Charge-Transfer Coupling of an Embedded Pentacene Dimer with the

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Charge-Transfer Coupling of an Embedded Pentacene Dimer with the Surrounding Crystal Matrix Piotr Petelenz* and Mateusz Snamina The K. Gumiński Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, Ingardena 3, 30-060 Kraków, Poland S Supporting Information *

ABSTRACT: Singlet exciton fission in pentacene is commonly described in terms of the dimer model, containing some adjustable parameters that implicitly account for the influence of the crystalline environment. Here we use state-of-the-art results published by other authors to base the dimer model on strictly ab initio input, and consistently extend it, explicitly including the interaction with the surrounding crystal matrix. In the multiscale approach that ensues, the molecular pair residing at the center of the model cluster is identified with the dimer for which the ab initio results are available. The main environmental effects are attributed to electrostatic stabilization of the dimer charge transfer (CT) states and to CT coupling between the dimer and the adjacent molecules. According to our calculations, this latter coupling stabilizes the lowest Frenkel state so that its energy becomes lower than that of the triplet-pair (tt) state, in dramatic contrast to the situation in an isolated dimer. The interpretationally successful level order assumed in semiempirical approaches may be restored by extending the purely electronic approach of the original ab initio calculations to include vibronic effects. The results show that extreme caution must be exercised when extrapolating dimer results to draw conclusions concerning the situation in the crystal.

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INTRODUCTION Since its prediction in 1968,1 singlet exciton fission in molecular crystals has been considered an intriguing phenomenon. Its early studies were focused mostly on the magnetic field dependence.2,3 Accordingly, its interpretation at that time was dealing with the processes operative in the spin subspace,4,5 which was based on the tacit assumption that the requisite model Hamiltonians were averaged over the spatial part of the wave function. This attitude has changed with the advent of organic electronics. In the new context, fission is viewed as a potential way to bypass the Shockley−Queisser limit for solar cell performance. The sensibility of the process to the energies of excited states involved offers a chance to maximize the triplet yield by appropriate tuning (e.g., by chemical substitution) of the electronic properties of the active ingredient. This prompts interest in the spatial part of the wave functions of the entities involved, which governs the relevant energetics. Pentacene, for which high fission efficiency was conclusively documented,6−10 seems a good candidate for studies in this direction. Theoretical description of the fission process in this system was developed in a number of papers where it was couched in terms of the dimer model representing the pair of molecules on which the resultant triplet excitons reside.11−13 The energies of the electronic configurations used as the basis in which to expand the eigenstates were usually taken from relatively simple quantum chemistry approaches and subsequently fine-tuned on intuitive grounds. In contrast, in the © XXXX American Chemical Society

recent paper by Zeng et al. the pertinent energies have been obtained from state-of-the-art ab initio calculations14 and have not been further tampered with, presumably representing the most advanced reference input to date. However, in the real system this pair is a part of the crystal, and there is no chance to describe the latter at a comparable level of approximation. Nonetheless, the crystal field does affect the states of the embedded dimer. As we have recently shown,15 the electrostatic interaction of the model dimer with its crystal surroundings has an unexpected influence on charge transfer (CT) state energies, substantially affecting their relative spacing and their separation from other states. Specifically, the energy gap between one of the CT states and the triplet pair state (tt) changes dramatically, which is likely to influence their mixing. Yet, the approach we initiated15 still failed to account for some salient features of the situation. In fact, electron exchange with the surroundings was consistently neglected: the underlying model described the dimer as if it were contained in a bubble with the walls penetrable to electrostatic interactions but impenetrable to electrons. Admittedly, the probability of the dimer to be ionized is marginal anyway. However, its CT states are bound to mix with those where one of the charges is located on a neighboring molecule. In other words, the energies of the dimer eigenstates are inevitably modified by the virtual Received: September 30, 2015 Revised: November 17, 2015

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DOI: 10.1021/acs.jpcc.5b09547 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C processes of electron exchange with surrounding molecules. This paper is intended to determine the extent to which this fact affects the earlier conclusions.14,15

(c) charge-transfer states with the electron located on the molecule displaced by the (longer) basic translation b relative to the molecule on which the hole is located. It should be stated for the sake of completeness that the crystal axes are presently labeled according to refs 16 and 17 so that the a and b crystal directions are interchanged with respect to the previous notation.12,18 In the following we will label the types of CT configurations by specifying the relative position of the electron, assuming that the hole resides at the center of the coordinates. It should be noted in this context that in the types where the cation and anion are shifted with respect to each other by one of the lattice periods charge interchange produces a configuration that is symmetry-equivalent to the initial one and has the same characteristics (for instance energy). In the cases where the cation and the anion belong to different sublattices [i.e., for (±1/2, ∓1/2,0) and (±1/2, ±1/2,0) types] the configuration generated by electron−hole interchange is not equivalent to the initial one and in some respects may substantially differ (as discussed in considerable detail in refs 13−15). The Hamiltonian used here to describe the pentacene excited (singlet) states differs from that introduced previously19 in the following respects 1. only one, instead of a few Frenkel excitons per molecule, is now taken into account, in order to link directly to the diabats of ref 14; 2. the b-directed CT states are added to the basis set [as stated in item (c) above)], which was already practiced in a different context;20 3. the tt state of ref 14 (the singlet component of triplet pair) is explicitly included. The original approach19,20 approximated the total manyelectron Hamiltonian of the crystal correct to first order in the (nearest and second-nearest neighbor) intermolecular overlap integrals and included only singly excited configurations. Here, the tt configuration is arbitrarily added since (owing to the low triplet energy) it is located in the region of lowest single excitations (in contrast to other doubly excited states). As far as overlap considerations are concerned, internal consistency is retained as long as the direct coupling of the triplet-pair state to singlet Frenkel excitons (entering in second order) is neglected. However, its inclusion would be desirable to maintain consistency with the paper of Zeng et al.14 which is our benchmark reference to assess the role of the CT coupling between the dimer and the rest of the crystal. On the other hand, extension of the Hamiltonian to second order in the overlaps would introduce a number of matrix elements that do not appear in the Hamiltonian of ref 14 and would influence the results in an uncontrollable way. On the basis of the values of similar integrals,21 their global effect is predictable, though; they would affect the tt state only slightly, possibly shifting it by a few meV. On this view, we generally adopted the Hamiltonian19,20 valid to first order in overlap integrals but tentatively complemented it by selective inclusion of those second-order terms that mediate the coupling of the tt state to Frenkel excitons. Then, by comparing the results to those obtained with the pertinent parameters set equal to zero we confirmed their marginal role. In their simplest realization, our calculations are performed for an array of 10 molecules. The model dimer of ref 14 is identified with the pair of molecules in the (0,0,0) and (1/2,−1/2,0) positions, located at the center of the cluster.

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THEORETICAL METHODS Hamiltonian. The model dimer of ref 14 is now assumed to represent the two molecules contained in the pentacene unit cell. The eg and ge dimer diabats14 are identified with a Frenkel exciton residing at the (0,0,0) and (1/2,−1/2,0) positions, respectively. The ac charge-pair diabat is represented by the CT configuration consisting of an electron located at (0,0,0) and a hole at the (1/2,−1/2,0) position, with the charges reversed for the ca diabat. In the triplet-pair diabat tt both positions are assumed to be occupied by triplet excitons. The model crystal is assumed to be rigid (vibrations are ignored) and to consist of similar unit cells infinitely repeated in three dimensions. The actual calculations are performed for a part of it, namely, for a finite 2-dimensional cluster extracted from the dense-packing crystal ab plane (cf Figure 1). This is

Figure 1. Model clusters under consideration (10, 24, and 44 molecules). The herringbone pattern depicts the molecules in the crystal ab plane. The different colors represent the gradually expanding coordination spheres around the (central, white) model dimer. The gray background corresponds to the crystalline surroundings, not included explicitly in the model.

plausible since the present study is focused on charge transfer coupling and the charge transfer integrals out of the ab plane are negligibly small. The influence of the rest of the crystal is simulated by adopting the diagonal energies of the localized (diabatic) CT configurations obtained from earlier calculations15 where the complete (3-dimensional) structure was accounted for. Using a version of the configuration interaction approach, the cluster excited states are expressed as linear combinations of localized configurations corresponding to the singlet component of the triplet-pair state of the model dimer (located at the center of the cluster), a set of Frenkel excitons (each consisting of the electron and the hole residing on one of the molecules forming the cluster), and several types of short-range CT excitons. The latter comprise (a) charge-transfer states consisting of the hole residing at (0,0,0) and the electron located on a near-neighbor molecule, i.e., at the (±1/2, ∓1/2,0) or (±1/2, ±1/2,0) position; (b) charge-transfer states with the electron located on the molecule displaced by the shortest basic translation a relative to the molecule on which the hole is located; B


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The Journal of Physical Chemistry C

Figure 2. Energy level diagrams in Version 1. The consecutive columns correspond to the model dimer and to the model clusters containing 10, 24, and 44 molecules, respectively. The color codes the fractional contributions from the tt state (green), the two Frenkel states residing at the centraldimer molecules (cyan and magenta), the two CT states located at the central-dimer molecules (black), and all other basis configurations (blue). The eigenstates deriving from (c) type CT configurations (which appear above 2.6 eV) are not shown.

method for electrostatic interaction energies in the CT states15) or cover some ground that is common with the Zeng et al. paper and are validated by the comparison (as is the case with exciton dissociation integrals calculated by Yamagata et al.22). No adjustable parameters are invoked. Most details concerning the parametrization are to be found in the Supporting Information, but the following issue deserves an in-depth comment. This paper is focused on charge transfer states in the crystal. Their diagonal energies E CT (r) may be schematically represented as

Larger clusters are constructed by adding more distant coordination spheres of the central dimer, as shown in Figure 1. Parametrization and Calculations. Following our earlier work, the Hamiltonian18 is formulated in terms of its matrix elements on the basis of the wave functions (a)−(c) that are abstract in the sense that their explicit form is not a priori specified. As is usually the case in multiscale approaches like the present one, the requisite matrix elements should be obtained from quantum chemistry calculations at a molecular level, which do require the explicit form of the molecular wave functions to be specified. The more accurate these functions are, the more realistic our results are expected to be. For this reason, wherever possible, this study is based on the parameters derived from the most advanced calculations performed to date for the pair of nearest-neighbor molecules in the pentacene crystal.14 Some quantities characterizing the crystal that are absent from the restrictive dimer model are evaluated using complementary approaches that either are the most accurate available (such as the self consistent charge field

E CT(r ) = I − A + C(r )

(1)

where r, I, and A are the electron−hole distance, the ionization potential, and electron affinity of the crystal molecules, respectively. C(r) stands for the effective electron−hole interaction energy, including their electrostatic (de)stabilization due to the crystalline environment (if any is present) and C


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Figure 3. Energy level diagrams in Version 2. The contents of the columns and the color code are the same as in Figure 2.

The test is performed in two alternative versions. In our earlier paper15 we calculated the stabilization energies of the CT states of the model dimer due to its electrostatic interaction with its crystalline surroundings but without charge exchange. By adding those electrostatic contributions to the state-of-theart ab initio diabat energies14 of the corresponding isolateddimer states we have now obtained their embedded-dimer counterparts, which are employed as the diagonal matrix elements of the Hamiltonian. This is our Version 1. However, the study15 also revealed that the influence of the surrounding crystal molecules results primarily from symmetrygoverned compensation of some intradimer electrostatic interactions, which in the sophisticated ab initio approach14 had been calculated in a different (although admittedly more accurate) approximation. By breaking symmetry, this inconsistency might have changed the compensation balance. Hence, as an alternative to Version 1 where the intradimer electrostatics are implicitly included at the ab initio level,14 we have now recalculated them in the same way as the respective terms describing the interaction of the model dimer with its crystalline surroundings15 and evaluated the embedded-dimer

dominated by polarization and charge-quadrupole contributions. The self consistent charge field (SCCF) method15 provides a reasonably good approximation to evaluate C(r) for a charge pair, no matter whether the latter is embedded in the crystal or not. Here we have taken advantage of this fact, evaluating the effective e−h interaction energies (in the two CT states with opposite polarities) also for the isolated model dimer for which Zeng et al.14 had earlier calculated the corresponding total energies (with respect to the ground state). By applying eq 1, comparison of the two sets of results enabled us to evaluate I−A as equal to 5.035 eV (which is averaged over the two CT state polarities, cf. Supporting Information). In this way we have aligned our zero of energy with that of Zeng et al. Our further strategy is to take at face value the state-of-theart results for Hamiltonian matrix elements obtained for the diabats in the isolated dimer14 and insert them into the cluster model described above. With general consistency established in this way, our aim is to probe to what extent the (virtual) charge exchange between the dimer and its crystalline environment would affect the final conclusions of ref 14. D


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⟨MTNT|H|M+N−⟩) set equal to zero (where M, N stand for the wave functions of individual molecules and the superscripts denote their excited or ionized state). In this situation the CT manifold was entirely isolated from Frenkel states. Of course, if in addition the charge transfer integrals (such as ⟨M±N|H|MN±⟩) involving only the unexcited molecules were eliminated, the calculations would return the input CT configurations as the corresponding eigenstates. Otherwise, all eigenstates of CT provenance are mixtures of the (a), (b), and (c) types of configurations enumerated earlier. Nonetheless, the eigenstates deriving primarily from configuration types (b) and (c) (with charge transferred along one of the lattice periods) are reasonably well separated on an energy scale from those with dominant nearest neighbor character. For this latter family we have calculated the center of gravity of the (±1/2,∓1/2,0) type contributions, which is defined as the mean of energy eigenvalues from this family, weighted by the fractional contributions of the (±1/2,∓1/2,0) type configurations to the corresponding eigenvectors. This has been done separately for the configurations with the hole at (0,0,0) and the electron at (±1/2,∓1/2,0) and for those with the charges reversed. The averages (for both versions of the parametrization) are collected in Table 1. They may be identified with hypothetical

energies from eq 1, using the averaged I−A value described above. The ensuing CT state energies, which in our opinion better represent the situation in the crystal, are used as the respective diagonal terms of the cluster Hamiltonian in Version 2 of our calculations. In both versions the diagonal energies of the CT states of (b) and (c) types (absent from the Zeng et al. dimer model) are obtained from eq 1 using the above I−A value and the appropriate C(r) values obtained from SCCF calculations. The details of the parametrization are available in the Supporting Information. Diagonalization of the Hamiltonian yields the eigenenergies and eigenvectors. The stability of the results has been verified by performing the calculations for clusters containing a gradually increasing number of molecules (10, 24, and 44). In order to pinpoint the role of some crucial interactions, the calculations were subsequently repeated with the corresponding coupling constants (embodied in the off-diagonal Hamiltonian matrix elements) set equal to zero (vide infra). In some cases a similar procedure was applied to test the sensitivity of the final conclusions to variation of specific less certain parameters.

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RESULTS AND DISCUSSION Sequence of Energy Levels. The resultant energy level diagrams for several cluster sizes are shown in Figure 2 (Version 1) and Figure 3 (Version 2). The colored horizontal bars code the contributions from the various basis states. Irrespective of the variant of diagonal energy estimates, the main difference in comparison with the outcome of isolateddimer models consists in the location and nature of the ttderived eigenstate(s). As soon as the model dimer is embedded in the crystal lattice (cf. columns 2−4 of Figures 2 and 3), this state is no longer the lowest eigenstate of the system but gets immersed in the singlet Frenkel exciton continuum (in the present finite model represented by a densely spaced system of cluster eigenstates), which turns it into a resonance level. Inevitably, there is no longer a single eigenstate of tt parentage, but the tt contribution is spread over a manifold of states with nonnegligible tt admixture. This is no surprise since the tt configuration is coupled via the intradimer CT configurations to other CT states of the crystal and hence indirectly to Frenkel excitations of the surrounding crystal bulk (these dominate among the “other basis configurations” in the 1900−2000 meV range, marked in blue). In effect, the eigenstates in the crystal differ enormously from their isolated-dimer counterparts. Specifically, the mixing between the tt state and the Frenkel states is dramatically increased; in fact, in most eigenstates the tt configuration is merely an admixture to a wave function dominated by singlet excitons. In addition to the density-of-states effect described above, the tt-Frenkel-manifold coupling, being mediated by superexchange-type interaction via the charge-transfer states, is further strengthened by stabilization of the latter. In fact, the CT configurations are particularly sensitive to the effect of crystalline environment. The differences were spectacular already at the level of diagonal energies: the energy gap between the configurations of opposite polarities was found to be reduced by an order of magnitude owing to crystal-field effects.15 Here this trend is amplified by the off-diagonal charge transfer terms. CT Manifold. In order to unravel the underlying physical mechanisms, we repeated the calculations with all exciton dissociation integrals (such as ⟨M S N|H|M ± N ∓ ⟩ or

Table 1. Calculated CT State Energy Shifts upon Inclusion of Charge Transfer Coupling with the Surrounding Crystal Matrixa position variant

hole

electron

diagonal energyb

center of gravityc

shift

1

(0, 0, 0) (1/2, −1/2, 0) (0, 0, 0) (1/2, −1/2, 0)

(1/2, −1/2, 0) (0, 0, 0) (1/2, −1/2, 0) (0, 0, 0)

2282 2451 2357 2376

2238 2371 2301 2316

−44 −80 −56 −60

2 a

All energies in meV. bPrior to charge transfer coupling. cCenter of gravity of the near-neighbor contribution in eigenstates of nearneighbor provenance.

energies of the two dimer CT configurations of opposite polarities (corresponding to ac and ca of ref 14), now embedded in the crystal. As the Frenkel exciton dissociation integrals have been earlier set equal to zero, the listed values refer to the dimer CT configurations prior to their mixing with intramolecular excitations and may be correlated with the corresponding diagonal energies (representing the dimer states subjected to electrostatic interactions with surrounding crystal molecules but without CT coupling). As was to be expected, the coupling to (b) and (c) type CT states, which have higher energies, pushes the effective intradimer CT configurations to lower energies. For parametrization Version 2 the resultant shift is on the order of 60 meV and is approximately the same for both CT state polarities. In contrast, for Version 1 the two polarities substantially differ in their sensitivity to this influence, which results from the different separation of the corresponding diagonal energies from those of the perturbing CT states of the crystal. In this case the upper level exhibits a larger shift of about 80 meV, to be compared to only 44 meV for the lower level. In this way, the CT coupling to the crystalline surroundings amplifies the trend resulting from purely E


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their sensitivity to the spatial extent of the basis orbitals,23,24 and may be seriously underestimated here. The same applies to the CT integrals of semiempirical provenance that have been borrowed from ref 13 and are quite close in size. This general tendency is confirmed by the results of Parker et al.25 which are rooted in a different methodology, using a far superior basis set. In order to probe the potential consequences of this effect, we have repeated the calculations with all charge-transfer integrals scaled up by 20%. This brought about no qualitative changes (the corresponding figure, analogous to Figure 3, is available in the Supporting Information), merely amplifying the trends indicated earlier: the CT-induced downward shift of the Frenkel exciton band bottom has increased by about 50 meV, and the eigenstates of tt parentage now get immersed even deeper in the quasicontinuum. Our conclusion is that the isolated dimer model, taken literally, is not a good representation of the situation in the crystal. The most spectacular discrepancy is concerned with the relative position of the tt level which in the dimer model is the lowest of all excited states, whereas in the consistently parametrized model crystal it is immersed in the Frenkel exciton band. The above conclusion is an issue of internal consistency rather than agreement with experiment. Paradoxically, if the published computational results do reproduce exactly the situation in the isolated dimer, the final conclusions of ref 14 extrapolated for the crystal are questionable. Then, as the tt state is no longer the lowest singlet excitation, the small fluorescence intensity of the pentacene crystal no longer follows as an inevitable consequence of the actual eigenstate sequence (which is now different), and the dependence of interstate mixing on intermolecular distance also has to be reconsidered. As a matter of fact, the resulting order of energy levels contradicts that underlying the existing interpretations of singlet fission.11−13 With this view, the conjecture that solid-state effects “will not affect the central conclusions regarding the mechanism of singlet fission in pentacene”, expressed in ref 14, does not seem to be tenable since it heavily relies on a different order of energy levels (with the tt state located below the Frenkel excitons). It is worth noting that the original level ordering14 is restored when vibronic effects are taken into account. In the strongvibronic-coupling approximation each molecular vibrational state (in the leading progression-forming mode) individually splits into a (vibronic) exciton band, narrowed (with respect to the purely electronic picture) by the appropriate product of vibrational overlap integrals involved. For the ground vibrational state this factor amounts to about exp(−1), as was pointed out by Zeng et al. (based on ref 22), leading to the lowest level of Frenkel parentage being raised well above the tt state. In this particular respect, vibronic interaction compensates the influence of crystalline surroundings, so that for the special case under consideration the purely electronic energy levels of the isolated dimer are qualitatively similar to the lowest vibronic (but not purely electronic) levels of the dimer in the crystal. This is accidental, since the parameters that govern these two effects are independent. If a priori design of materials optimally suited for singlet exciton fission is ever to be achieved, the mechanism of the phenomenon must be dissected into independent components. We hope that this paper provides a step in that direction.

electrostatic interactions, further reducing the spacing between the two configurations. The above results enable one to rationalize the effect of CT coupling with the surrounding molecules on the eigenstates of the model dimer. The neutral Frenkel and tt configurations are coupled to the CT states by dissociation integrals of similar size, but in the tt case the integrals governing electron and hole transfer have opposite signs,14 so that the corresponding interactions interfere destructively, in contrast to constructive interference for singlet Frenkel excitons. Moreover, the higher diagonal energies of these latter states make them more sensitive to the position of the nearby CT states with which they interact. Ultimately, the eigenstate of tt parentage is less strongly stabilized than the lowest Frenkel-derived states, which locates it in the exciton band (despite its lower diagonal energy). Apart from their interpretational role, the center-of-gravity energies of nearest-neighbor excitons may be useful in the following context. A detailed description of, e.g., fission dynamics (such as that presented in refs 11 and 12), feasible for a dimer representing the pair of molecules in which the process occurs, would be practically impossible if the pair’s crystalline neighborhood were to be explicitly accounted for. This difficulty may be circumvented if the crystal states are mapped onto the states of the model dimer, which is then viewed as a hypothetical effective entity, and the characteristics (including energies) of its electronic states (to be used as the basis set) are from the outset renormalized with respect to their isolated-dimer counterparts. Here we have proposed a way to estimate these renormalized energies.

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CONCLUSIONS The dimer model has been extensively used in the singlet fission problem,11−13 but (until ref 14 appeared) its parametrization always relied on some intuitive posit that introduced some adjustable ingredient. We believe that the dimer model viewed as an ef fective expedient to reduce the complexity of the situation in the crystal (e.g., to deal with the dynamics of the fission process, as in refs 11 and 12) is perfectly justified and should by no means be abandoned. In this context, however, its input parameters must not be those directly calculated for the isolated dimer but must implicitly account for the influence of the crystalline environment. The CT configurations, where the separated charges strongly interact with surrounding molecules, are especially sensitive to this effect, and their energies must be judiciously renormalized. In this study we have proposed a possible approach that enables one to do that and calculated the corresponding effective energies based on ab initio input. It should be emphasized that the parametrization used in this study is as close as possible to that derived by Zeng et al. for the isolated dimer diabats, so that it is based on the best state-ofthe-art ab initio calculations. In no way do we challenge the latter. Our objective has been merely to assess the influence of the crystalline environment on the model dimer, painstakingly keeping our approach consistent with the above-mentioned computational results. Inevitably, though, the accuracy of the input data provided by ref 14 is one of the factors limiting the ultimate physical validity of our results. It has to be borne in mind that the underlying basis set, although as extended as feasible for a large dimer, by general standards is not a very advanced one. This may affect especially the charge-transfer integrals, which are known for F


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(13) Beljonne, D.; Yamagata, H.; Brédas, J. L.; Spano, F. C.; Olivier, Y. Charge-Transfer Excitations Steer the Davydov Splitting and Mediate Singlet Exciton Fission in Pentacene. Phys. Rev. Lett. 2013, 110, 226402. (14) Zeng, T.; Hoffmann, R.; Ananth, N. The Low- Lying Electronic States of Pentacene and Their Roles in Singlet Fission. J. Am. Chem. Soc. 2014, 136, 5755−5764. (15) Petelenz, P.; Snamina, M.; Mazur, G. Charge-Transfer States in Pentacene: Dimer versus Crystal. J. Phys. Chem. C 2015, 119, 14338− 14342. (16) Mattheus, C. C. Polymorphism and Electronic Properties of Pentacene, Ph.D. thesis, Rijksuniversiteit Groningen, The Netherlands, 2002. (17) Qi, D.; Su, H.; Bastjan, M.; Jurchescu, O. D.; Palstra, T. M.; Wee, A. T. S.; Rübhausen, M.; Rusydi, A. Observation of Frenkel and Charge Transfer Excitons in Pentacene Single Crystals Using Spectroscopic Generalized Ellipsometry. Appl. Phys. Lett. 2013, 103, 113303. (18) Campbell, R. M.; Monteath Robertson, J.; Trotter, J. The Crystal and Molecular Structure of Pentacene. Acta Crystallogr. 1961, 14, 705−711. (19) Petelenz, P.; Slawik, M.; Yokoi, K.; Zgierski, M. Z. Theoretical Calculation of the Electroabsorption Spectra of Polyacene Crystals. J. Chem. Phys. 1996, 105, 4427−4440. (20) Stradomska, A.; Kulig, W.; Slawik, M.; Petelenz, P. Intermediate Vibronic Coupling in Charge Transfer States: Comprehensive Calculation of Electronic Excitations in Sexithiophene Crystal. J. Chem. Phys. 2011, 134, 224505. (21) Tiberghien, A.; Delacote, G. Intégrales de Transfert des Charges et des Excitons Triplets, dans les Cristaux Aromatiques. Cas du Naphthalène et de l’Anthracène. J. Phys. (Paris) 1970, 31, 637−656. (22) Yamagata, H.; Norton, J.; Hontz, E.; Olivier, Y.; Beljonne, D.; Brédas, J. L.; Silbey, R. J.; Spano, F. C. The Nature of Singlet Excitons in Oligoacene Molecular Crystals. J. Chem. Phys. 2011, 134, 204703. (23) Blumberger, J. Recent Advances in the Theory and Molecular Simulation of Biological Electron Transfer Reactions. Chem. Rev. 2015, 115, 11191−11238. (24) Kubas, A.; Hoffmann, F.; Heck, A.; Oberhofer, H.; Elstner, M.; Blumberger, J. Electronic Couplings for Molecular Charge Transfer: Benchmarking CDFT, FODFT, and FODFTB against high-level ab initio calculations. J. Chem. Phys. 2014, 140, 104105. (25) Parker, S. M.; Seideman, T.; Ratner, M. A.; Shiozaki, T. Model Hamiltonian Analysis of Singlet Fission from First Principles. J. Phys. Chem. C 2014, 118, 12700−12705.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b09547. Parametrization: the details of Hamiltonian parametrization are described. Sensitivity of the results to CT integrals variation: the results are shown for increased values of CT integrals (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +48 12 6632212. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We express our gratitude to G. Mazur for his continued inestimable advice on software development. This research was performed with equipment purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (contract no. POIG.02.01.00-12-023/08).

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