Article
pubs.acs.org/JACS
The Low-Lying Electronic States of Pentacene and Their Roles in Singlet Fission Tao Zeng, Roald Hoffmann, and Nandini Ananth* Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York, 14853, United States
*S Supporting Information
ABSTRACT: We present a detailed study of pentacene monomer and dimer that serves to reconcile extant views of its singlet fission. We obtain the correct ordering of singlet excited- state energy levels in a pentacene molecule (E (S1) ■ INTRODUCTION states, the subscript 1 is used to denote the first excited state in fi the respective spin manifolds, the subscript 0 indicates the Singlet ssion (SF) is the process by which one singlet exciton, “···” generated in a material by absorbing one photon, splits into two ground state, and indicates the intermediate steps in SF 1,2 that are the subject of our study. As the number of rings in triplet excitons. The low-lying singlet-coupled state of the 11 two triplet excitons may be accessible and close in energy to the polyacenes increases, they satisfy the energy criterion better; the endoergicities of SF for anthracene,12,13 tetracene,14 and initially excited singlet, making SF a spin-allowed process. The 15−17 ensuing doubling of photon-to-electron conversion efficiency pentacene crystals have been determined to be 0.53, 0.18, − appears to be a promising pathway to surpass the Shockley− and 0.11 eV respectively; pentacene (1) is thus the smallest Queisser conversion limit (about 30%)3 for single junction acene where SF is exoergic. Experimental studies using − solar cells.4 6 transient absorption spectroscopy reveal that SF in pentacene 18−21 SF has been extensively studied in polyacenes, a family of occurs on a time scale of ∼100 fs. hydrocarbons that are formed by linearly fusing benzene rings. In 1965, SF was first proposed to explain the relatively weak and delayed fluorescence in anthracene (three rings) single crystals.7 It was invoked again in the next few years to explain fluorescence quenching and triplet exciton generation in crystalline tetracene (four rings).8,9 These early examples fi fi In identifying the mechanism of SF, a rst step is the clearly demonstrated the potential bene t of SF in photovoltaic investigation of the low-lying excited states of a pentacene materials: SF competes with, and indeed quenches the fi fl molecule. Despite signi cant advances in electronic structure uorescence of the photogenerated singlet exciton, and yields methods, the large size of pentacene limits the accuracy and two triplet excitons. Fluorescence of these triplet excitons to reliability of methods that can be used to obtain the detailed the singlet ground state is spin forbidden, making them longer- 1 ff electronic-state information necessary to characterize SF. lived and allowing them greater di usion length scales that in Previous studies include work by Paci et al. where time- turn help them reach the interface between the light-harvesting dependent density functional theory (TD-DFT) is used to material and an electron acceptor (e.g., C60). At the interface, calculate the relative energetics of the S1, T1, and T2 states of a each exciton potentially transfers an electron to the acceptor, pentacene molecule.5 Pabst and Köhn used an approximate resulting in the generation of two free charge carriers. coupled-cluster singles and doubles model (CC2) to calculate From a thermodynamic point of view, to facilitate SF: → 22 the T1 Tn excitation energies of pentacene. However, more hv accurate results require multireference electronic structure S01→→···→ST2 1 (1) methods.23,24 In the first such study,25 Zimmerman et al. used a material should satisfy the exoergicity criterion − ≥ 10 E (S1) 2E (T1) 0, or at least not fall far from it. Here, Received: January 26, 2014 S and T are used to label singlet and triplet excited or excitonic Published: March 16, 2014 © 2014 American Chemical Society 5755 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article Figure 1. (a) Symmetrically unique C−C bond lengths of our CASSCF/12π12e optimized structure for the ground (blue) and the lowest triplet (green) states in comparison with experimental values (black) taken from ref 52. (b) HOMO and (c) LUMO of pentacene. multireference Møller−Plesset perturbation theory tetracene trimer;44 Berkelbach et al. approximated the − (MRMP)26 28 to investigate the low-lying states of pentacene. couplings between the diabats using Fock matrix elements of Their work identifies the lowest-lying excited state as a doubly the HOMO and LUMO on two pentacene molecules;41 excited state corresponding to two triplet excitons effectively Beljonne et al. employed a similar scheme to investigate the generated within one molecule. This finding, however, Davydov splitting and SF in pentacene.42 Feng et al. contradicts experimental observations.1,2,29,30 investigated the possible nonadiabatic coupling between the The mechanism of SF in acenes, i.e., the detailed steps in eq single- and multiexciton states for tetracene and pentacene 1, requires a model that includes two or more molecules. An dimers.45 More recently, the same group proposed a simple early kinetic model proposed by Merrifield and Johnson and 46 − kinetic model for SF that elucidates the role of entropy in SF. elaborated by Suna,31 33 aimedtoexplainthemutual Kuhlman et al. used a combination of TD-DFT and quantum annihilation of triplet excitons, describes SF as a stepwise mechanics/molecular mechanics (QM/MM) to investigate the process: excited states of a pentacene dimer in a crystalline environment 1 and proposed an internal conversion mechanism for SF in this hv k2 k1 11 S01→+STTTTHIoo()11 Hoo I 1 1 system. Zimmerman et al. performed complete active space k−−2 k 1 (2) self-consistent field (CASSCF) calculation on a symmetric 1 pentacene dimer to identify a low-lying multiexciton state and where (T1T1) corresponds to a singlet state formed by the ffi suggested a nonadiabatic mechanism for the transition from the intimate spin-coupling of two T1 excitons. The rate coe cients 1 25 in eq 2 k (k− ) describe the formation (annihilation) of the S1 state to the (T1T1) state. In a follow-up study, the 2 2 proposed nonadiabatic mechanism was further examined for a coupled triplet excitons and k1 (k−1) measure the rates of the disentanglement and separation (entanglement and approach) dimer structure from crystalline pentacene, with restricted fl 47,48 of the two triplets. However, recent time-resolved two-photon active-space spin- ip method (RAS-SF) calculation. fi photoemission (TR-2PPE) spectroscopic experiments on The rst objective of this paper is to investigate the low-lying crystalline tetracene34 and pentacene35 suggest the simulta- excited states of the pentacene molecule. The second objective 1 is to reconcile the quite differing extant views of the mechanism neous formation of the (T1T1) and S1 states via a coherent mechanism:36 of SF in a pentacene dimer using multireference methods based on wave function theory that include both nondynamical and hv 1 dynamical correlations. Our third objective is to accurately S01→⇔[()STT11] (3) calculate diabatic states and provide a clear physical picture of In an effort to qualitatively explain these findings, recent the roles of the three different kinds diabats thought to theory has focused on three kinds of diabatic states involved in participate in pentacene SF. In this work, we confine our SF for dimers: local single-excitations on one molecule, charge- picture of SF to the steps from photoabsorption to the transfer states, and singlet-coupled multiexciton states with two occupation of the singlet-coupled triplet−triplet state, 1(T T ). ff 37−42 1 1 triplets located on di erent molecules. An expression for The further decoherence of the coupled triplets is left for a the Hamiltonian matrix elements between these states has been future quantum dynamics study. formulated by Smith and Michl using a minimum configuration 1 fi fl space. Calculations for speci c systems include that of Di ey ■ ENERGY ORDERING OF EXCITED STATES OF A and Van Voorhis who employed different DFT schemes (TD- PENTACENE MOLECULE and constrained DFT) to approximate the diabats and calculate their coupling matrix element, for the dimer of triphenylene In choosing the active space for our multireference calculations and 1,3,5-trinitrobenzene;43 Havenith et al. employed a we are guided by the work of Kawashima et al.24 who calculated nonorthogonal configuration interaction method to calculate the valence π→π* excitation spectra for anthracene and the diabatic states and their Hamiltonian couplings for a tetracene using a 12π12e (12 π orbitals, 12 electrons) active 5756 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article 2 space and the MRMP method to obtain good agreement with 0.02 EH. These invariances suggest no intruder state problem experimental measurements. All computations in this section in our calculations. are done in a state-specific manner unless further specified. We The lowest singlet excited state of the pentacene molecules is 1 optimize the monomer structures for the lowest singlet and the the B2u state mainly (84%) formed by a HOMO-to-LUMO lowest triplet states at the level of CASSCF/12π12e. As Figure (Figure 1b,c) single excitation. There is a second singlet excited − 1 1a shows, the C C bond lengths of the ground state are in state, Ag, about 0.3 eV higher in energy, that arises from reasonable agreement with experiment, with the largest double excitations: 48% HOMO to LUMO, 7% HOMO−1to deviations occurring for the “cross-ring” bonds. We then LUMO, and 6% HOMO to LUMO+1. Since excitation to this calculate the excitation energies using the MRMP/12π12e state is symmetry forbidden, it is a dark state and has been 25 scheme. Singlet energies are calculated using the S0 optimized labeled D in the literature; this state is a common feature of 57,58 structure and triplet energies obtained using the T1 structure. polyene spectroscopy. The energy ordering of these two This selection of structures is consistent with the experimental singlet excited states from our calculations is E(S1) 5757 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article fi between E(S1) and 2E(T1) is retained. However, if eq 1 con gurations to describe the essence of SF when both S1 and proceeds through a coherent mechanism (vide infra), the T1 involve only HOMO-to-LUMO excitation. These prototype ≥ fi thermodynamic condition of E(S1) 2E(T1) may not be as con gurations are schematically shown in Figure 3a and include 1 important as it was thought in the population of (T1T1) dimer state. ■ ROLES OF LOW-LYING STATES IN PENTACENE SINGLET FISSION The study of SF in pentacene requires a model with more than one molecule; the minimal model for the process is a dimer, which we now proceed to analyze. To build a dimer model, we π combine our CASSCF/12 12e optimized S0 monomer geometry with the TOPAS (the trade name for Ticona cyclo- olefin copolymers) substrate thin film intermolecular config- uration.52 This configuration is also used in the theoretical study of Kuhlman et al.11 We call this structure, shown in Figure 2, the reference dimer structure, with the two monomers Figure 3. (a) Schematic illustration of the six diabats (prototypes) involved in the pentacene SF. Vertical bars stand for electrons, while horizontal bars for molecular orbitals. “A” and “B” denote the two monomers in Figure 2. (b) A diabatic molecular orbital (DMO) of the dimer that has clear character of the HOMO on one monomer (compare with Figure 1b). (c) A canonical molecular orbital of the dimer, mainly composed of HOMOs on the two monomers. the ground state, gg, local single-excitons, eg and ge, charge- Figure 2. Arrangement of two monomers in a unit cell of the transfer configurations, ca and ac, and the singlet-coupled pentacene crystal thin film structure on a TOPAS substrate.52 Black triplet−triplet state, tt. In the diabatic state labels, the first letter and white spheres represent carbon and hydrogen atoms. Also shown corresponds to monomer unit A and the second to monomer is the RC−C distance of the dimer. unit B, and we use the notation g for ground state (like S0), t for triplet (like T1), e for a local excitation of one monomer (like labeled A and B. Similar models to study pentacene SF, without S1), c for cation, and a for anion formation of a monomer. the surrounding molecules in real crystals, have been used in The “real” diabatic states in our calculation are obtained from refs 41, 42, and 45. We note that pentacene SF is observed to the adiabatic singlet states of the dimer by applying a unitary occur within a very short time (∼80 fs) after initial transformation that maximizes the character of one prototype photoexcitation,35,36 allowing us to assume that the monomer configuration in one diabat.60 Specifically, we employ the four- structure is frozen during SF. fold scheme of Nakamura and Truhlar.61,62 Such a direct Although our dimer model captures the most essential diabatization includes both one- and two-electron contributions structural features of the pentacene crystal unit cell, it does not in the diabatic couplings. Our diabatic molecular orbitals account for solid-state effects that can lead to greater separation (DMOs), one of which is shown in Figure 3b, are obtained by between monomers and a different dielectric constant. We rotating all active orbitals, one of which is shown in Figure 3c, believe, however, that this will not affect our central conclusions to satisfy the maximum overlap reference molecular orbitals regarding the mechanism of SF in pentacene. criterion.61,62 The reference MOs are obtained from restricted In the previous section we established the sufficiency of a open-shell Hartree-Fock (ROHF) calculations on the dimer at 2o2e active space for each monomer. Accordingly, state- a large (20 Å) intermolecular distance; each DMO can be averaged extended multiconfigurational quasi-degenerate per- unambiguously associated with a frontier orbital of one turbation theory (XMCQDPT) calculations with an active monomer; the monomer DMOs are used to generate the space of 4o4e (HOMOs and LUMOs on the two monomers) diabatic configurations in Figure 3a. are performed to obtain the adiabatic energies and wave Feng et al. have argued that it is not possible to represent the functions for the six lowest singlet states for a given dimer low-lying adiabatic states of the dimer by a few diabats, as used structure. ISA shifting is employed. The six singlet adiabatic here, since their contributions do not add up to 100%.45 All d d states are labeled S0, ..., and S5, where the superscript d denotes practical diabatization schemes, however, only yield quasi- that they are dimer states (distinct from the monomer states diabatic states that change smoothly with nuclear configuration discussed above), and the subscript denotes the ordinal energy and maintain their chemical characters as much as possi- ordering. ble.60,63,64 In almost all calculations shown here, the We use only six singlets for the dimer, assuming that (and contributions of the six prototype configurations in Figure 3a here we follow Smith and Michl)1 it takes six prototype diabatic add up to more than 92% of the six adiabatic states, leading us 5758 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article d− d to believe that our diabatization scheme can provide a reference structure, there are four excited states (S1 S4), quite quantitative understanding of SF. different in character, within 0.5 eV of each other. If more Analyzing Interactions between the Diabatic States at pentacene units were included in the model, a band of states the Reference Dimer Structure. We investigate the would evolve. In consideration of SF, it is unlikely that one interactions between diabatic states of the pentacene dimer at could focus on a single excited state. ff d d its reference geometry to understand their roles in SF. The The energy di erence between S2 and S3, the two states with Hamiltonian matrix in the basis of the six diabatic states labeled substantial eg and ge components, is 0.062 eV. Following a by their dominant prototype configurations is referee’s suggestion, we note that this energy, after being scaled by 4exp(−1), gives 0.091 eV and is not far from the 0.12 eV experimental Davydov splitting of pentacene crystal.65,66 The factor of 4 comes from an increase in the splitting due to the four nearest neighbors for each pentacene in the crystal. The exp(−1) is a reduction factor comes from vibronic coupling, with the approximate Huang−Rhys factor 1.67 We calculate the d ← d d ← d transition dipole moments for S2 S0 and S3 S0 to be (5.75, −0.16, −0.24) and (0.62, 3.46, 0.37) debyes, respectively, in the crystal abc-frame. This is consistent with the earlier finding that the transitions to the Davydov-split pair of excitation are mainly where all energies are in units of meV, and the diagonal polarized in the crystal ab-plane, with the lower component elements are shifted such that E(gg) is 0 meV. Qualitatively along the a-axis and the higher along the b-axis.67 We note the similar matrices were constructed in refs 41 and 42, including a- and b-axes are oppositely defined in the present paper (and only one-electron coupling. From our XMCQDPT calculation, ref 52) and ref 67; our a-axis contains the nearest equivalent the magnitudes of the tt-ca, tt-ac, eg-ca, eg-ac, ge-ca, and ge-ac neighbors and corresponds to the b-axis in ref 67. couplings are all below 100 meV and are, as expected, smaller As eq 1 indicates, the first step of the overall SF process is than the 100−160 meV values obtained from the Hartree− d photoexcitation from S0 to one or more low-lying singlet states. Fock approximation used in ref 41. The couplings that stem To gauge the likelihood (intensity) of this transition, we from two-electron interactions, such as tt-eg, tt-ge, eg-ge, and ac- d fi consider the oscillator strength between S0 and the other ve ca, are <6 meV, validating previous assumptions that they are adiabatic states. The gg diabat forms the overwhelming negligible in comparison with the couplings that arise mainly d 1,2 component of the S0 state, as shown in Table 2. At the from one-electron interactions. The ac (ca) diabat lies about reference geometry, the energy of the gg diabat is separated 300 (1000) meV above the tt, eg, and ge. The large energy from the others by more than 1.5 eV, as shown in eq 4, and this difference between ca and ac arises from the asymmetric is true at all geometries. We can, therefore, associate the Sd-to- geometrical alignment of the two monomers and was also 0 Sd oscillator strengths with transitions between gg and the other noted by Beljonne et al.42 We provide a detailed explanation of i five diabatic states. this energy difference in Section S.2 in the SI. The transition dipole moment between gg and tt is zero since The adiabatic states are linear combinations of the six diabats, a double excitation is involved and the dipole operator is a one- electron operator. The transition dipole moments between the gg and the two charge-transfer diabats are expected to be small because they involve transitions between the HOMO and LUMO on different monomers as shown in Figure 3a. Clearly, the gg-to-eg and gg-to-ge diabatic transitions are the main d d contributors to the S0-to-Si oscillator strengths. Therefore, populating the multiexciton state via a coherent mechanism such as the one proposed by Chan et al.35,36 requires either direct or indirect coupling between the single- and multiexciton The percentage weights of the diabatic states comprising each diabats, i.e., an excited adiabatic state with substantial adiabatic state at the reference geometry are provided in contributions from both the tt and eg/ge diabats. Table 2, along with the adiabatic energies. Note that at the The negligibly small tt-eg and tt-ge Hamiltonian couplings ∼ a ( 1 meV) and the relatively large energetic separation Table 2. Energies (in eV) and Percentage Weights (%) of (∼ 110 meV) between the tt and eg/ge diabats exclude the the Six Diabats in the Six Singlet Adiabatic States at the likelihood of a direct coupling between them. Indirect coupling Reference Structure through the charge-transfer diabats seems to be the only viable 1,2,44 energy gg tt eg ge ac ca scheme. The ca and ac states have similar magnitude d couplings with the tt, eg, and ge states, but the energy of the ac S0 0.000 99.8 0.0 0.0 0.0 0.1 0.1 d state is lower than the ca state, indicating that the indirect tt-eg/ S1 1.875 0.0 85.6 4.0 2.2 8.2 0.0 d ge couplings should be mainly mediated by the ac diabat. S2 1.956 0.0 10.3 50.0 31.6 6.9 1.3 d Further, based on the decomposition of adiabatic states S3 2.018 0.0 0.0 38.4 61.6 0.0 0.0 d Sd 2.337 0.0 3.8 7.2 4.0 85.0 0.0 presented in Table 2, we note that S5 is derived primarily 4 d d from the ca diabat, and S is a nearly pure excitonic state, with S5 3.087 0.0 0.3 0.5 0.7 0.0 98.5 3 very little tt and charge-transfer character. The states with a d d d All energies are relative to the ground-state energy at the reference substantial mixing between tt, eg, and ge, such as S1, S2, and S4, structure. all have a sizable ac component. We conclude that only one of 5759 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article the charge-transfer states, the ac diabat, mediates the coupling adiabat is more than 93% described by this complementary between the multi- and single-exciton diabats. state at each RC−C, giving it nearly pure local-exciton character. d d d We note that the indirect tt-eg/ge coupling is switched on by The S1, S2, and S4 states deviate from the diabatic states that lowering the symmetry of the dimer. For example, were the contribute to them at short RC−C distances, demonstrating the dimer to have a parallel sandwich D2h structure, the tt state and importance of couplings between diabats in this region. The d the Davydov split pair of single-exciton states, energy curve of the ground state S0, which is not shown here 1/2 | ⟩ ±| ⟩ (1/2) ( eg ge ), would transform as Ag, B3g and B1u, (see Figure S.6), has a minimum at RC−C = 5.5 Å, much like the 45 respectively. As long as the D2h symmetry is maintained, there excited states. This distance is shorter than the separation of will be no tt-eg/ge coupling, direct or indirect. The dimer in the 5.9 Å in the film structure, because our dimer model lacks the pentacene crystal unit cell has a C1 symmetry. Consequently, all attraction between the surrounding pentacenes and the dimer diabatic states transform as the same irreducible representation, ones, which increases RC−C. A, and can undergo mixing. The relatively large slopes of all excited state energies at large ff Structural E ects on the Pentacene Singlet Fission: RC−C (Figure 4a) suggest substantial electrostatic contributions Intermolecular Distance. We define the intermonomer to these energies. The two charge-transfer states should have distance, RC−C, as the distance between the middle carbon more pronounced electrostatic interactions and we see this in atoms at the far edges of the monomers shown in Figure 2, their steeper slopes at large RC−C, compared to the other 42,47 following previous work. Our reference dimer geometry has diabatic states. At RC−C = 5.6 Å, the ac diabat crosses the eg and RC−C = 5.9 Å. The energies of the adiabatic and diabatic excited ge ones and becomes degenerate with the tt state at ≤ states are shown in Figure 4a and the important Hamiltonian RC−C 5.3 Å (see Figure 4a). The superexchange mechanism of indirect tt-eg/ge coupling proposed by Berkelbach et al.41 assumes an energy ordering of E(tt) fi fi Figure 5. DMOs related to the Hamiltonian couplings of (a) eg-ac; (b) Figure 4. (a) Energies of the ve diabatic states and ve excited tt-ac; (c) ge-ac. The dimer takes the reference geometry. The green adiabatic states. (b) Important couplings between the diabats along ff d dashed lines highlight the directions of orbital interactions. RC−C. All energies in (a) are di erential energies with respect to the S0 energy at the reference geometry, RC−C = 5.9 Å. Note that the energy scale in (b) is quite different from (a). suggested by Smith and Michl.2 In this picture, the three couplings are approximated by the integrals of d d ⟨ | ̂| ⟩ 1/2⟨ | ̂| ⟩ ⟨ | ̂| ⟩ ̂ couplings between diabats in Figure 4b. We note that S5 and S3 hA F hB , (3/2) hA F lB , and lA F lB , where F is the Fock have energy curves close to the ca and eg/ge diabats, operator of the gg diabat. The orbital labels use h and l to respectively, over almost the entire range of monomer represent the HOMO and LUMO, respectively, and the d separations (Figure 4a). For S5, this is due to its dominant ca subscript indicates the monomer associated with the orbital. d character. The explanation for S3 closely following the eg/ge The orbital alignments in Figure 5 demonstrate that both hB “σ ” diabatic energy curves is a little more involved. Out of the two- and lB present a -face (single phase in the region of dimensional space of quasi-degenerate and weakly coupled eg interaction) to interact with hA and lA. In Figure 5(a), hB is “ ” and ge states, one can construct a state that maximally interacts pointing toward the upper edge of monomer A, where hA has with the ac and tt states, and a complementary orthogonal state a large amplitude. Constructive interaction between hA and hB d ⟨ | ̂| ⟩ that has negligible interactions with the two diabats. The S3 is responsible for the large magnitude of hA F hB . On the 5760 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article ff d fi Figure 6. (a) E ective oscillator strength from the S0 state to the tt diabat (black circle) and contributions to it from the ve adiabatic states (red d d d d fi d plus, S1; green cross, S2; blue star, S3; pink open square, S4; cyan lled square, S5); (b) The distance in energy between the eg and ac diabats as functions of RC−C. other hand, the protruding lobe of lB toward monomer A points the oscillator strengths, the sum extending and partitioned over fi d lower, more toward the nodal surface of hA (Figure 5(b)) and the ve low-lying S states, where lA has small amplitude (Figure 5(c)). This leads to a ⟨ | ̂| ⟩ 5 5 small magnitude of lA F lB and an even smaller magnitude of eff i 2 ⟨ | ̂| ⟩ fi f ==∑∑fCitt, f hA F lB . We con rm this qualitative picture by calculating the tt tt i ⟨ | ̂| ⟩ i==1 i 1 (6) three matrix elements, h A F h B = 112 meV, ⟨ | ̂| ⟩ ⟨ | ̂| ⟩ hA F lB = 80 meV, and hA F lB = 91 meV at the reference 2 d 1/2 Here Ci,tt is the contribution of the tt diabat in the Si adiabatic geometry. Including the (3/2) factor of the tt-ac coupling we d state, f i the oscillator strength from the ground to the Si state, obtain values of 112, 98, and 91 meV for the magnitudes of the i and f tt the component of adiabatic oscillator strength between one-electron approximate eg-, tt-, and ge-ac couplings. Trans- d d S0 and Si associated with the tt state. The oscillator strengths f i lating monomer B along the RC−C vector does not change the are obtained by replacing the XMCQDPT treatment by the topology of orbital interactions in Figure 5, allowing the order formally equivalent general multiconfigurational quasidegener- in magnitude of these couplings to be maintained along the ate perturbation theory (GMC-QDPT) for ease of implemen- entire range of RC−C. tation. We note that these two MRMP schemes exhibit similar In contrast to the pair of charge-transfer diabats, the pair of accuracy, as verified by the <0.01 eV difference in excitation local-exciton states eg and ge stay close to each other, with their ff energies obtained from the two methods. di erence in energy <0.06 eV along the whole RC−C range. The eff i ’ The f tt and its component f tt s are plotted in Figure 6a as a extent of their coupling with the ac state is therefore the only i function of RC−C. A large value of f tt requires both large values factor in determining which of them is more involved in the 2 fi of Ci,tt and signi cant magnitude for f i. As described previously, pentacene SF. Based on the analysis in the previous paragraph, d f i is substantial when the extent of eg/ge admixture is large in Si . one can see that the eg state is more strongly coupled to the ac A delicate balance between the contributions of the eg, ge, and tt state. The adiabatic Sd, Sd, and Sd states in eq 5 and Table 2, d i 1 2 4 states to a given Si state is thus essential to obtain large f tt and which contain considerable tt contributions, consistently have eff 1 2 eff f tt values. As RC−C decreases, the magnitudes of f tt, f tt, and f tt more contributions from the eg state than from the ge state. start increasing at the reference geometry, RC−C = 5.9 Å (the The rapidly decreasing ac diabat and the increase in its actual geometry in the film), and reach a maximum at about coupling with the tt, eg, and ge states result in the adiabatic RC−C = 5.6 Å. Figure 6a clearly shows that the major − d energies splitting apart as RC C decreases (Figure 4a). The contribution to SF occurs by excitation to S2, a state in which crossing (or avoided crossing) of the adiabatic energy curves the eg and ge diabats dominate (more than 80%) at the 47 described in a previous work is not observed in our reference structure. This state acquires more of the tt fi ff calculations. Rather, we nd that the e ective mixing of the component as RC−C shortens (a detailed comparison of the d tt, eg, and ge, as shown by the S1,2,4 states in eq 5, supports the decompositions at R − = 5.9, 5.6, and 5.1 Å is provided in 35,36 C C coherent mechanism for SF proposed by Chan et al. Table S.3). ff eff E ective Oscillator Strengths for Transitions from the f tt maximizes at RC−C = 5.6 Å, where there is the maximum Ground State to the Multiexciton State. Although the tt mixing between the multi- (44.9%) and single-exciton (47.3%) d diabatic state cannot be directly occupied through photo- diabats in S2. As mentioned above, the ac and eg/ge diabatic excitation due to its zero transition dipole moment with the gg energy curves cross in this region (Figure 4(a)), yielding the d “ ” (S0) state, in a coherent mechanism, the dark tt state is strongest ac-mediated tt and eg/ge mixing. As RC−C further occupied through a coherent superposition with the “bright” eg shortens, the ac diabat is separated in energy from eg/ge (Figure eff or ge states. Beljonne et al. pointed out that this is equivalent to 6(b)). Consequently, the mediated mixing and f tt decrease. an intensity borrowing by the tt diabat from the eg or ge.42 To Evidently, a shorter intermolecular distance does not guarantee quantify the coherent occupation of tt upon absorption of a a more effective tt occupation. photon, we define an “effective oscillator strength” from the With the lattice parameters of the pentacene crystal structure ’ 52 ground state to the tt state as the sum over this diabat s share of in use, where RC−C = 5.9 Å, we calculate the projections of the 5761 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article fi ϕ ϕ ff d Figure 7. (a) De nition of the rotational angles A and B. (b) E ective oscillator strength from the S0 state to the tt diabat (black circle) and fi d d contributions to it from the ve adiabatic states, as functions of the rotation of monomer B around its long axis (red plus, S1; green cross, S2; blue d d fi d star, S3; pink open square, S4; cyan lled square, S5). (c) Similar to (b), but for the rotation of monomer A. eff ϕ RC−C unit vector to be 0.5, 0.87 (∼ 3/2), and 0.15 on the a-, interesting feature is that f tt drops down to close to 0 as A b-, and c-axes. We therefore predict that slightly compressing increases to about 20°, as the two monomers become ̂ ϕ the pentacene crystal along the â+ 3 b direction, to an perpendicular to each other, and rises up again as A increases ≈ further. RC−C 5.6 Å, is likely to enhance the tt occupation after photon absorption and consequently, the efficiency of the Small clockwise rotations of monomer A or B around their long axes are, thus, shown to improve the efficiency of SF in pentacene SF. fi Structural Effects on the Pentacene Singlet Fission: pentacene. In practice, the orientation might be modi ed by Intermolecular Orientation. The highly anisotropic π-type introducing some bulky substituent along the long edge of frontier orbitals shown in Figure 5 suggest that intermolecular pentacene molecule, however, this may at the same time orientation is likely to significantly affect SF in pentacene. The enlarge the intermolecular distance (i.e., RC−C) and conversely orientational degree of freedom most likely to play a role is the reduce the tt occupation. It would appear that adjusting the intermolecular distance may provide a better way to tune SF rotation of the two monomers around their respective long- ffi axes, shown in Figure 7a, since such rotation substantially e ciency. changes the π−π overlap.11,69 To investigate this effect, we first rotate monomer B around its long axis and calculate the ■ CONCLUSIONS adiabatic and diabatic energies, diabatic couplings, and effective We have presented a thorough quantum chemistry study of the oscillator strength from the ground state to the tt state, as pentacene monomer and dimer. Our calculations are based on functions of the rotational angle. We define the rotational angle, the well-developed MRMP scheme, which can treat both ϕ B, such that positive values indicate a counter-clockwise nondynamical and dynamical electron correlations accurately rotation. The rotational angle is constrained to vary from −30° and cost-effectively. For the monomer, we put our effort into to 30°, limiting our study to dimer configurations proximal to obtaining theoretically the correct energy order of pentacene’s the reference geometry. excited states. It is likely that the previously reported25 (and eff ϕ 1,2 Figure 7b shows a dramatic dependence of f tt on B. The tt subsequently disputed) energy ordering of the lowest two ff ϕ ° e ective oscillator strength vanishes for B >5 and maximizes singlet excited states is a result of an intruder state problem; ϕ ≈− ° for B 10 . The maximum value of 0.06 is smaller than the with a proper active space, an order of E(S1) 5762 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article mechanism proposed by Smith and Michl1,2 in pentacene. The ■ ASSOCIATED CONTENT ff involvement of the charge-transfer state induces e ective *S Supporting Information mixing between the multi- and single-exciton diabats in the Figures S.1−S.10; Tables S.1−S.3; all calculation results of low-lying adiabatic states, supporting the coherent mechanism pentacene monomer with detailed discussion; discussion on the for rapid pentacene SF proposed by Chan et al.35 We find that energy difference between the ac and ca diabats and on the ca ff only one of the two charge-transfer states, ac, is engaged in the diabat along RC−C; analysis of the tt e ective oscillator strength ϕ ϕ SF in pentacene; it is the low-lying charge-transfer state that as functions of B and A; coordinates and absolute energies gets closer to the multi- and single-exciton states. Moreover, the for the petacene monomer S0 state, the pentacene monomer T1 ac diabat can move into degeneracy with the single-exciton state, and the pentacene dimer reference structure. states, more effectively mediating the mixing of the bright This material is available free of charge via the Internet at single- to and dark multiexciton diabats. This finding is different http://pubs.acs.org. from the basic assumption of high-lying charge-transfer states in ■ AUTHOR INFORMATION the superexchange model, emphasizing the need to adapt the general SF model to specific cases. Corresponding Author We quantify the population of the singlet coupled multi- [email protected] exciton state after absorbing a photon by defining an effective Notes fi oscillator strength for this state and computing it as a function The authors declare no competing nancial interest. of dimer structure. Dimer structures with exceptionally high ■ ACKNOWLEDGMENTS values of the effective oscillator strength are identified. Among these, a structure with a reduced intermolecular distance is We are grateful to Professor Paul Zimmerman for sharing with especially appealing for improving SF efficiency, since it can be us his GAMESS-US inputs and pentacene monomer realized by compressing the pentacene crystal along a certain coordinates. We need to thank Professor Neil Ashcroft for direction. enlightening discussions. We also thank Professor Mark Gordon and Dr. Michael Schmidt for their continuing development of the GAMESS-US program suite. T.Z. is ■ COMPUTATIONAL METHODS especially thankful to Dr. Schmidt for his implementation of The necessity of treating both the multireference nature of their wave the fourfold diabatization scheme into GAMESS-US. T.Z. functions and dynamical correlation for acene molecules has been expresses his gratitude to the Natural Sciences and Engineering established by Hirao et al.23,24 Following their calculations for the Research Council of Canada for the Banting postdoctoral shorter acenes and Zimmerman et al.’s work on pentacene,25 we apply fellowship (201211BAF-303459-236530). R.H. is grateful to MRMP-based methods to calculate the wave functions and energies of the National Science Foundation for its financial support the low-lying electronic states of the pentacene molecule. We employ (CHE-1305872). N.A. is grateful for the support of a start-up 70 the cc-pVTZ basis set in our monomer calculations; the basis set is grant from Cornell University. trimmed to reduce computational cost, i.e., the f basis functions of carbon and d functions of hydrogen are removed.25 The MRMP ■ REFERENCES calculations for pentacene monomer are carried out using the GMC- − 54,71,72 (1) Smith, M. B.; Michl, J. Chem. Rev. 2010, 110, 6891 6936. QDPT developed by Nakano et al. For the MRMP/2o2e (2) Smith, M. B.; Michl, J. Annu. Rev. Phys. Chem. 2013, 64, 361−386. calculation of vertical excitation energies of pentacene molecule, the (3) Shockley, W.; Queisser, H. J. J. Appl. Phys. 1961, 32, 510−519. π CASSCF/12 12e optimized ground state structure is used. Intruder (4) Hanna, M. C.; Nozik, A. J. J. Appl. Phys. 2006, 100, 074510. state avoidance (see text) is employed with the conventional shifting (5) Paci, I.; Johnson, J. C.; Chen, X. D.; Rana, G.; Popovic,́ D.; David, 2 56 parameter of 0.02 EH. The same parameter is used in all ISA D. E.; Nozik, A. J.; Ratner, M. A.; Michl, J. J. Am. Chem. Soc. 2006, 128, treatments in this work. The zeroth-order states are prepared in a 16546−16553. state-averaged CASSCF manner. (6) Congreve, D. N.; Lee, J.; Thompson, N. J.; Hontz, E.; Yost, S. R.; The larger size and lower symmetry (C1) of the dimer require us to Reusswig, P. D.; Bahlke, M. E.; Reineke, S.; Van Voorhis, T.; Baldo, M. use a smaller basis set. We employ the SBKJC73,74 pseudopotential and A. Science 2013, 340, 334−337. its associated double-ζ basis set for the dimer calculations. The d (7) Singh, S.; Jones, W. J.; Siebrand, W.; Stoicheff, B. P.; Schneider, function from the 6-31G(d) basis set of carbon is added to account for W. G. J. Chem. Phys. 1965, 42, 330−342. polarization. This combined basis set is of similar quality to the 6- (8) Swenberg, C. E.; Stacy, W. T. Chem. Phys. Lett. 1968, 2, 327−328. 31G(d) basis which was used in a previous dimer study,47 for the (9) Groff, R. P.; Avakian, P.; Merrifield, R. E. Phys. Rev. B 1970, 1, − valence electrons. The carbon core 1s electrons are replaced by a 815 817. pseudopotential. The same basis set and pseudopotential are used in (10) Greyson, E. C.; Stepp, B. R.; Chen, X.; Schwerin, A. F.; Paci, I.; the MRMP/2o2e calculation for the vertical excitation energies of Smith, M. B.; Akdag, A.; Johnson, J. C.; Nozik, A. J.; Michl, J.; Ratner, M. A. J. Phys. Chem. B 2010, 114, 14223−14232. pentacene monomer. (11) Kuhlman, T. S.; Kongsted, J.; Mikkelsen, K. V.; Møller, K. B.; The XMCQDPT method of Granovsky75 is used to handle both the Sølling, T. I. J. Am. Chem. Soc. 2010, 132, 3431−3439. nondynamical and dynamical correlation for the dimer. Since (12) Wolf, H. C. Solid State Phys. 1959, 9,1−81. dynamical electron correlation is explicitly included, we do not adjust (13) Avakian, P.; Abramson, E.; Kepler, R. G.; Caris, J. C. J. Chem. the state energies as is done in refs 45 and 47. All GMC-QDPT Phys. 1963, 39, 1127−1128. ’ calculations in this work also use Granovsky s XMCQDPT zeroth- (14) Tomkiewicz, Y.; Groff, R. P.; Avakian, P. J. Chem. Phys. 1971, order Hamiltonian to have reasonable interactions between the zeroth- 54, 4504−4507. order states. (15) Sebastian, L.; Weiser, G.; Bassler,̈ H. Chem. Phys. 1981, 61, All calculations are carried out using the quantum chemistry 125−135. program package GAMESS-US.76,77 Its associated graphical software (16) Lee, K. O.; Gan, T. T. Chem. Phys. Lett. 1977, 51, 120−124. MacMolPlt78 is used to prepare all the molecular geometry and orbital (17) Burgos, J.; Pope, M.; Swenberg, C. E.; Alfano, R. R. Phys. Status images in this paper. Solidi. B 1977, 83, 249−256. 5763 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764 Journal of the American Chemical Society Article (18) Rao, A.; Wilson, M. W. B.; Hodgkiss, J. M.; Albert-Seifried, S.; (54) Ebisuzaki, R.; Watanabe, Y.; Nakano, H. Chem. Phys. Lett. 2007, Bassler,̈ H.; Friend, R. H. J. Am. Chem. Soc. 2010, 132, 12698−12703. 442, 164−169. (19) Rao, A.; Wilson, M. W. B.; Albert-Seifried, S.; Di Pietro, R.; (55) Choe, Y.-K.; Witek, H. A.; Finley, J. P.; Hirao, K. J. Chem. Phys. Friend, R. H. Phys. Rev. B 2011, 84, 195411. 2001, 114, 3913−3918. (20) Wilson, M. W. B.; Rao, A.; Clark, J.; Kumar, R. S. S.; Brida, D.; (56) Witek, H. A.; Choe, Y.-K.; Finley, J. P.; Hirao, K. J. Comput. Cerullo, G.; Friend, R. H. J. Am. Chem. Soc. 2011, 133, 11830−11833. Chem. 2002, 23, 957−965. (21) Wilson, M. W. B.; Rao, A.; Ehrler, B.; Friend, R. H. Acc. Chem. (57) Schulten, K.; Karplus, M. Chem. Phys. Lett. 1972, 14, 305−309. Res. 2013, 46, 1330−1338. (58) Hosteny, R. P.; Dunning, T. H.; Gilman, R. R.; Pipano, A.; (22) Pabst, M.; Köhn, A. J. Chem. Phys. 2008, 129, 214101. Shavitt, I. J. Chem. Phys. 1975, 62, 4764−4779. (23) Hashimoto, T.; Nakano, H.; Hirao, K. J. Chem. Phys. 1996, 104, (59) Akimov, A. V.; Prezhdo, O. V. J. Am. Chem. Soc. 2014, 136, 6244−6257. 1599−1608. (24) Kawashima, Y.; Hashimoto, T.; Nakano, H.; Hirao, K. Theor. (60) Ruedenberg, K.; Atchity, G. J. J. Chem. Phys. 1993, 99, 3799− Chem. Acc. 1999, 102,49−64. 3803. (25) Zimmerman, P. M.; Zhang, Z.; Musgrave, C. B. Nat. Chem. (61) Nakamura, H.; Truhlar, D. G. J. Chem. Phys. 2011, 115, 10353− 2010, 2, 648−652. 10372. (26) Hirao, K. Chem. Phys. Lett. 1990, 190, 374−380. (62) Nakahara, H.; Truhlar, D. G. J. Chem. Phys. 2002, 117, 5576− (27) Hirao, K. Chem. Phys. Lett. 1992, 196, 397−528. 5593. (28) Hirao, K. Chem. Phys. Lett. 1993, 201,59−66. (63) Ichino, T.; Gauss, J.; Stanton, J. F. J. Chem. Phys. 2009, 130, (29) Amirav, A.; Even, U.; Jortner, J. Chem. Phys. Lett. 1980, 72,21− 174105. 24. (64) Köppel, H.; Domcke, W.; Cederbaum, L. S. Adv. Chem. Phys. (30) Griffiths, A. M.; Friedman, P. A. J. Chem. Soc., Fraday Trans. 2 1984, 59,57−246. 1982, 78, 391−398. (65) Prikhotko, A. F.; Tsikora, L. I. Opt. Spectrosc. 1968, 25, 242. (31) Merrifield, R. E. J. Chem. Phys. 1968, 48, 4318. (66) Zanker, V.; Preuss, J. Z. Angew. Phys. 1969, 27, 363. (32) Johnson, R.; Merrifield, R. Phys. Rev. B 1970, 1, 896−902. (67) Yamagata, H.; Norton, J.; Hontz, E.; Olivier, Y.; Beljonne, D.; (33) Suna, A. Phys. Rev. B 1970, 1, 1716−1739. Bredas,́ J. L.; Silbey, R. J.; Spano, F. C. J. Chem. Phys. 2011, 134, (34) Chan, W.-L.; Ligges, M.; Zhu, X.-Y. Nat. Chem. 2012, 4, 840− 204703. 845. (68) This assumption is only valid if the two monomers are (35) Chan, W.-L.; Ligges, M.; Jailaubekov, A.; Kaake, L.; Miaja-Avila, symmetrically positioned. L.; Zhu, X.-Y. Science 2011, 334, 1541−1545. (69) Marciniak, H.; Fiebig, M.; Huth, M.; Schiefer, S.; Nickel, B.; (36) Chan, W.-L.; Berkelbach, T. C.; Provorse, M. R.; Monahan, N. Selmaier, F.; Lochbrunner, S. Phys. Rev. Lett. 2007, 99, 176402. − R.; Tritsch, J.; Hybertsen, M. S.; Reichman, D. R.; Gao, J.; Zhu, X.-Y. (70) Dunning, T. H. J. Chem. Phys. 1989, 90, 1007 1023. Acc. Chem. Res. 2013, 46, 1321−1329. (71) Nakano, H.; Uchiyama, R.; Hirao, K. J. Comput. Chem. 2002, 23, − (37) Greyson, E. C.; Vura-Weis, J.; Michl, J.; Ratner, M. A. J. Phys. 1166 1175. Chem. B 2010, 114, 14168−14177. (72) Miyajima, M.; Watanabe, Y.; Nakano, H. J. Chem. Phys. 2006, (38) Yamagata, H.; Norton, J.; Hontz, E.; Olivier, Y.; Beljonne, D.; 124, 044101. Bredas,́ J. L.; Silbey, R. J.; Spano, F. C. J. Chem. Phys. 2011, 134, (73) Binkley, J. S.; Pople, J. A.; Hehre, W. J. J. Am. Chem. Soc. 1980, − 204703. 102, 939 947. (39) Teichen, P. E.; Eaves, J. D. J. Phys. Chem. B 2012, 116, 11473− (74) Stevens, W. J.; Basch, H.; Krauss, M. J. Chem. Phys. 1984, 81, − 11481. 6026 6033. (40) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. J. Chem. (75) Granovsky, A. A. J. Chem. Phys. 2011, 134, 214113. Phys. 2013, 138, 114102. (76) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; (41) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. J. Chem. Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A., Jr. J. Comput. Phys. 2013, 138, 114103. − (42) Beljonne, D.; Yamagata, H.; Bredas,́ J. L.; Spano, F. C.; Olivier, Chem. 1993, 14, 1347 1363. Y. Phys. Rev. Lett. 2013, 110, 226402. (77) Gordon, M. S.; Schmidt, M. W. In Advances in electronic structure (43) Difley, S.; Van Voorhis, T. J. Chem. Theory Comput. 2011, 7, theory: GAMESS a decade later; Dykstra, C. E., Frenking, G., Kim, K. S., 594−601. Scuseria, G. E., Eds.; Elsevier: Amsterdam, 2005. (78) Bode, B. M.; Gordon, M. S. J. Mol. Graphics Modell. 1998, 16, (44) Havenith, R. W. A.; de Gier, H. D.; Broer, R. Mol. Phys. 2012, − 110, 2445−2454. 133 138. (45) Feng, X.; Luzanov, A. V.; Krylov, A. I. J. Phys. Chem. Lett. 2013, 4, 3845−3852. (46) Kolomeisky, A. B.; Feng, X.; Krylov, A. I. J. Phys. Chem. C 2014, 118, 5188−5195. (47) Zimmerman, P. M.; Bell, F.; Casanova, D.; Head-Gordon, M. J. Am. Chem. Soc. 2011, 133, 19944−19952. (48) Zimmerman, P. M.; Musgrave, C. B.; Head-Gordon, M. Acc. Chem. Res. 2013, 46, 1339−1347. (49) Heinecke, E.; Hartmann, D.; Muller, R.; Hese, A. J. Chem. Phys. 1998, 109, 906−911. (50) Thorsmølle, V. K.; Averitt, R. D.; Demsar, J.; Smith, D. L.; Tretiak, S.; Martin, R. L.; Chi, X.; Crone, B. K.; Ramirez, A. P.; Taylor, A. J. Phys. Rev. Lett. 2009, 102, 017401. (51) Thorsmølle, V. K.; Averitt, R. D.; Demsar, J.; Smith, D. L.; Tretiak, S.; Martin, R. L.; Chi, X.; Crone, B. K.; Ramirez, A. P.; Taylor, A. J. Physica B 2009, 404, 3127−3130. (52) Schiefer, S.; Huth, M.; Dobrinevski, A.; Nickel, B. J. Am. Chem. Soc. 2007, 129, 10316−10317. (53) Nakano, H.; Nakayama, K.; Hirao, K. J. Chem. Phys. 1996, 106, 4912−4917. 5764 dx.doi.org/10.1021/ja500887a | J. Am. Chem. Soc. 2014, 136, 5755−5764