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Free Triplets versus Bound Triplet-Triplet Biexciton in Intramolecular Singlet Fission Materials: Structure-Property Correlations

Recent advances in singlet-fission research make it imperative that structure-property correlations that determine optical signatures of the triplet-triplet spin biexciton as well as its binding energy be understood precisely. We report many-body calculations of excited state absorptions from the triplet exciton and the triplet-triplet biexciton from two transversally linked dimers of pentacene derivatives. Comparison of experiment against theory leads to new interpretations of experiments performed earlier. We show that in the para-linked isomer the triplet-triplet does not dissociate to free triplets through the duration of the measurements. In contrast, even as calculated and experimental transient absorptions agree in the meta-isomer, the experimental observations here are more difficult to interpret, indicating the strong role structural variations can play in determining the rate and yield of free triplets. We also report many-body calculations of the spin gap, the energy difference between the spin quintet versus spin singlet triplet-triplet, as well as the binding energy of the spin singlet triplet-triplet, defined as the energy difference between two free triplets and the bound biexciton. The spin gap and the binding energy of the spin singlet triplet-triplet are different quantities in all but coupled two-level systems. The experimental behavior in the transversally linked dimers as well as previously studied longitudinally linked dimers agree with the trends that would be predicted from the computed biexciton binding energies.


I. INTRODUCTION
Singlet fission (SF) involves the generation of two spin-triplet excitons (T 1 ) from a single optically accessible singlet exciton (S 1 ) in an organic molecule. 1 It was argued by Hanna and Nozik 2 that this photophysical process can be utilized to overcome the Shockley-Quessier limit 3 for the efficiencies of single junction organic solar cells, ushering in a new generation of organic photovoltaic devices. While early work on SF was abandoned due to poor efficiencies, recent experiments on pentacene crystals and thin films, as well as on dimers of acene molecules were thought to generate triplets with efficiencies approaching 200%, and have revived the interest in SF. [4][5][6][7][8][9] More recent theoretical [10][11][12] and experimental [13][14][15][16][17] works have, however, questioned whether SF does indeed reach completion in these experiments, at least in the hundreds of femtoseconds timescales claimed originally. This is also the subject matter of this work.
Experimental detection of SF involves time-resolved spectroscopy and requires the observations of, (i) instantaneous pump-induced appearance of singlet S 1 → S N photoinduced absorptions (PAs), where S N are higher energy spin-singlet excitations, (ii) paired decay of the singlet PA and the rise of triplet T 1 → T N PAs, and (iii) persistence of the triplet PAs. While this sequence of observations was claimed in earlier measurements on pentacene films [4][5][6][18][19][20] as well as on acene dimers, [7][8][9]21,22 recent experiments have cast doubts on the interpretations of these experiments. It is broadly agreed that the SF process can be schematically written as, S 0 + S 1 → 1 (TT) 1 → T 1 + T 1 , where S 0 is the chromophore molecule in its ground state and 1 (TT) 1 is a bound quantum-entangled biexciton of two triplets. The 1 (TT) 1 , nominally a double excitation within molecular orbital theory, often occurs energetically below the optical exciton as a consequence of strong Coulomb repulsive electrons among the π electrons of the chromophore molecule. [23][24][25][26] It is then noted that demonstration of the completion of the SF process requires understanding the difference in the PAs from the 1 (TT) 1 and the free triplets T 1 , if any. This is not an easy task in the absence of theoretical or experimental work on the ultrafast photophysics of the 1 (TT) 1 that characterized this field until recently. While excited state absorptions (ESAs) from the free triplets had been studied previously, similar information on ESAs was not available for the 1 (TT) 1 .
In a series of recent papers, we reported the first ever computational results for ESAs from S 1 , T 1 and 1 (TT) 1 for crystalline pentacene, 11 TIPS-pentacene dimers (hereafter BPn, see Fig. 1 (a)), 10 and the asymmetric dimer of TIPS-pentacene and TIPS-tetracene (hereafter PTn, see Fig. 1 (b)). 12 Here, TIPS refers to the triisopropylsilyl group (see Fig. 1). Our main results for BPn and PTn can be summarized as: (i) PA measurements at visible wavelengths are not suitable for distinguishing between T 1 and 1 (TT) 1 , as the ESAs from these two species are practically indistinguishable here; however, (ii) the 1 (TT) 1 ESA spectrum exhibits multiple absorptions in the near infrared (NIR) and at even longer wavelengths, primarily in the shortwave infrared (SWIR) region, that are completely absent in free triplets.
The occurrence of PAs in the IR, and their persistence at long times would therefore indicate that the SF process was incomplete at short time scales. Concurrent as well as subsequent PA studies in the IR have indeed indicated long-lived PAs in the IR, 13,14,27 lasting up to nanoseconds, or even microseconds in certain cases, indicating that femtosecond PAs in the visible that had earlier been ascribed to free triplets were actually from the 1 (TT) 1 .
In the present theoretical study, we focus on the novel TIPS-pentacene dimers shown in Figs. 1 (c) and (d), in which also very efficient intramolecular SF has been claimed from experimental transient absorption studies. 28 The monomer acene fragments in these compounds are linked substantially differently from those studied previously by us, BPn and PTn. We adopt the same short-hand nomenclatures, p − 2 and m − 2, used by Zirzlmeier et al., 28 to label the dimers linked through para- (Fig. 1(c)) and meta-( Fig. 1(d)) linkages via a phenyl group, respectively. There are multiple goals of the present study. Our first goal is to investigate whether there is a common thread that links the photophysics of all the acene dimers shown in Fig. 1, viz., whether the PA spectra of 1 (TT) 1 are nearly independent of the nature of the linkage between the acene monomers. The answer to this is not immediately obvious, as from a cursory glance, the extent of delocalization in compounds (c) and (d) appear to be larger than in compounds (a) and (b). Second, we investigate the precise role of the linker phenyl group in the photophysics of the para-versus the meta-compounds. In the experimental literature very efficient SF has been claimed for both the pentacene compounds, 28 while in more recent work Korovina et al. have claimed that for the corresponding tetracene dimers the singlet exciton in the para-isomer relaxes to the 1 (TT) 1 state (which then dissociates into free triplets following molecular rotation), but the meta-isomer decays radiatively to the ground state. 29 The different behaviors have been ascribed to the weakness of the coupling between the monomers that is expected from general grounds in the case of meta linkage. In addition, recent theoretical work that has claimed that the triplet-triplet binding energy, defined as 2 × E(T 1 ) − E( 1 (TT) 1 ), where E(· · · ) is the energy of the state · · · , is positive for the para compound but negative for the meta compound. 30 This brings us to our third goal, to examine the issue of binding energy of the 1 (TT) 1 biexciton in all the acene dimers of Fig. 1 vis-à-vis other related compounds (such as linear polyenes). The issue of binding energy of the 1 (TT) 1 as a function of the molecular structure is actually of broader interest going beyond the present dimer molecules 25,26,29,31,32 and we examine this in detail here.

II. THEORETICAL MODEL AND METHOD
Model Hamiltonian, molecular geometry and parameters.
We consider only the carbon (C) atoms and their π electrons for both compounds. We use the π-electron only Pariser-Parr-Pople (PPP) Hamiltonian. 33,34 where c † iσ creates an electron with spin σ on the p z orbital of carbon (C) atom i, n iσ = c † iσ c iσ is the number of electrons with spin σ on atom i, and n i = σ n iσ is the total number of electrons on the atom. We retain electronic hoppings t ij only between nearest neighbors i and j. U is the Coulomb repulsion between two electrons occupying the p z orbital of the same C-atom, and V ij is long range Coulomb interaction.
The average bond lengths within an acene unit are different for the perpheral (1.40Å) and internal (1.46Å) bonds. 12 Based on a widely used bond length-hopping integral relatonship 35 we have chosen intra-acene peripheral (internal) hopping integrals t ij as −2.4 (−2.2) eV. For the C-C triple bond in the TIPS group we have chosen t ij = −3.0 eV. 35 Since the steric repulsion is expected to be small in both the p − 2 and m − 2 dimers, we have chosen planar geometries for both and therefore interunit hopping integrals −2.2 eV 35 between the TIPSpentacene monomers and the phenyl linker. We use the screened Ohno parameterization for the long range Coulomb repulsion, V ij = U/κ 1 + 0.6117R 2 ij , where R ij is the distance inÅ between C-atoms i and j and κ is an effective dielectric constant. 36 The Coulomb parameter U is chosen by quantitative fittings of the TIPS-acene monomer and dimer lowest singlet and triplet excitation energies. As discussed before, 10-12 the ground state absorption spectra and energies in the spin singlet sector are best obtained with U = 6.7 eV and κ = 1.0, while the energy of the lowest triplet, as well as the triplet and triplet-triplet ESA spectra are best obtained with U = 7.7 eV and κ = 1.3.

Exciton Basis.
Our calculations are done using a localized exciton basis that allows for physical, pictorial representations of excited state eigenstates. [10][11][12] The Hamiltonian (Eq. 1) is rewritten as, where H intra consists of all terms within Eq. 1 that are purely intramolecular (ı.e., all intersite hoppings and Coulomb interactions are between atoms within a TIPS-pentacene monomer or the phenyl linker) and H inter consists of the remaining intermolecular terms within the Hamiltonian. The advantage of this procedure is that it allows classifications of basis functions as not only single, double etc excitations, but also as intramolecular versus intermolecular.
Computations using the exciton basis are performed in two steps. In the first step we solve H intra at the Hartree-Fock (HF) level to obtain the HF molecular orbitals ( The MRSDCI is an iterative procedure whose first step involves a double-CI calculation starting from a trial set of 1e-1h and 2e-2h excitations, whose choice is determined by the nature of the eigenstate we are trying to determine.

III. COMPUTATIONAL RESULTS
Ground State Absorption. We further note that difference in the nature of the optical excitations begins to emerge between p − 2 and m − 2 already from the study of the ground state absorption. Aside from the obvious differences in symmetry, the primary reason behind the observed differences is the relatively weak coupling between the acene monomers in the case of the meta linkage, which we will see leads to far greater differences in the triplet and triplet-triplet sectors.
Beyond the energy region shown in Figs. 2, there exist calculated monomeric absorptions at ∼ 3.5 eV which are in agreement with the experimental results (see Figs. S14, S15 of reference 28), but will not be discussed further.

Triplet States and ESA
The triplet states of p − 2 and m − 2, as well as their calculated ESAs, are remarkably different from one another, as we now show.
The calculated ESA spectrum from the lowest triplet T 1 in p − 2 is shown in Fig. 3(a).
The relevant wavefunctions in the spin triplet subspace are shown in Fig. 3(c). The calculated triplet ESA spectrum of m − 2 ( Fig. 3(b)) is different from that of p − 2.
The relevant wavefunctions here are shown in Fig. 3(d). Unlike in p − 2 we find a pair of nearly degenerate triplets T 1 and T 1 in the m − 2 dimer, which are decoupled FE excitations on the individual TIPS-pentacene monomers, as shown in Fig. 3(d). This is very similar to what we found previously for PTn dimers, where also there occurred two low energy non-degenerate triplets that were predominantly FE excitations localized on the individual acene monomers. 12 The decoupling of T 1 and T 1 is a direct consequence of the weak CT to the CT state we have labeled 1 (TT) 2 in Fig. 4(a). We have labeled this as triplet-triplet for the sake of convenience only.
Finally, there occurs in p − 2, but not in m − 2 ESA in the IR to the CT state S 2 of Fig 2(c), as is shown in Fig. 4(b). Its presence in p − 2 could have been predicted from our computational results for BPn and PTn, while its absence in m − 2 is anticipated from our result for triplet ESA in Fig. 3(b). Although this absorption is calculated to be relatively weak, the corresponding ESA has been seen in BPn, 14 for which we concluded that the SF process was incomplete in femtosecond times. We conclude the same here, as noted below.

Triplet-Triplet Binding Energy
There are two distinct energy differences that are relevant in SF. The first is the energy difference between the singlet exciton S 1 and 1 (TT) 1 , E(S 1 )− E( 1 (TT) 1 ) = ∆E S−T T . The second is the binding energy E b of the 1 (TT) 1 , defined as E b = 2 × E(T 1 )−E 1 (TT) 1 . ∆E S−T T determines whether or not the relaxation to the triplet-triplet state is downhill. In our calculations of pentacene dimers in Fig. 1, with the Hubbard parameter ranging between 6.7 to 7.7 eV (that are necessary to fit ground state spectra and ESA) we have in all cases found the 1 (TT) 1 to occur at or below S 1 (nearly zero or decidedly positive ∆E S−T T ) For the one case where we have calculated E(S 1 ) and E( 1 (TT) 1 ) for a tetracene dimer 42 we found negative ∆E S−T T ( 1 (TT) 1 above S 1 ). To first approximation, the relative energies of S 1 versus 1 (TT) 1 can be guessed from the known relative energies of S 1 and T 1 .
The second quantity, E b , determines in principle whether or not the bound triplet-triplet separates into two free triplets. This quantity is far harder to determine quantitatively, both because of its small magnitude, and because when certain conditions are met E b can be negative (ı.e., E 1 (TT) 1 > 2 × E(T 1 )). Additional complications arise from the fact that in the existing literature E b has sometimes been defined or estimated from the energy difference between the lowest triplet-triplet with total spin S = 2 and the lowest triplettriplet that is overall spin singlet, 43,44 (E( 5 (TT) 1 )-E( 1 (TT) 1 )). Below we refer to this energy difference as the spin gap. Such calculations have sometimes found E b in SF materials to be as large as 0.4 eV (thus approaching the electronic exciton binding energy in π-conjugated polymers). Furthermore, negative binding energies have also been calculated 44 and predicted from general considerations. 30 We find E b in all cases significantly smaller than the highest numbers obtained previously. The triplet-triplet biexciton is bound by spin-spin interactions and its binding energy is necessarily several times smaller than the singlet charged exciton binding energy, that arises from direct Coulomb interactions. We also disagree with the previous prediction 30 of negative E b in m − 2, and present MRSDCI computational results that find positive E b here.
In Table I we have given the calculated S = 0 versus S = 2 energies of the triplettriplets as well as E b for linear trans-octatetraene and trans-dodecahexaene, which are well understood theoretically from previous work by many groups. The spin gap here is enormous, due to the highly discrete nature of the HF MOs in these systems with few electrons. The size dependence of E b is easy to understand: in the triplet-triplet wavefunction of transoctatetraene the two individual triplet excitons are strongly overlapping, giving a highly confined triplet-triplet, whose energy is raised because of the confinement. 45 This overlap between the triplets is of course smaller in trans-dodecahexaene, and E b is larger. (actually, with larger screening parameters κ, calculated E b can even become negative 41 in octatetraene because of overlapping triplets in 1 (TT) 1 ) Conversely, in systems where the triplet excitons are nonoverlapping, negative E b is not expected. Table I also lists the calculated MRSDCI E( 5 (TT) 1 ), E( 1 (TT) 1 ) and E b for BP1, p − 2 and m − 2. In all cases we find from our exciton basis wavefunctions (Fig. 5) that the individual triplets in the triplet-triplet occupy different acene monomers. We thus find E b for the dimers larger than the spin gap, opposite to what is found in the short polyenes.
We agree with reference 30 that the effective intermonomer hopping for the meta linkage is much weaker than for ortho and para-linkages (which is precisely why the triplet excitons  . Highly efficient SF to the extent of ∼ 156% in both p − 2 and m − 2 dimers were reported on the basis of the observation of rapid appearance in the visible region of what the investigators thought to be triplet PAs. 28  are slightly different from those in the experimental paper. 29 Based on previous work on a different but structurally related tetracene dimer, 42 we believe that the 1 (TT) 1 here is above S 1 . Whether or not the greater delocalization in the para isomer brings the 1 (TT) 1 below S 1 can only be known from direct computation. In both cases, however, we expect E b to be positive.