Diagrammatic Exciton Basis Theory of the Photophysics of Pentacene

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Diagrammatic Exciton Basis Theory of the Photophysics of Pentacene Dimers Souratosh Khan† and Sumit Mazumdar*,†,‡,§ †

Department of Physics, ‡Department of Chemistry and Biochemistry, and §College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, United States S Supporting Information *

ABSTRACT: Covalently linked acene dimers are of interest as candidates for intramolecular singlet fission. We report many-electron calculations of the energies and wave functions of the optical singlets, the lowest triplet exciton, and the triplet−triplet biexciton, as well as the final states of excited state absorptions from these states in a family of phenyl-linked pentacene dimers. While it is difficult to distinguish the triplet and the triplet−triplet from their transient absorptions in the 500−600 nm region, by comparing theoretical transient absorption spectra against earlier and very recent experimental transient absorptions in the near- and mid-infrared, we conclude that the end product of photoexcitation in these particular bipentacenes is the bound triplet−triplet and not free triplets. We predict additional transient absorptions at even longer wavelengths, beyond 1500 nm, to the equivalent of the classic 21A−g in linear polyenes.

T

Figure 1. BPn dimer, with pentacene monomer molecules linked by n = 0, 1, and 2 phenyl groups. The curved arrows denote rotations about interunit bonds.

systems18−21 are of strong current interest as candidates for intramolecular singlet fission (iSF), as we briefly discuss below. SF is the process22 in which an optical spin-singlet exciton S1 dissociates into two spin triplet excitons T1. If each photogenerated triplet dissociates with 100% efficiency at the donor− acceptor interface of an organic solar cell, the photoconductivity is doubled. Enhanced external quantum efficiencies using SF have been reported for pentacene/C60 solar cells.23,24 The bulk of the theoretical25−34 and experimental35−44 literature until now had focused on intermolecular SF (xSF), in which the two triplets are generated on neighboring weakly coupled monomer molecules in a thin film or crystal. Recent investigations of dimer molecules,13−21 in which the monomers are linked by covalent bonds, is driven by the belief that the stronger coupling between the monomer components will give higher SF efficiency. We present here the results of theoretical investigations of the photophysics of BPn, focusing on electronic states reached by ground state as well as transient absorption.13 The methodology we adopt is very similar to those used to determine the precise excited state ordering in linear polyenes.1−4 There is a one-to-one correspondence between the scopes of the present research and these early investigations that introduced the modern era of the study of carbon-based systems with strong electron correlations. In the Supporting Information we have given a brief summary of the polyene literature for readers unfamiliar with it. We have performed high-order configuration interaction (CI) calculations within the π-electron

Following the original investigators, we will refer to these molecules as bipentacenes (BPn), with n = 0, 1 and 2, respectively. The photophysics of these and other bipentacenes13−17 and related covalently linked dimeric molecular

Received: July 14, 2017 Accepted: August 30, 2017 Published: August 30, 2017

he consequences of strong electron correlations in π-conjugated systems have been investigated most intensively at the two extremes of system sizes: (a) small molecules such as linear polyenes, and (b) extended systems, such as π-conjugated polymers and single-walled carbon nanotubes. The occurrence of the lowest two-photon 21A−g state below the one-photon 11B+u optical state in the former was of strong interest in the past.1−4 The 21A−g plays a weak role in the photophysics of most extended systems,5 where the phenomena of interest are exciton formation6,7 and the consequence thereof on nonlinear optical spectroscopy.8 Understanding the photophysics of discrete but large molecular systems of intermediate size poses new challenges.9−12 We consider members of one such family of large π-conjugated molecules here, dimers of bis-(triisopropylsilylethynyl) (TIPS) pentacene, covalently linked by 0, 1, and 2 phenyl groups (see Figure 1).13

© XXXX American Chemical Society

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Pariser−Parr−Pople (PPP) Hamiltonian.45,46 The key difference between earlier theoretical work on bipentacenes14−16 and our work is that we retain much larger active space, 24 molecular orbitals (MOs) overall, and incorporate CI with up to quadruple excitations from the Hartree−Fock (HF) ground state. We thus obtain energies and wave functions of all relevant eigenstates from direct CI calculations on PPP Hamiltonians with dimension ∼106. We have investigated the optically accessible spin-singlet states Sn (n > 0, n = 0 referring to the ground state), the lowest triplet exciton T1, as well as the triplet−triplet state 1(TT)1 state that is accepted to be the key intermediate in the SF process (here the superscript refers to the spin multiplicity, while the subscript indicates that it is the lowest triplet−triplet state). As in the 21A−g in the polyenes, the spin triplets in 1(TT)1 are quantum-entangled to give an overall spin singlet state (see Supporting Information, section I), and thus this state could have been written as Sn. Our nomenclature, which labels only one-photon allowed singlet states as Sn, makes a distinction between optically allowed and dark states simpler (note that this implies that n is not a quantum number). Additionally, barring complete CI calculation, which is not possible even for BP0, different computational methodologies even within the PPP model will determine different quantum numbers for the same excited eigenstate; our labeling of states will allow straightforward comparisons to existing and future theoretical work. As in our recent work on pentacene crystals,47 we obtain physical pictorial descriptions of S1, T1 and 1(TT)1, and also report calculations of excited state absorptions (ESAs) from these states in BP0, for direct comparisons of theoretical results against experimental ultrafast transient absorption study.13 A primary motivation is to elucidate the key differences between xSF and iSF. To this end, we investigate (i) the roles of the phenyl linkers in BP1 and BP2, (ii) the role of intramolecular charge-transfer (CT) between the pentacene units, and (iii) the difference between ESA from the 1(TT)1 and free T1. The last is particularly important, since this difference, if any, is the only reliable means of determining whether fission is indeed occurring to generate free triplets. We are not aware of existing calculations of transient absorptions. Our calculations of ground and excited state absorptions are within the rigid molecule approximation and do not include nuclear relaxation, as the accurate calculations of the highly correlated 1(TT)1 and the final states of triplet−triplet ESA are already formidable problems (see below). Additionally, the electron correlation parameters as well as the absorption energies extend to several electronvolts and are much larger than the molecular vibrational frequencies. Inclusion of nuclear relaxation will either merely shift the absorption wavelengths or give sidebands, without causing significant deviations from the computed results, as evidenced by the good fits we obtain with the experimental absorptions from all initial states (see below).

Although our focus is on BPn, our computational methodology can be easily extended to other pentacene dimers, and it is likely that the overall results are applicable even to other dimeric systems. PPP Hamiltonian.45,46 The PPP Hamiltonian is written as H=

∑ tij(cμ†iσcμjσ + c†jσciσ ) + U ∑ ni↑ni↓ ⟨ij⟩, σ

+

i

∑ Vij(ni − 1)(nj − 1) (1)

i800 nm) region, a point we will return to. We have not shown the T1 wave functions for BP1 and BP2, which are nearly identical to that of BP0. As with the singlet excitations, the benzene MOs do not contribute to the wave functions at this energy range. The Triplet−Triplet States. We performed MRSDCI calculations for 1(TT)1 for both BP0 and BP1, and the modified BP2. In both BP0 and BP1, we found 1(TT)1 energy to be lower than that of S1 (see Table 2 and also section IV of Supporting Information). As with S1, S2 and T1, we find that the benzene MOs play a weak role in the 1(TT)1 wave functions of BP1 and BP2, in spite of the “bimolecular” 2e−2h character of 1(TT)1. Figure 7a shows the diagrammatic exciton basis wave functions of the 1(TT)1 states for BP0, BP1 and the

modified BP2. These states are of even parity. In all cases, in addition to the lowest triplet−triplet configuration, there occur nonnegligible contributions from higher triplet−triplet 2e−2h configurations. CT contribution to the wave functions is negligible in all cases, unlike in the pentacene crystal, where CT configurations make significant contribution47 to 1(TT)1. Triple−Triplet Excited State Absorption. Figure 7b shows the exciton basis wave functions of the final states of the ESAs from 1 (TT)1 in BP0, while Figure 8a,b shows the corresponding calculated ESA spectrum. Note that the 1(TT)1 → 1(TT)3 transition is nearly identical in character to the monomeric T1 → T4 transition, in that both consist of LUMO → LUMO+1 and HOMO−1 → HOMO transitions. Hence this triplet−triplet ESA and the free triplet transition both occur in the same energy region. Note also that the final states in Figure 7b are not all triplet−triplet in character; they are labeled as such for convenience only, as discussed before. 1(TT)2 is a 2e−2h CT state, that is reached from the fundamental triplet−triplet configuration of 1(TT)1 by intermonomer CT. This CT absorption occurs at shorter wavelength (higher energy) than the T1 → T2 CT absorption in triplet ESA (see Figure 6b), a result that is also true for pentacene crystal.47 Finally, Figure 8b shows triplet−triplet ESA in the mid-infrared region where there is no absorption from the free triplet. The final state of absorption here is the one-photon allowed S2 state of Figure 3a. Similar absorption cannot occur in the triplet subspace (see Figure 6c, where we have shown the absence of triplet ESA in the mid infrared). Comparison to Experiments and Implication. We now point out the excellent semiquantitative agreements between the calculated and earlier13 as well as very recent54 experimental work on ground state and transient absorptions, and also make 4474

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Figure 7. (a) Calculated 1(TT)1 wave functions for BP0, BP1, BP2 with U = 6.7 eV and κ = 1.0 and θ = 0°. (b) The final states of the ESA from the 1 (TT)1 in the visible and near-infrared region. There is an additional ESA to S2 in mid-infrared.

excitations but excluded ne−nh excitations with n > 1. As shown in Figure S3 in the Supporting Information, red shift of the dimer absorption as well as larger intensity of the absorption to S2 are both obtained now. Our singles-CI calculation also uses the exciton basis, and the characterization of S2 as a CT excitation remains valid. Previously, the experimental spin-singlet transient absorption spectrum of BP0 has shown PA in 450−550 nm region (Figure 3 in ref 13), corresponding to our calculated singlet ESA in the 500−650 nm region in Figure 5a. Subsequent transient absorption measurements have found the PA corresponding to the calculated ESA at ∼1200 nm in Figure 5a.54 We predict additional PA in the mid-infrared region beyond 1500 nm (see inset in Figure 5a). PA in this region has been found previously in π-conjugated polymers and single-walled carbon nanotubes.8 More interesting in the present context are the triplet and triplet−triplet ESAs. The calculated ESA spectrum of the triplet−triplet in BP0 is shown in Figure 8a,b. Whether or not there is a difference between transient absorptions from the triplet and triplet−triplet in materials exhibiting SF has been a longstanding question. In pentacene crystal, the difference is subtle41,43,47 because of the weak coupling between the monomers. There, the “intramonomer” absorptions from T1 and 1(TT)1 occur at nearly the same wavelength, a feature that remains true in our calculations for BP0 (see absorptions near ∼550 nm in Figures 6b and 8a). Thus, based on the strong PAs in the visible alone, it is difficult to conclude whether the end product of the photoexcitation is the triplet−triplet or free triplet. Figures 3 and 5 of ref 13, however, find a second moderately strong transient absorption in BP0 in the ∼675− 775 nm region in “triplets obtained from singlet fission”, but not in photosensitized triplets. There occurs a much weaker transient absorption at slightly longer wavelength in the photosensitized triplet.13 Based on this experimental information and our computed ESA spectra in Figures 6b and 8a, we are led to believe that primary photoproduct in the dimer is the triplet−triplet and not the free triplet. This feature occurrence of a CT absorption in the triplet that is weaker than the CT absorption in the triplet−triplet at slightly longer wavelengthis also true in the crystal.47 In addition to the absorption at the edge of the visible and the near IR, we have calculated an additional ESA from 1 (TT)1 to S2 in BP0 and BP1, in the mid infrared region

Figure 8. Calculated triplet−triplet ESA in BP0 for θ = 0° and 30° and U = 7.7 eV, κ = 1.3 in (a) visible and (b) mid infrared. The mid infrared ESA in BP1 (solid green) has been included in panel b for comparison. The intensity is reduced significantly with the addition of a phenyl linker, in agreement with ref 54.

theoretical predictions. We then present the implications of the theoretical results for iSF. The calculated ground state absorption spectra of Figure 3a agree very well with the experimental spectra in Figure 2 of Reference 13, except for the blue shift (instead of red shift) of the BP0 and BP1 absorptions to S1, relative to the monomer absorption, and the relatively weak intensity of the short wavelength absorption to S2. We ascribe this to the limited active space in our MRSDCI calculations. It is well-known that the one-photon optical states are predominantly 1e−1h with relatively little contribution from 2e−2h configurations.3,49 We speculate that the exclusion of the higher energy 1e−1h configurations in our choice of the active space is the reason behind these quantitative inaccuracies. This is confirmed from our singles-CI calculation reported in section III of the Supporting Information, where we have retained all 1e−1h 4475

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large active space that retains 24 MOs is insufficient for obtaining the correct intensities of the absorption to S2 in Figure 3a, or for performing ESA calculations for BP1 and BP2. High-order CI calculations while retaining an even larger active space will be necessary for obtaining ESA spectra of the latter. Inclusion of nuclear relaxation, that may give better quantitative fitting of ESAs is also a future goal.

(see Figure 8b). The counterpart of this ESA is absent in the free triplet T1. This PA has been experimentally observed in BP0 and BP1 recently,54 albeit at slightly shorter wavelength (∼1200 nm). The implication of this observation is clear: direct photoexcitation of BPn generates the 1(TT)1 two-photon state and not free triplets. This does not preclude SF since triplets may be generated at longer times driven by intermolecular couplings. Our conclusion is in agreement with that reached by Sanders et al. in their more recent investigation of pentacene− tetracene heterodimers.18 The most likely reason behind the occurrence of the calculated 1(TT)1 → S2 infrared absorption at longer wavelength than in the experiment is that the true 1 (TT)1 occurs at an energy lower than calculated. Recall that in the polyenes, the energy of the triplet−triplet 21A−g decreases with length faster than the optical 11B+u exciton.55 Recall also that in the pentacene crystal, the calculated 1(TT)1 occurs slightly below S1.47 It is then likely that the 1(TT)1 occurs at an even lower energy than S1 in the dimer.

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APPENDIX A: GROUND STATE ABSORPTION SPECTRUM OF BP2 Computational constraints prevent us from including the TIPS group in our calculations of BP2. Hence, we performed an MRSDCI calculation of the ground state absorption spectrum of two pentacene molecules tethered to each other with the help of two phenyl linkers (modified BP2) as well as the monomer (pentacene) with the same set of parameters as has been used for the other TIPS-dimers (BP0, BP1) (Table A.1). Table A.1. Calculated Energies (in eV) of the Three Lowest Optical, Triplet, and Triplet−Triplet States of BP2 and U = 6.7 eV, κ = 1.0

Direct photoexcitation of BPn generates the 1(TT)1 two-photon state and not free triplets.

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compound

S1

S1*

S2

T1

BP2

2.1

3.11

3.26

1.06

1

(TT)1 2.23

On comparing the spectrum with that of BP0 and BP1, we notice that the optical signatures are not only blue-shifted, but the higher energy CT state (S2) splits into two states (S*1 and S2) (Figure A.1). Both these states have a weak dipole

CONCLUSION AND OUTLOOK In summary, we have performed correlated-electron calculations that include CI with up to dominant 4e−4h excitations for the optical singlet, lowest triplet and the triplet−triplet states in BPn, n = 0−2, using an exciton basis, over an active space of 24 MOs. We have also performed similar high order CI calculations of ESAs from S1, T1, and 1(TT)1 for BP0 over extended wavelength region, for comparison to experimental results.13,54 We are able to give physical intepretations of all excitations within a pictorial exciton basis desciption of eigenstates. We find that the benzene MOs contribute very weakly to the S1, T1, and 1(TT)1 wave functions in BP1 and BP2. We find singlet ESAs at wavelengths longer than the observed PA in the 500−600 nm range,13 in the 1000−1200 nm and at even longer wavelength (>2000 nm). PA at ∼1000 nm has been observed very recently.54 Most importantly, we find significant difference in the calculated ESAs from the T1 and the 1(TT)1 in BP0, in contrast to pentacene crystal, where the difference is subtle. Based on the calculated difference, earlier PA spectra,13 and very recent observation of long wavelength (>1000 nm) PA from a state different from S1,54 we conclude that the primary product of photoexcitation in BPn is 1(TT)1 and not free triplets. In principle, strong interdimer coupling may lead to further dissociation of the 1(TT)1 leading to the generation of free triplets. Finally, much of the theoretical work on both xSF and iSF until now has been performed using a very small active space and limited CI, sometimes including only the configurations shown in Figure 3b and a few others. Our calculations clearly show the need to have both a large active space and to perform high-order CI calculations. To begin with, unbiased determination of 1(TT)1 wave function is not possible without performing such calculations. As seen in Figure 7a, only about 80% of the true 1(TT)1 is a simple product of two triplets in the two pentacenes. More importantly, calculations of ESAs that allow us to distinguish between T1 and 1(TT)1 are not possible without retaining a large active space. Actually, even our relatively

Figure A.1. (a) Ground state absorption spectrum of a pentacene monomer (solid red) and modified BP2 (dashed violet - Here, BP2 refers to two pentacene molecules covalently linked with the help of two phenyl spacers) and (b) dominant excitonic configurations to the final states in the absorption spectrum: S1, S1*, and S2. 4476

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polymers. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 49, 10102− 10112. (6) Leng, J.; Jeglinski, S.; Wei, X.; Vardeny, Z.; Benner, R.; Guo, F.; Mazumdar, S. Optical probes of excited states in poly-(p-phenylenevinylene). Phys. Rev. Lett. 1994, 72, 156−159. (7) Wang, F.; Dukovic, G.; Brus, L. E.; Heinz, T. F. The Optical Resonances in Carbon Nanotubes Arise from Excitons. Science 2005, 308, 838−841. (8) Zhao, H.; Mazumdar, S.; Sheng, C.-X.; Tong, M.; Vardeny, Z. V. Photophysics of excitons in quasi-one-dimensional organic semiconductors: Single-walled carbon nanotubes and π-conjugated polymers. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 075403. (9) Raghu, C.; Anusooya Pati, Y.; Ramasesha, S. Density-matrix renormalization group study of low-lying excitations of polyacene within a Pariser-Parr-Pople model. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 66, 035116. (10) Aryanpour, K.; Roberts, A.; Sandhu, A.; Rathore, R.; Shukla, A.; Mazumdar, S. Subgap Two-Photon States in Polycyclic Aromatic Hydrocarbons: Evidence for Strong Electron Correlations. J. Phys. Chem. C 2014, 118, 3331−3339. (11) Aryanpour, K.; Shukla, A.; Mazumdar, S. Electron Correlations and Two-Photon States in Polycyclic Aromatic Hydrocarbon Molecules: A Peculiar Role of Geometry. J. Chem. Phys. 2014, 140, 104301. (12) Chakraborty, H.; Shukla, A. Pariser-Parr-Pople Model Based Investigation of Ground and Low-Lying Excited States of Long Acenes. J. Phys. Chem. A 2013, 117, 14220−14229. (13) Sanders, S. N.; Kumarasamy, E.; Pun, A. B.; Trinh, M. T.; Choi, B.; Xia, J.; Taffet, E. J.; Low, J. Z.; Miller, J. R.; Roy, X.; et al. Quantitative Intramolecular Singlet Fission in Bipentacenes. J. Am. Chem. Soc. 2015, 137, 8965−8972. (14) Zirzlmeier, J.; Lehnherr, D.; Coto, P. B.; Chernick, E. T.; Casillas, R.; Basel, B. S.; Thoss, M.; Tykwinski, R. R.; Guldi, D. M. Singlet fission in pentacene dimers. Proc. Natl. Acad. Sci. U. S. A. 2015, 112, 5325−5330. (15) Lukman, S.; Musser, A. J.; Chen, K.; Athanasopoulos, S.; Yong, C. K.; Zeng, Z.; Ye, Q.; Chi, C.; Hodgkiss, J. M.; Wu, J.; et al. Tuneable Singlet Exciton Fission and Triplet-Triplet Annihilation in an Orthogonal Pentacene Dimer. Adv. Funct. Mater. 2015, 25, 5452− 5461. (16) Fuemmeler, E. G.; Sanders, S. N.; Pun, A. B.; Kumarasamy, E.; Zeng, T.; Miyata, K.; Steigerwald, M. L.; Zhu, X. Y.; Sfeir, M. Y.; Campos, L. M.; et al. A Direct mechanism of Ultrafast Intramolecular Singlet Fission in Pentacene Dimers. ACS Cent. Sci. 2016, 2, 316−324. (17) Sakuma, T.; Sakai, H.; Araki, Y.; Mori, T.; Wada, T.; Tkachenko, N.; Hasobe, T. Long-Lived Triplet Excited States of Bent-Shaped Pentacene Dimers by Intramolecular Singlet Fission. J. Phys. Chem. A 2016, 120, 1867−1875. (18) Sanders, S. N.; Kumarasamy, E.; Pun, A. B.; Steigerwald, M. L.; Sfeir, M. Y.; Campos, L. M. Intramolecular Singlet Fission in Oligoacene Heterodimers. Angew. Chem., Int. Ed. 2016, 55, 3373− 3377. (19) Korovina, N. V.; Das, S.; Nett, Z.; Feng, X.; Joy, J.; Haiges, R.; Krylov, A. I.; Bradforth, S. E.; Thompson, M. E. Singlet Fission in a Covalently Linked Cofacial Alkynyltetracene Dimer. J. Am. Chem. Soc. 2016, 138, 617−627. (20) Liu, H.; Nichols, V. M.; Shen, L.; Jahansouz, S.; Chen, Y.; Hanson, K. M.; Bardeen, C. J.; Li, X. Synthesis and photophysical properties of a face-to-face stacked tetracene dimer. Phys. Chem. Chem. Phys. 2015, 17, 6523−6531. (21) Margulies, E. A.; Miller, C. E.; Wu, Y.; Ma, L.; Schatz, G. C.; Young, R. M.; Wasielewski, M. R. Enabling singlet fission by controlling intramolecular π stacked covalent terrylenediimide dimers. Nat. Chem. 2016, 8, 1120−1125. (22) Singh, S.; Jones, W. J.; Siebrand, W.; Stoicheff, B. P.; Schneider, W. G. Laser Generation of Excitons and Fluorescence in Anthracene Crystals. J. Chem. Phys. 1965, 42, 330−342.

coupling with the ground state. While the relative weights of the CT diagrams in S2 are larger, S*1 with the same configurations as S2 has strong contributions from intramonomer transitions. It is therefore conceivable that with the increase in the separation of the pentacene units, the higher energy CT state would further split into a new Frenkel and CT state. It is apparent that the modified spectrum above closely resembles the ground state absorption spectrum obtained for BP0 and BP1. As before, the benzene orbitals play an insignificant role in the description of the elctronic states in BP2.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b01829. 21A−g in polyenes, monomer, and dimer calculations, Figures S1−S9, linear absorption spectrum (SCI), excited state absorption spectra, Nref and Ntotal (PDF)

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AUTHOR INFORMATION

ORCID

Sumit Mazumdar: 0000-0002-1010-4044 Notes

The authors declare no competing financial interest. Biographies Souratosh Khan is a graduate student at the Unversity of Arizona. He completed his M.Sc in Physics (2010) from IIT Bombay, India. His Ph.D. thesis is on the photophysics of low band gap polymers and the harvesting of solar energy through the process of singlet fission. Sumit Mazumdar received his Ph.D. in Chemistry in 1980 from Princeton University under the supervision of Professor Zoltan Soos. He is a Professor of Physics, Chemistry, and Optical Sciences at the University of Arizona. His research interests include theory of strongly correlated electrons, broken symmetry and superconductivity in narrow-band systems, photophysics of carbon-based semiconductors, and organic optoelectronic devices.

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ACKNOWLEDGMENTS We acknowledge many helpful discussions with Professor Alok Shukla (IIT Bombay) and partial support from NSF-CHE1151475 and Arizona TRIF photonics. We are thankful to Dr. Matthew Sfeir (Brookhaven National Laboratory) for sharing with us unpublished transient absorption data that have since been published (see ref 54). We became aware of ref 54 only recently, when it was published after submission of our manuscript. This experimental work agrees with our conclusions arrived at independently.

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REFERENCES

(1) Hudson, B. S.; Kohler, B. E.; Schulten, K. Linear Polyene Electronic−Structure and Potential Surfaces. Excited States 1982, 6, 1− 95. (2) Ramasesha, S.; Soos, Z. G. Correlated States in Linear Polyene, Radicals, And Ions - Exact PPP Transition Moments and Spin Densities. J. Chem. Phys. 1984, 80, 3278−3287. (3) Tavan, P.; Schulten, K. Electronic Excitations in Finite and Infinite Polyenes. Phys. Rev. B: Condens. Matter Mater. Phys. 1987, 36, 4337−4358. (4) Schmidt, M.; Tavan, P. Electronic excitations in long polyenes revisited. J. Chem. Phys. 2012, 136, 124309. (5) Guo, F.; Guo, D.; Mazumdar, S. Intensities of two-photon absorptions to low-lying even-parity states in linear-chain conjugated 4477

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DOI: 10.1021/acs.jpclett.7b01829 J. Phys. Chem. Lett. 2017, 8, 4468−4478