exchange energies here (J and K) are related to the

correlation between the excited electron and the remnant electron. The wavefunction of the hole is, however, exactly the same as that of the remnant electron, and the hole has an opposite charge such that J and K for electron-hole pair are expected to be equal in magnitude but opposite in sign to those for electron-electron pair. Since it is considered to be more straightforward to describe an exciton as a hole-electron pair and the magnitudes of J and K are of more interest, we associate the terms, J and K, with electron-hole pair, hereafter. 50

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Electron correlation effect is also expected to play an important role in ∆EST. In

our calculation, while it tends to stabilize singlet state further, it destabilizes triplet state; see Table S1. This is presumably because HF/CIS level of theory take to some extent into account the electron correlation of triplet state, which is seemingly overestimated. We do not discuss this effect in further detail, however, since this effect is beyond the scope of this study. 53

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Further single-point calculations with larger basis sets, i.e., aug-cc-pVTZ, were

also carried out for acene monomers (Table S2). The results state that the effects of larger basis sets remain marginal; in particular, ∆EST values are nearly unaltered. 57

As mentioned in the previous section, according to CIS theory, the exchange

energy between hole and electron for triplet transitions is computed to be zero, while that


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for singlet ones is calculated to be 2K. Therefore, K for triplet state is approximated by the half of 2K for the singlet state of the similar nature. 58

We can also consider the renormalization effect of wavefunctions. As the

wavefunctions expand, their amplitude at a given position should be reduced, so that the overlap between hole and electron wavefunctions, ψhψe , also diminishes. However, the space over which the exchange integral should be calculated is enlarged; this is expected to cancel out the aforementioned effect. As a result, even if it exists, the effect of the wavefunction renormalization is likely to be marginal. 59

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65

Shirai, S.; Iwata, S.; Tani, T.; Inagaki, S. Ab initio Studies of Aromatic Excimers

Using Multiconfiguration Quasi-Degenerate Perturbation Theory J. Phys. Chem. A 2011, 115, 7687-7699. 66

Huenerbein, R.; Grimme, S. Time-Dependent Density Functional Study of

Excimers and Exciplexes of Organic Molecules Chem. Phys. 2008, 343, 362-371.


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