Singlet Fission: Optimization of Chromophore Dimer Geometry

CHAPTER SEVEN

Singlet Fission: Optimization of Chromophore Dimer Geometry Eric A. Buchanan*, Zdeněk Havlas*,†, Josef Michl*,†,1 *University of Colorado, Boulder, CO, United States † Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic 1 Corresponding author: e-mail address: [email protected]

Contents 1. Introduction 1.1 Why Is Theoretical Work on SF Important? 1.2 A Closer Look at SF 2. The HOMO/LUMO Model 2.1 Definition 2.2 Expressions for TRP 2.3 A Simple Approximation for TRP 2.4 Local Maxima of jTRPj2 in the 6-D Space of Rigid Dimer Geometries 2.5 State Energies in the 6-D Space of Rigid Dimer Geometries 3. Applications 3.1 Two Ethylene Molecules 3.2 Two Clipped Cibalackrot Molecules 4. Outlook Acknowledgments Appendix References

176 176 179 186 186 189 195 199 202 205 205 206 211 212 212 223

Abstract After a brief review of electronic aspects of singlet fission, we describe a systematic simplification of the frontier orbital (HOMO/LUMO) model of singlet fission and Davydov splitting in a pair of rigid molecules. In both instances, the model includes electron configurations representing local singlet excitation on either chromophore, charge transfer in either direction, and triplet excitation in both chromophores (biexciton). The resulting equations are simple enough to permit complete searches for local extrema of the square of the electronic matrix element and to evaluate the effect of intermolecular interactions on the exoergicity of singlet fission and on the biexciton binding energy in the six-dimensional space of rigid dimer geometries. The procedure is illustrated on results for the six best geometries for dimers of ethylene and of an indigoid heterocycle with 24 carbon, nitrogen, and oxygen atoms.

Advances in Quantum Chemistry, Volume 75 ISSN 0065-3276 http://dx.doi.org/10.1016/bs.aiq.2017.03.005

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1. INTRODUCTION The present chapter represents a generalization of our earlier efforts1 and describes a simplified method for the complete approximate evaluation of the effect of rigid dimer geometry on the rate of singlet fission (SF). It is illustrated by application to two examples. The purpose of the procedure is not the prediction of absolute rates of SF, but the prediction of all approximate regions of dimer geometries at which the rate is locally maximized. Some references to other work on SF are provided, but a comprehensive review is not attempted.

1.1 Why Is Theoretical Work on SF Important? In recent years, SF has become a popular subject of investigation, since it provides an intriguing intellectual challenge and at the same time offers a possible path toward overcoming the Shockley–Queisser limit2 to the efficiency of a single-junction solar cell while not requiring any expensive current matching typical of multijunction cells. In the simplest description, SF3,4 is a process in which a singlet excited molecule transfers some of its energy to a ground-state neighbor and both end up in their triplet states, which then separate and become independent. Since it occurs in the excited singlet state (usually the lowest one), whose lifetime is typically measured in ns, SF is easily detectable only if it occurs on a time scale of ns as well, and dominant only if it occurs on a time scale that is much shorter. Initially, the two triplets formed in SF are coupled into an overall singlet, making the process spin-allowed and potentially as fast as other types of spinallowed energy transfer (ET) between molecules in contact. It thus is the fastest way to produce triplet from singlet excited states without relying on very strong spin–orbit coupling (i.e., without flipping spins). In a few compounds, half a dozen years ago SF was found to outcompete all other decay channels and produce a nearly 200% yield of triplet states.5–8 Used in a solar cell, such ideal material could in principle produce twice the current at half the voltage, which in itself does not offer any advantage. However, when lowenergy photons transmitted by the SF-capable material are captured by a subsequent layer of an ordinary solar cell material, the theoretical maximum efficiency rises from the Shockley limit of roughly 1/3 to a value close to 1/2.9 Unfortunately, only a handful of materials are known to perform SF with a triplet yield anywhere near 200% and even fewer are practical, primarily because most are too sensitive to the combined action of light and

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atmospheric oxygen. A useful SF material must not only have a triplet yield very close to 200% but also needs to satisfy many additional requirements. Some of the most obvious ones are a long-lived and mobile lowest triplet state, preferably at 1–1.1 eV above the singlet ground state, and a lowest excited singlet state at somewhat more than twice the triplet excitation energy (2–2.3 eV), with strong absorption everywhere above this threshold. Redox properties must match the junction material used for charge separation in order to incur minimal losses. There should be no charge separation from the excited singlet, only from the triplet. The material must have desirable adhesive and/or mechanical properties and must be inexpensive. Poisonous elements are to be avoided. The list could be continued. Clearly, it would be best if a large selection of SF materials were available. The desire to deduce structural guidelines for the design of new efficient SF materials thus adds a practical motivation to the interesting intellectual challenge posed by the presently rather limited understanding of the mechanistic details of the SF process. Useful guidelines for finding superior SF materials need to address at least two types of principal problems: (i) selection of an optimal chromophore and (ii) selection of an optimal mutual disposition of the chromophores in space. Facile separation of the two triplets could be added to the list. (i) A search for optimal chromophores is based on the requirement that SF be isoergic or preferably10–12 slightly exoergic in order to be fast and competitive, and this implies a rarely satisfied relation between the singlet (S1) and triplet (T1) excitation energies, E(S1)  2 E(T1). It was recognized early on13 that members of two overlapping groups of chromophores can be theoretically expected to meet this condition: biradicaloids (derived from a perfect biradical by a relatively weak covalent perturbation14) and large benzenoid hydrocarbons. The case of biradicaloids has been elaborated in considerable detail.15,16 It is unfortunate that biradicaloid structures frequently have high chemical reactivity and sensitivity to traces of oxygen. The classical SF chromophores, tetracene and pentacene, as well as the first successful chromophore derived from theoretical considerations,13 1,3-diphenylisobenzofuran,6,17–19 have biradicaloid character and are highly reactive. Additional biradicaloids identified by computations as potentially interesting for SF20–22 have so far not been tested experimentally. Most of them were captodatively stabilized and carry an acceptor and a donor group on each radical center.

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In recent years the family of highly efficient SF materials with a triplet yield above an arbitrarily set limit of 170% has grown beyond the carotenoid, all-trans-3R,30 R-zeaxanthin,7 the biradicaloid, 1,3-diphenylisobenzofuran,6 and the classical parent and variously substituted or doubled polyacenes, tetracene8 and pentacene,23–28 to shorter (triisopropylsilylethynylanthracenes29) and longer (hexacene12) members of the series, and to derivatives of terrylene,30,31 diketopyrrolopyrrole,32,33 and a quinoidal bithiophene.34 (ii) The optimal arrangement of the chromophores in space has been of considerable interest and is well recognized as critical for high triplet yields.33,35–40 Theoretical guidance clearly is very valuable. For example, general consideration of the simplified diabatic HOMO/LUMO model3,4 described in more detail later and numerical calculations in adiabatic framework41 suggested that an exactly stacked pair of chromophores would be less favorable than a slip-stacked arrangement with the slip in the direction of the HOMO–LUMO transition moment. (this slip removes a plane of symmetry relative to which the S1 electronic function is antisymmetric and the 1(TT) electronic function is symmetric, making it impossible for a totally symmetric electronic Hamiltonian to perturb the former into the latter not only in the HOMO/LUMO model but at any level of approximation unless an antisymmetric vibration is present in the total wave function of the initial or the final state; this vibronic effect is likely to be weak). An approximate general formula based on the HOMO/LUMO model has since appeared4,11 for the electronic matrix element TRP (R ¼ S1 is the reactant and P ¼ 1(TT) is the product) as a function of the mutual disposition of two chromophores in space. The formula was subsequently elaborated in a paper that also formulated a simple explicit rule for the optimal geometries in the model system of two ethylenes,1 cf. Section 3.1. In the present chapter, we first provide a qualitative description of the overall SF process as it is presently understood. Subsequently, we focus on two aspects of SF related to electronic structure, leaving aside many important issues such as dynamics.10,11,42 Specifically, we consider the optimization of the mutual disposition of two SF chromophores in space, both with respect to maximizing the electronic matrix element for SF and to minimizing the potentially deleterious effect of Davydov splitting. We do this within the framework of the HOMO/LUMO approximation and in an Appendix provide full exact expressions for the matrix elements of the interaction Hamiltonian within this model, with inclusion of intermolecular overlap. We summarize the derivation of the approximate formula1 for the SF

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electronic matrix element TRP in a dimer. Its simplicity permits complete searches of the relevant parts of the six-dimensional (6-D) space defined by the location and orientation of two adjacent rigid bodies, even though the identification of all local maxima of jTRPj2 required its evaluation at more than half a billion geometries. We next derive a similar formula for the effect of intermolecular interactions on the exothermicity of the SF process and show how these interactions can discriminate against some of the local maxima of jTRPj2. We also note that an analogous approximate formula can be written for the binding energy of the biexciton. Finally, we provide examples of the application of the results to SF in a model system of ethylene dimer and in a realistic system of a dimer of a heterocycle related to indigo. Although in most cases the present limitation to only two chromophore molecules does not describe reality, it seems to us that there is considerable merit in attempts to understand dimers, which could provide a useful qualitative guide to structures that are optimal for SF in general.

1.2 A Closer Look at SF There are two distinct ways of viewing SF. In the adiabatic description,43 one starts with electronic states of the dimer or a larger conglomerate, and the transition to the final states is caused by the non-Born-Oppenheimer terms in the Hamiltonian that induce nonradiative transitions from one surface to another. In the diabatic framework,3,10,44 which we are using for the present description of the SF process, one starts with electronic states of the individual partners, in the simplest case obtained by diagonalization of the Hamiltonian of a single molecule. Then, a transition from the initial to the final state is induced by the intermolecular terms in the electronic Hamiltonian of the total system. The diabatic description is more commonly used for the discussion of experimental results and its merits have been discussed in detail.10 It has the practical advantage that calculations of the starting and final states are performed for a system that is only half the size of the total system in the case of a dimer, and less in the case of a larger conglomerate. Also, it permits the use of qualitative concepts that are familiar from theories of charge45,46 and energy47–50 transfer, such as Marcus theory and F€ orster theory. A disadvantage of the diabatic description is that the definition of the initial state for SF usually is not as simple as in a solution, where diffusion brings two initially truly isolated partners together. In most instances, e.g., in a crystal, the partners participating in SF interact from the very start. A diabatic treatment then often begins with an adiabatic calculation of a dimer or higher aggregate followed by a diabatization to produce the desired diabatic basis.51–53

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In the diabatic treatment, the expression for the rate of SF W(RP) is the Fermi golden rule, W ðRPÞ ¼ h1 TRP 2 ρðE Þ

(1)

where the electronic matrix element TRP reflects the intermolecular electronic interaction between the initial reactant state R and the product double triplet biexciton state P and ρ(E) is the density of states in the product state at the energy E at which it is generated. Both terms are critically important, but in the present text, we primarily deal with TRP and only at the end consider ρ(E). In most cases the description of SF in terms of just two interacting molecules and a single-step process is grossly oversimplified. A more realistic schematic picture is provided in Fig. 1. It is still incomplete in that it shows only two chromophores, and this limitation applies throughout the present text. The effects of delocalization of the initial singlet excited state over a larger number of molecules in larger aggregates and in crystals by excitonic and charge-transfer interactions have been investigated by many authors and we provide only a few leading references.42,54–61 D+ A

B

A

B

D+ S1 D– T1 S0

D+ D+ S1 1.2.1 S1 D– D– hv T1 T1 S0 S0

D+ S1 D– T1 S0

A D+ S1 D– T1 S0

S1

D–

T1

S0

A B

D+ D+ S1 S1 A B A 1.2.1 A B D– D– D+ D+ D+ D+ D+ T1 T1 S1 S1 1.2.2 S0 S0 1.2.2 S1 S1 1.2.4 S1 D– D– D– D– D– T1 T1 T1 T 1 T1 D+ D+ S0 S0 S1 S1 S0 S0 S0 B 1.2.1 D– D– D+ 1.2.3 T1 T1 S1 S0 S0 hv D– T1 1.2.5 S0

B D+ S1 D– T1 S0

Fig. 1 Top right: symbolic representation of electronic states of partners A and B in SF. Center: Sequence of events (competing decay paths not shown). Possible (black) and actually occupied (red) electronic states in frames (blue, real species; red, a species that can be virtual or real). Narrow frames: separated partners, wide frames: partners in contact. Top path: SF in solution; bottom path: SF in crystal, aggregate, or dimer (in covalent dimers, the last step on the right is impossible). All steps are reversible and the sections in which they are discussed are stated.

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Fig. 1 indicates that each of the chromophores could be in the ground state S0, in the lowest triplet state T1, in the lowest excited singlet state S1, in the lowest doublet state of the radical cation D+, or in the lowest doublet state of the radical anion D. For chromophores of interest, the excitation energy of the S1 state is about twice that of the T1 state. To conserve the total electron count, if one of the partners is in its D+ state, the other one must be in its D state. Fig. 1 shows the sequence of events during SF and refers the reader to the sections where they are discussed, and it also shows a schematic pictorial representation of the dominant electronic configuration in each state (other contributing configurations are not shown). It applies in the limit of complete localization of initial electronic excitation on one of the partners. It is easily modified to describe the other limit, in which electronic excitation energy is delocalized equally over the two partners. They can then be in the lower or the upper singlet excimer state (S1S0  S0S1), in the lower or the upper triplet excimer state (T1S0  S0T1), in one of the biexcitonic states 1,3,5 (T1T1), in the charge-separated states D+D DD+, or in a more or less complicated superposition of these states. 1.2.1 Initial Excitation The initial excitation normally occurs by absorption of a photon, which converts S0 to S1 (or a higher excited singlet, which then usually rapidly decays to S1). Fig. 1 shows that this process may excite a single chromophore, e.g., in solution, where diffusion may later provide an opportunity for an encounter and subsequent SF. Most SF experiments deal with solids, where the initial excitation is normally delocalized over several chromophores. In a pair with delocalized states, the allowed excimer state (S1S0 + S0S1) is reached initially. It is the higher of the two excimer states if the two partners are in a geometry typical of H aggregates (transition dipole moment vectors stacked) and the lower one if they are in a geometry typical of J aggregates (transition dipole moment vectors approximately parallel to a straight line). In dimers and small aggregates, excitation may be localized or delocalized, depending on geometry and the strength of interchromophore coupling.62 Coherent excitation of two or more states (not shown in Fig. 1) has been proposed in an effort to interpret the results of time-resolved two-photon photoionization experiments on crystals of polyacenes.54 An ultrashort laser pulse excites a coherent superposition of the initial S1S0, final 1(TT), and mediating D+D states. After a short time interaction with the phonon bath destroys the quantum coherence to produce the 1(TT) biexciton state or the S1S0 state, which then behaves as if it were produced in an ordinary

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absorption process by itself and proceeds to perform SF to yield an additional portion of the 1(TT) state. 1.2.2 The First SF Step (or Two?) The conversion of the excited singlet state S1 or the lower excimer state, (S1S0 + S0S1) at the J geometry or (S1S0  S0S1) at the H geometry, and at times even a higher electronic excited state, into the 1(TT) biexciton is the key process in SF. Depending on its rate, it may proceed before the excess vibrational energy usually present in the initially prepared electronically excited state is lost to the surroundings, concurrently with this loss, or only after vibrational equilibration. This provides ample opportunity for multiexponential kinetics. A simplified schematic representation is provided in Fig. 2. Fig. 2 is only symbolic in that it shows the dominant electron configurations, while in fact others contribute as well. The bottom shows frontier orbital occupancies and spins in the localized singlet configurations with one or the other chromophore excited, and in the biexciton configuration in the center. On the top, the charge-separated configurations are shown. D+

D–

D–

9

1

S1

2

5

3

S0

4

6

7

D+

T1

T1

8

S0

S1

›

›

›

›

›

›

›

›

10 1 = 〈S0S1|H|D+D–〉, 3 = 〈S1S0|H|D+D–〉, 5 = 〈D+D–|H|T1T1〉, 7 = 〈S1S0|H|T1T1〉

›

›

2 = 〈S1S0|H|D–D+〉, 4 = 〈S0S1|H|D–D+〉, 6 = 〈D–D+|H|T1T1〉, 8 = 〈S0S1|H|T1T1〉, 9 = 〈D+D–|H|D–D+〉, 10 = 〈S1S0|H|S0S1〉

Fig. 2 Electron configurations of a dimer important for SF, and interactions between them. Solid lines: potentially strong interactions; broken lines: weak interactions, dependent on repulsions between charge densities defined by products of orbitals located on different molecules.

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Potentially strong interactions between configurations are shown as solid lines and weak interactions as broken lines. The associated interaction Hamiltonian matrix elements are shown at the bottom. There are two ways (mechanisms) in which formation of the biexciton can proceed. (i) The usual process involves a single step and no intermediate. In the diabatic description and starting in a localized excited singlet state, its rate is proportional to TRP 2 (cf. Eq. 1), where in the usual simple approximation TRP contains three contributions. A usually small one is due to a direct interaction of the initial with the final state and is indicated by a red arrow in Fig. 2. Two often large ones, which can be of mutually opposite signs, are due to mediated interactions indicated by cyan and blue arrows. The two mediating states that provide the interaction (superexchange) paths are the charge-transfer configurations D+D and DD+ shown in Fig. 2 at the top. In these, an electron has been removed from one or the other chromophore and added to its partner. To the best of our knowledge, the mediated path was first proposed half a century ago.63 Although some authors still argue that it is not clear whether the direct or the mediated contribution to the matrix element is more important, there is only one case that we are aware of in which the two have been carefully separated in a computation and the direct contribution was found to dominate over the algebraic sum of the mediated ones (HOMO/LUMO approximation for a stacked pair of tetracenes slipped significantly along the long axis and only 0.2 A˚ along the short axis; the total value of the electronic matrix element was only 10 meV); we have not been able to reproduce this result and find that the mediated term exceeds the direct term by three orders of magnitude).64 The interference between the two mediated paths has interesting implications for the dependence of TRP on the choice of the mutual disposition (distance and orientation) of the two chromophores; at the best geometries the interference is constructive. A way to suppress one of the paths and thus minimize a destructive interference is to make the two partners inequivalent and the D+D and DD+ states thus very different in energy (cf. Section 1.2.5). (ii) In rare instances the energy of one or both charge-transfer configurations can be so favorable that they no longer represent virtual states but become observable real states with finite lifetimes (minima in the lowest excited singlet potential energy hypersurface). Then, the formation of a biexciton can proceed by a two-step mechanism with one of the charge-transfer species as an observable intermediate.65,66 This

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intermediate appears to have deactivation channels such as intersystem crossing to yield one triplet molecule and internal conversion to yield the ground state. It is therefore unlikely that two-step SF conversions will be optimal when one aims for a high triplet yield. 1.2.3 The Biexciton Since SF is not complete until the two resulting triplets have become independent, the initially formed triplet pair (biexciton) generally needs to be viewed as a possibly detectable intermediate 1(TT) that can proceed in several directions. If its binding energy is sufficient or if it occurs in a covalent dimer, it may be separately observable and kinetically significant, making the SF process a two-step event. First of all, the initially reached 1(TT) state of the biexciton is only one of the nine levels that result from the coupling of two triplets. They can be thought of as a singlet 1(TT), the three components of a triplet 3(TT), and the five components of a quintet 5(TT). They are mixed by a small tensor term in the spin Hamiltonian, magnetic dipole–magnetic dipole interaction, familiar from electron spin resonance of triplet states and jointly with spin–orbit coupling responsible for their zero-field splitting (D and E terms in EPR spectroscopy). Quantum beats in delayed fluorescence are observed67 and the quintet state has been observed directly by time-resolved EPR spectroscopy.68 The description of the S1S0–1(TT)–3(TT)–5(TT) interconversion cannot be provided by standard kinetic expressions but requires a density matrix approach. When the two partners are equivalent by symmetry, the situation is simplified in that 1(TT) does not directly couple to 3(TT) but only to 5(TT). A recent summary of the coupling of spins of four electrons is available.69 In the presence of an outside magnetic field, a Zeeman term also contributes to the mixing of the nine sublevels. This provides an opportunity for significant magnetic field effects on SF, and their study played an important role in investigations of SF when it was first discovered.70,71 The biexciton can return to the initial S1S0 state but it can also complete the SF process by overcoming its binding energy and dissociating to two independent triplets T + T (Fig. 1). Their spins may remain coherent for some time and they can encounter each other again either before or after complete loss of spin coherence by spin–lattice interaction. Little is known about structural effects on the biexciton binding energy, which is related to the energy difference between the three substates, singlet, triplet, and quintet. Simple expressions for the usually only slightly different energies of the 1 (TT), 3(TT), and 5(TT) configurations are available,3 but the effect of interaction with higher energy configurations of the three multiplicities, which

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would presumably stabilize the singlet the most and the quintet the least by energy arguments alone, is difficult to guess without a detailed computation. The importance of inclusion of usually unknown biexciton binding energy in the evaluation of the overall yield of free triplets in SF was emphasized in a simple kinetic model proposed a few years ago.72 Internal conversion of the biexciton to the only lower singlet, S0S0, looks improbable because of a large energy gap, but it sometimes occurs quite rapidly and reduces the yield of free triplets in a presently unpredictable way. Its mechanism is not well understood and it is possible that it occurs via a specific “photochemical” decay path through a conical intersection (e.g., those used in 2 + 2 or 4 + 4 dimerization). Moreover, to the degree that the biexciton has developed 3(TT) character, fast internal conversion to the lower energy TS0 state is always a possibility. Acquisition of 5(TT) character normally does not open a similar additional decay channel via locally excited Q1S0 states, which are typically too high in energy (agreement with this statement is not universal68). In general, it seems to us fair to say that the structural factors that dictate the outcome of the competition between dissociation of the biexciton state into two independent triplets and other decay paths are very important for the overall yield of free triplets in SF but are not well understood. 1.2.4 Separated Triplet Pair The 1(TT) to T + T separation (Fig. 1) is usually competitive in crystals, aggregates, or polymers, due to facile triplet hopping from chromophore to chromophore, or even in solution, due to diffusion. It is generally favored by entropy.72,73 In isolated covalent dimers, the separation cannot occur and the SF process stops at the stage of a biexciton, which ultimately follows one of its other decay channels. Among the properties of the free triplets, long lifetime and high mobility are essential for any practical applications in solar cells. 1.2.5 Heterofission So far, we have tacitly assumed that the two chromophores involved in SF are the same, and in most cases studied so far that was indeed so (homofission). The two partners can also be different. Even if they are the same chemical species, the geometry of the pair may make them inequivalent. They can however also be different chemical species altogether (heterofission). Unless the excitation energies of the partners are matched, the SF process will then involve some conversion of electronic into vibrational energy and finally into heat, and cannot be fully efficient. If the partners do have the same singlet as well as triplet excitation energies, there will be no efficiency penalty, and this is the case shown in Figs. 1 and 2.

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The most interesting case is encountered when one of the partners acts as an electron donor and the other as an electron acceptor, such that the D+D and DD+ states have very different energies. This can result in a significant advantage for the rate of SF, since the possibly destructive interference between the two paths mediated by these charge-transfer states discussed in Section 1.2.2 can be greatly diminished (cf. “charge-transfer polymers”74,75).

2. THE HOMO/LUMO MODEL 2.1 Definition The HOMO/LUMO model of SF3,4 treats explicitly only electrons in the frontier orbitals on each partner (HOMO is the highest occupied and LUMO is the lowest unoccupied MO in the ground configuration). It is assumed that the initial singlet state on each partner is well described as a HOMO–LUMO excitation, and that electrons in molecular orbitals of lower energy than the HOMO on each partner can be treated as a rigid core. This assumption is most easily fulfilled when the HOMO to LUMO excited state is S1, but it could also be met when this state is S2 (or an even higher excited singlet), if the absorbed photon has sufficient energy and if SF is faster than internal conversion to S1, as is the case in certain carotenoids.76 The HOMO/LUMO model has seen much use over the past decades for many purposes; see for instance Refs.63,77–82 Even under the best of circumstances, the description of the S1 state of a chromophore as a HOMO–LUMO excitation and the limitation of the active space to only two orbitals on each partner are only approximate. Although the HOMO/LUMO model therefore cannot be exactly correct, it is appealing conceptually and it has seen wide use in SF studies. In particular, it was employed very successfully in the first microscopic dynamical calculations of SF in polyacenes.10–12 It is also supported by the results of a treatment of tetracene and pentacene dimers by active space decomposition,44 and we believe that it represents a good starting point for even simpler treatments that are needed if thorough searches of the sixdimensional space of relative geometries of two rigid bodies are to be made, as described later. It is assumed that the two interacting chromophores A and B have equal excitation energies, but they may differ in their reduction and oxidation potentials. The singlet ground state of the chromophore pair is represented by a Slater determinant constructed from doubly occupied orbitals, S0 A S0 B ¼ S0 S0 . In this notation the state of chromophore A is on the left and that of chromophore B is on the right (cf. Fig. 1). We consider the ground

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state S0, the lowest excited singlet state S1, and the lowest triplet state T1 of each partner, and the two lowest-energy charge-separated states 1D+D and 1DD+, in which an electron is transferred from A to B or from B to A, respectively (Fig. 1). The singlet excited states of A and B result from electron promotion from the HOMO hA or hB into the LUMO lA or lB, respectively. When we need to refer to an unspecified general state of partner A or B, we use UA and UB. The six-dimensional singlet state subspace treated explicitly in the model allows local excitations in either chromophores, a simultaneous triplet excitation in both chromophores, and the charge-transfer (CT) 1D+D and 1  + D D excitations: S0 S0 ¼ NS0S0 jhA α hA β hB α hB βj

(2)

S1 S0 ¼ NS1S0 21=2 ðjhA α lA β hB α hB βj  jhA β lA α hB α hB βjÞ

(3)

S0 S1 ¼ NS0S1 21=2 ðjhA α hA β hB α lB βj  jhA α hA β hB β lB αjÞ

(4)

D + D ¼ N + 21=2 ðjhA α lB β hB α hB βj  jhA β lB α hB α hB βjÞ

(5)

D D + ¼ N + 21=2 ðjhA α hA β lA α hB βj  jhA α hA β lA β hB αjÞ

(6)

1 1

1

T1 T1 ¼ NT1T1 31=2 ½jhA α lA α hB β lB βj + jhA β lA β hB α lB αj 1⁄2ðjhA α lA β hB α lB βj + jhA α lA β hB β lB αj + jhA β lA α hB α lB βj + jhA β lA α hB β lB αjÞ

(7)

The normalization factor 1/√(N!) for a Slater determinant is included implicitly (N ¼ 4 is the number of electrons in the active space). In the wave functions (2)–(7), core electrons are not shown. Because orbitals hA and lA are not orthogonal to orbitals hB and lB, the normalization factors Nx depend on intermolecular overlap integrals:
 ⁄ (8) NS0S0 ¼ ðShAhB  1Þ2    ⁄ (9) NS1S0 ¼ 2ShAhB 2 SlAhB 2  ShAhB 2  SlAhB 2 + 1    ⁄ (10) NS0S1 ¼ 2ShAhB 2 ShAlB 2  ShAhB 2  SlAhB 2 + 1   ⁄ (11) N + ¼ ShAhB 2 + ShAlB 2 + 1   ⁄ (12) N + ¼ ShAhB 2 + SlAhB 2 + 1 1

2

1

1

1

1

2

2

2

2


  1⁄2 NTT ¼ ðShAhB SlAlB  ShAlB SlAhB Þ2 + 1⁄2 ShAhB 2 + ShAlB 2 + SlAhB 2 + SlAlB 2 + 1

ð where Sab ¼ aðr1 Þbðr1 Þdr1 .

(13)

The Hamiltonian matrix is D D D E 3        1 +  E  1  + E  1       ^ S0 S0 ^ S1 S0 ^ S0 S1 ^ D D ^ D D ^  ðT 1 T 1 Þ S0 S0 H S0 S0 H S0 S0 H S0 S0 H S0 S0 H S0 S0 H 6 D E 7 D D 6  7       1 +  E  1  + E  1      6 S S H 7  ^ S1 S0 ^ S0 S1 ^ D D ^ D D ^  ðT 1 T 1 Þ S1 S0 H S1 S0 H S1 S0 H S1 S0 H S1 S0 H 0 0 ^ S1 S0 6 7 6 D E 7 D D 6  7        1 +  E  1  + E  1      6 S0 S0 H 7 ^ S0 S1 ^ S0 S1 ^ S0 S1 ^ D D ^ D D ^  ðT 1 T 1 Þ S1 S0 H S0 S1 H S0 S1 H S0 S1 H S0 S1 H 6 7 6D E7  1 +  E D  1 +  E D  1 +  E D1 +   1 +  E D1 +   1  + E D1 +   1 6 7 6 S0 S0 H ^ D D ^ D D ^ D D ^ D D ^ D D ^  ðT1 T1 Þ 7 S1 S0 H S0 S1 H D D H D D H D D H 6 7 6D E7  1  + E D  1  + E D  1  + E D1 +   1  + E D1  +  1  + E D1  +  1 6 7 ^ D D ^ D D ^ D D ^ D D ^ D D ^  ðT1 T1 Þ 7 6 S0 S0 H D D H D D H D D H S1 S0 H S0 S1 H 6 7 4D E D E D E D E D E D E5  1  1  1       1 1 1 1 +  ^  1  + ^  1 ^  ðT 1 T 1 Þ ^  ðT1 T1 Þ ^  ðT1 T1 Þ ^  ðT 1 T 1 Þ S0 S0 H S1 S0 H S0 S1 H D D H ðT1 T1 Þ D D H ðT1 T1 Þ ðT1 T1 ÞH 2

(14)

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189

and the model permits an approximate description of processes such as ET, CT, and SF. This requires an evaluation of matrix elements in terms of integrals over the one-electron and two-electron parts of the Hamiltonian and of overlap integrals ShAhB, SlAlB, ShAlB, and SlAhB (note that ShAlA ¼ ShBlB ¼ 0). The complete results, obtained using the program MAPLE,83 are shown in Appendix. Those expressions that are needed for a description of SF were published previously.1

2.2 Expressions for TRP In the first approximation,3 the initial state in SF is represented by the configuration S1S0 if it is localized, and either S1S0 + S0S1 or S0S1  S1S0 if it is fully delocalized. The final state is represented by the configuration 1(T1T1). Very similar formulas result in both cases.3 Here we state results for TRP applicable for a localized initial state. Those for the delocalized initial state are given in Section 2.5, Eq. (51). The matrix element TRP ¼ HRP  SRPE becomes48 TRP ¼ TS1S0/T1T1¼ ^ 1S0i, HS1S0/T1T1  SS1S0/T1T1ES1S0, where HS1S0/T1T1 ¼ h1(T1T1)jHjS 1 ^ 1S0i. The matrix eleSS1S0/T1T1 ¼ h (T1T1)jS1S0i, and ES1S0 ¼ hS1S0jHjS ment TS1S0/T1T1 that approximates TRP in this treatment is commonly referred to as the “direct” term. In a better approximation, the initial state is assumed to be a linear combination of S1S0 with a small admixture of 1D+D and 1DD+, and the final state is a similar linear combination of 1(T1T1) with 1D+D and 1DD+. If the initial and final states are degenerate and the 1D+D and 1DD+ states are also degenerate and higher in energy by ΔE, and if we use first-order perturbation theory and neglect terms containing products of two small numbers, the matrix element TRP becomes4,11   TRP ¼ TS1S0=T1T1  TS1S0= +  T +=T1T1 + TS1S0= + T + =T1T1 =ΔE (15) where the subscript + stands for 1D+D and the subscript + stands for 1  + D D . The expression for TRP now consists of the direct term TS1S0/T1T1 and a term mediated by the virtual states 1D+D and 1DD+, which contains division by the energy difference ΔE. Formula (15) is the standard first-order expression for TRP that is applicable when the energies of the chargeseparated states 1D+D and 1DD+ are the same. It has seen much use and it shall also be used in the following. If the energies of the charge-transfer states 1D+D and 1DD+ are different, the result changes to

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  TRP ¼ TS1S0=T1T1  ½ TS1S0= +  T +=T1T1 =ΔE 1 D + D + TS1S0= + T + =T1T1   =ΔE 1 D D + (16)

If the 1D+D and 1DD+ states are very close in energy to the S1S0 and (T1T1) states, the first-order expressions (15) and (16) will overestimate the magnitude of the mediated term. Then, an explicit diagonalization within the three-dimensional [1D+D, 1DD+, S1S0] and [1D+D, 1DD+, 1 (T1T1)] spaces is necessary and a more complicated formula for TRP results. Formulas (15) and (16) also assume that the coupling between the initial S1S0 and final 1(T1T1) states is weak relative to the effects of the phonon bath and that the initial excitation does not produce their coherent superposition. If this condition is not satisfied, a diagonalization in the full four-dimensional space [1D+D, 1DD+, S1S0, 1(T1T1)] is needed. This appears to be the case for the very fastest SF events, observed in crystalline pentacene54 (cf. Section 1.2.1). Only the two-electron part of the Hamiltonian contributes to the direct ^ 1S0i when interchromophore overlap is term TS1S0/T1T1 ¼ h1(T1T1)jHjS neglected and it is very small (typically on the order of meV), because the two-electron integrals involved represent electrostatic interactions between overlap densities at least one of which is very small (it originates in the multiplication of a molecular orbital located on A with a molecular orbital located on B). It is generally reasonable to state that the rate of SF is primarily determined by interactions of starting and final states mediated by virtual charge-transfer states.4,10,84 The mediated (indirect) term on the right-hand side of Eq. (13), (TS1S0/CATCA/T1T1 + TS1S0/ACTAC/T1T1)/ΔE, contains contributions both from the two-electron and the one-electron part of the Hamiltonian. It typically amounts to hundreds of meV at realistic geometries, even though it contains a division by a potentially large energy difference between the initial and the charge-separated states and also sometimes suffers from destructive interference of the two paths mediated by the virtual charge-transfer states ^ 1D+D + 1DD+i ¼ (Fig. 2). The interference is reflected in the term hS1S0jHj ^ 1D+Di + hS1S0jHj ^ 1DD+i, where the two matrix elements on the hS1S0jHj right could be comparable in size and opposite in sign. 1

2.2.1 Intermolecularly Nonorthogonal Orbitals When the four interchromophore overlap integrals ShAhB, SlAlB, ShAlB, and SlAhB are not neglected, the expressions for the terms that occur in Eq. (15)

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are quite lengthy and complicated. They have been published, partly in supporting information.1 Full results for all elements of the Hamiltonian matrix (14) and the associated overlap integrals are given presently in Appendix, and permit not only an evaluation of the expression (15) for SF but also a similar treatment of processes such as ET and CT. The closest analog are the previously reported48 expressions for ET matrix elements ^ 1S0i, derived after setting ShAlB ¼ SlAhB ¼ 0. We see no physical hS0S1jHjS justification for such selective neglect of these two of the four overlap integrals. For realistic dimer geometries, the interchromophore overlaps are usually smaller than 0.1. Terms that are higher than first order in overlap can therefore be safely neglected. To first order in overlap, the results are     ^ S1 S0 ¼NS1S0 2 fFhAhA + FlAlA + 2FhBhB  ðhA hA jhA hA Þ  ðhB hB jhB hB Þ S1 S0 H  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + 2ðhA lA jhA lA Þ  ðhA hA jlA lA Þ  2SlAhB ½FlAhB  ðhA hA jlA hB Þ + 2 ðhA lA jhA hB Þ  2ShAhB ½FhAhB  ðhA hA jhA hB Þ + ðhA hB jlA lA Þ + ðhA lA jlA hB Þg D

 1 +  E ^  D D ¼ NS1S0 N + fFlAlB  ðhA hA jlA lB Þ + 2 ðhA lA jhA lB Þ S1 S0 H

(17)

+ ShAlB ½FhAlA  ðhA hA jhA lA Þ + SlAlB ½FhAhA + 2FhBhB   ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhB hB jhB hB Þ  SlAhB ½FhBlB + 2ðhA hB jhA lB Þ  ðhA hA jhB lB Þ  ShAhB ½2ðhA hB jlA lB Þ + ðhA lA jhB lB Þ + ðhA lB jlA hB Þg D  1  + E ^  D D ¼ NS1S0 N + fFhAhB  ðhA hB jlA lA Þ + 2ðhA lA jlA hB Þ S1 S0 H

(18)

+ 2SlAhB ½FhAlA  ðhA lA jhB hB Þ  ShAhB ½FhBhB + FlAlA + FhAhA  3ðhA hA jhB hB Þ  ðhA hA jlA lA Þ  ðhA hB jhA hB Þ + 2 ðhA lA jhA lA Þ  ðhB hB jhB hB Þ + 2ðlA hB jlA hB Þ  ðlA lA jhB hB Þ  ðhA hA jhA hA Þg

(19)

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D E  1 ^  ðT1 T1 Þ ¼ NS1S0 NT1T1 ð3=2Þ1=2 f ðlA hB jlA lB Þ S1 S0 H  ðhA hB jhA lB Þ + SlAlB ½FlAhB  ðhA hA jlA hB Þ  ShAlB ½FhAhB  ðhA hA jhA hB Þ + ðhA hB jlA lA Þ  ðhA lA jlA hB Þ + SlAhB ½FlAlB  ðhA hA jlA lB Þ + 2ðhA lA jhA lB Þ  ðlA hB jhB lB Þ  ðlA lB jhB hB Þ  ShAhB ½FhAlB  ðhA hB jhB lB Þ + ðhA lA jlA lB Þ  ðhA lB jhB hB Þ + ðhA lB jlA lA Þ  ðhA hA jhA lB Þg D

E   1 +   ^ 1 D D H ðT1 T1 Þ ¼ N + NT1T1 ð3=2Þ1=2 fFlAhB  ðhA hA jlA hB Þ

(20)

+ ðlA hB jlB lB Þ  SlAlB ðhA hB jhA lB Þ + SlAhB ½FhAhA + FhBhB + FlBlB  ðhA hA jhA hA Þ  3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhA hA jlB lB Þ + 2ðhA lB jhA lB Þ  ðhB hB jlB lB Þ  ðhB hB jhB hB Þ + ShAlB ½2ðhA lB jlA hB Þ  ðhA hB jlA lB Þ  ShAhB ½FhAlA  ðhA hA jhA lA Þ + ðhA hB jlA hB Þ  ðhA lA jhB hB Þ + ðhA lA jlB lB Þ + ðhA lB jlA lB Þg D

1

(21)

E

 1 ^  ðT1 T1 Þ ¼ N + NT1T1 ð3=2Þ1=2 fFhAlB  ðhA lB jhB hB Þ + ðhA lB jlA lA Þ D D +  H  SlAlB ðhA hB jlA hB Þ + ShAlB ½FhAhA + FhBhB + FlAlA  ðhB hB jhB hB Þ  3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðlA lA jhB hB Þ + 2ðlA hB jlA hB Þ  ðhA hA jlA lA Þ  ðhA hA jhA hA Þ + SlAhB ½2ðhA lB jlA hB Þ  ðhA hB jlA lB Þ  ShAhB ½FhBlB  ðhB hB jhB lB Þ + ðhA hB jhA lB Þ  ðhA hA jhB lB Þ + ðlA hB jlA lB Þ + ðlA lA jhB lB Þg

(22)     S1 S0 j1 D + D ¼ NS1S0 N + ShAhB 2 SlAlB  ShAhB ShAlB SlAhB + SlAlB (23)     (24) S1 S0 j1 D D + ¼ NS1S0 N + ShAhB ShAhB 2  SlAhB 2  1   (25) S1 S0 j1 ðT1 T1 Þ ¼ NS1S0 NTT ð3=2Þ1=2 ðSlAhB SlAlB  ShAhB ShAlB Þ   1 +  1  D D j ðT1 T1 Þ ¼ N + NTT ð3=2Þ1=2 ShAhB ShAlB SlAlB + ShAlB 2 SlAhB + SlAhB (26)

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  1  + 1  D D j ðT1 T1 Þ ¼ N + NTT ð3=2Þ1=2 ShAhB SlAhB SlAlB + SlAhB 2 ShAlB + ShAlB (27)

where F^ is the Fock operator for the ground state configuration S0S0. This operator includes the mutual interactions of electrons in the active space but also their interaction with those in the inactive core. The symbols ðabjcdÞ ¼ ð aðr1 Þbðr1 Þð1=r12 Þc ðr2 Þd ðr2 Þdr1 dr2 represent the two-electron (electron– electron repulsion energy) integrals in the basis of molecular orbitals of the partners. Three comments can be made: (i) The direct term in Eq. (15) contains off-diagonal elements of the Fock operator such as FlAhB and FhAhB. They are generally small relative to the diagonal elements such as FhAhA, because they contain one molecular orbital on each partner. Moreover, they enter multiplied by an overlap integral. Nevertheless, their contribution to the direct term might still exceed the contribution provided by the minute twoelectron integrals, and this may lead to situations in which the direct term need not be entirely negligible in Eq. (15) relative to the mediated term. (ii) The mediated term in Eq. (15) contains not only these off-diagonal one-electron integrals but also the much larger diagonal ones, albeit multiplied by overlap integrals. The latter contribution could be comparable to those provided by off-diagonal one-electron integrals and could have a significant effect on the structural dependence of the mediated term. (iii) The third comment does not refer to SF itself, but to the possible decay of the real (not virtual) 1D+D or 1DD+ intermediate that intervenes when the conversion of the initial S1S0 state to the biexciton 1T1T1 state occurs in two steps (Section 1.2.2). We have noted under (ii) that the presence of overlap influences the mediated term in Eq. (15) through its effect on the indirect coupling of the S1S0 state and the 1T1T1 state via the virtual 1D+D and 1DD+ charge-transfer states. It introduces similar diagonal one-electron integrals into the coupling of the 1D+D and 1DD+ states with the S0S0 ground state and thus might play a role in expressions for back-electron transfer through which these charge-transfer states are deactivated to the ground state. Because of the potential importance of overlap-containing

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terms, in exact solutions of the HOMO/LUMO model we do not neglect intermolecular overlap even though it is small. In the development of simple formulas, we neglect it. The expressions (17)–(27) that include overlap are useful for qualitative considerations such as the above points (i)–(iii), but for actual numerical computations of exact solutions of the HOMO/LUMO model, we prefer to perform a L€ owdin orthogonalization of the orbitals located on different partners, and subsequently use the simple formulas for orthogonal orbitals. 2.2.2 Intermolecularly Orthogonalized Orbitals The neglect of intermolecular overlap in the simplified procedure or the use of intermolecularly L€ owdin orthogonalized molecular orbitals in the exact solution of the HOMO/LUMO model simplifies expressions (17)–(27) considerably. The normalization factors Nx in (8)–(13) become equal to unity, the overlap integrals in (23)–(27) vanish, and the expressions (17)– (22) for the Hamiltonian matrix elements simplify to the previously published3 simple formulas (28)–(32), written in terms of the Fock operator ^ which includes interactions with core electrons: for the ground state F,   ^ 1 ðT1 T1 Þ ¼ ð3=2Þ1=2 ½ðlA hB jlA lB Þ  ðhA hB jhA lB Þ S1 S0 jHj (28)     ^ 1 D + D ¼ lA jFjl ^ B + 2 ðhA lA jhA lB Þ  ðhA hA jlA lB Þ (29) S1 S0 jHj     ^ 1 D D + ¼  hA jFjh ^ B + 2 ðhA lA jlA hB Þ  ðhA hB jlA lA Þ S1 S0 jHj (30)   1 +  1   ^ ðT1 T1 Þ ¼ ð3=2Þ1=2 lA jFjh ^ B + ðlA hB jlB lB Þ  ðhA hA jlA hB Þ D D jHj 1

(31)    1=2 ^ ðT1 T1 Þ ¼ ð3=2Þ ^ B + ðlB hA jlA lA Þ  ðhA lB jhB hB Þ D D jHj hA jFjl 

+

1

(32) The Fermi golden rule formula for the SF rate then is   ^ 1 D + D ^ 1 ðT1 T1 Þi  ½ S1 S0 jHj W ðSFÞ ¼ h1 jfhS1 S0 jHj     ^ 1 T1 T1 =ΔE 1 D + D  1 D + D jHj      ^ 1 D D + 1 D D + jHj ^ 1 T1 T1 =ΔE 1 D D + gj2 ρðE Þ + S1 S0 jHj (33)

where the matrix elements of the Hamiltonian are given by expressions (28)–(32).

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2.3 A Simple Approximation for TRP For a rapid search for local maxima of jTRPj as a function of the relative disposition of two chromophores, it is desirable to simplify the expressions derived so far. The approximations to be introduced need to be fairly accurate at dimer geometries at which the absolute value of TRP is large, but they do not need to be valid at all at geometries at which jTRPj is small. We shall assume the energy differences between the charge-separated states and the initial and final states are large enough for the charge-separated state to be virtual and not a separate minimum on the S1 surface and for permitting the use of the first-order perturbation approximation and Eq. (15). If needed, first-order perturbation can be replaced by exact diagonalization, which makes the resulting formulas more complicated but does not involve much penalty in computation time. Starting with the exact solution of the HOMO/LUMO model for a pair of chromophores A and B, the following approximations were introduced1,85: 2.3.1 Neglect of Intermolecular Overlap This simplifies the expressions for matrix elements that are needed for the rate Eq. (33) to expressions (28)–(32). As is common in semiempirical theories, atomic orbitals are considered to be intramolecularly L€ owdin orthogonalized and yet retain their atomic properties. 2.3.2 Zero Differential Overlap Neglect of all electron repulsion integrals that involve charge densities resulting from products of orbitals located on different partners makes the direct term vanish and simplifies the mediated term greatly. The matrix elements needed for Eq. (33) now are:   ^ 1 ðT1 T1 Þ ¼ 0 S1 S0 jHj (34)     ^ 1 D + D ¼ lA jFjl ^ B S1 S0 jHj (35)     ^ 1 D D + ¼  hA jFjh ^ B S1 S0 jHj (36) 1 +  1    ^ ðT1 T1 Þ ¼ ð3=2Þ1=2 lA jFjh ^ B D D jHj (37)   1  + 1  ^ ðT1 T1 Þ ¼ ð3=2Þ1=2 hA jFjl ^ B D D jHj (38) The validity of the zero differential overlap (ZDO) approximation has been verified numerically for many different geometries of a pair of ethylenes.1 As an example, Fig. 3 shows the three contributions to the matrix

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^ 1(T1T1)i for two Fig. 3 Eq. (31): the three contributions to the matrix element h1D+DjHj parallel ethylene molecules separated by z ¼ 3.5 Å, as a function of slip along directions x and y. A similar example was published previously.1

^ 1(T1T1)i in Eq. (31) for two slip-stacked ethylenes. It element h1D+DjHj demonstrates that in the region of geometries where this matrix element is large, the two contributions neglected in the ZDO approximation (lAhBjlBlB) and (hAhAjlAhB) are indeed negligible relative to the contribution   ^ B that is kept in Eq. (37). (3/2)1/2 lA jFjh The formula for the SF rate now simplifies to          W ðSFÞ ¼ ð3=2Þ  h1  lA F^lB lA F^hB =ΔE 1 D + D         hA F^hB hA F^lB =ΔE ð1D D + Þj2 ρðE Þ (39)

2.3.3 A Minimum Valence Basis Set of Natural Atomic Orbitals In the next step, we express the HOMO and LUMO in a minimum basis sets of natural AOs (μ or κ on partner A and ν or λ on partner B). On each partner, this basis set is orthonormal (L€ owdin orthogonalized AOs), but the AOs on partner A may have a nonzero overlap with those on partner B. Fock ^ elements between AOs on different partners are equated to resoperator (F) onance (hopping) integrals βμν, which are then related to AO overlaps through the Mulliken approximation86,87:

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    μF^ν ¼ βμν ¼ kSμν

(40)

where the proportionality constant k is a function of the nature of atoms μ and ν via the Wolfsberg–Helmholtz formula86 used in Extended H€ uckel 87 theory (EHT),   k ¼ K Hμμ + Hνν =2 (41) and Hμμ is the standard EHT parameter describing the electron binding energy of orbital μ. In EHT calculations the value of K is usually set to 1.75 and this works well when the atoms μ and ν are separated by the usual intramolecular bonding distances. However, for distances close to or exceeding the sum of van der Waals radii, the value needs to be reduced. For ˚ , the Linderberg formula88 yields (2p–2p)π and (2p–2p)σ interactions at 3–5 A K ¼ 1.0. We adopt K ¼ 1, for which the jTRPj values obtained from the simplified and the exact solution of the HOMO/LUMO model for ethylene dimer at a variety of geometries are nearly identical. The final expression for the rate of SF now is       W ðSFÞ ¼ 3k4 =2h ρðE Þ  Σ μν clμ chν Sμν ðΣ κλ clκ clλ Sκλ Þ=ΔE 1 D + D     (42)  Σ μν chμ clν Sμν ðΣ κλ chκ chλ Sκλ Þ=ΔE 1 D D + 2 This general result can sometimes be simplified further. For qualitative considerations, it is often possible to neglect the dependence of ΔE(1D+D) and ΔE(1DD+) on the geometry of the A,B pair, since A and B need to be in contact. Typical values are 30–50 kcal/mol, but values outside this range are easily possible. If CT is equally likely in both directions, for instance, if the two partners are symmetry-related identical chromophores, it is possible to assume ΔE(1D+D) ¼ ΔE(1DD+) ¼ ΔE, and (42) then simplifies to   W ðSFÞ ¼ 3k4 =2hΔE2 ρðEÞ  2     Σ μν clμ chν Sμν ðΣ κλ clκ clλ Sκλ Þ  Σ μν chμ clν Sμν ðΣ κλ chκ chλ Sκλ Þ (43) If the two partners are different and electron transfer from A to B is much easier than from B to A, ΔE ¼ ΔE(1D+D) ≪ ΔE(1DD+), the second term in the brackets in Eq. (42) can be neglected and the expression simplifies to  2    W ðSFÞ ¼ 3k4 =2hΔE 2 ρðE Þ Σ μν clμ chν Sμν ðΣ κλ clκ clλ Sκλ Þ (44)

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The simple expressions (43) and (44) are easily programmed but in many cases they can be understood qualitatively upon visual inspection. After all, even for large molecules any one AO on partner A normally overlaps strongly with only very few AOs on partner B (often, none, but sometimes up to three or four), and most of the terms in the sums in Eqs. (42)–(44) are negligible. Although our treatment deals with cases in which the two partner molecules are distinct, it can be used for qualitative insight even if they are covalently bound. E.g., when a donor A and an acceptor B are connected through a single bond that links atom 1 on A with atom 10 on B, the only significant overlap integral is S110 and the brackets in Eq. (44) equal S110 2 cl1 2 ch10 cl10 . Then, to maximize jTRPj, the LUMO of A should have a large coefficient at its link atom 1, both the LUMO and the HOMO of B should have a large coefficient at its link atom 10 , and the linking bond should not be twisted too much. Our search for a mutual disposition of partners A and B that optimizes the rate of the S1S0 to 1TT conversion might thus appear to have been reduced to the maximization of the square in the brackets of Eqs. (42), (43), or (44), requiring only the knowledge of the expansion coefficients on HOMO and LUMO and of the overlap integrals between AOs on one and the other partner. However, only half of the work has been done, since the density of states factor ρ(E) in expressions (42)–(44) also depends on the choice of geometrical disposition of the partners. The chief reason for that is that the dimer geometry affects the energetics of the SF process by Davydov interaction that frequently stabilizes the lowest excited singlet state and leaves the energy of the lowest triplet nearly intact. Its effect on relative SF rates at various partner dispositions may be unimportant in practice if SF is sufficiently exothermic and rapid at all geometries, but it may be essential if SF is isoergic or endoergic at some of them. The factor ρ(E) has been treated by various authors, for example via microscopic dynamics,10–12 in the Marcus theory approximation,40 or using a simple kinetic model.72 In Section 2.5, we address the evaluation of the energetics as a function of geometrical arrangement of the partners in a fashion that resembles our treatment of jTRPj2. We take into account both factors that determine the magnitude of the Davydov splitting of the lowest singlet state, the direct interaction between transition densities and the contribution mediated by virtual CT states.89,90

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2.4 Local Maxima of |TRP|2 in the 6-D Space of Rigid Dimer Geometries The mutual disposition of two rigid molecules is described by six degrees of freedom, three translations (Tx, Ty, Tz) and three rotations (Rx, Ry, Rz). The value of the TRP matrix element is a function of these six variables, TRP ¼ TRP (Tx, Ty, Tz, Rx, Ry, Rz). Since in the rate expressions TRP appears in the second power, our search for the mutual dispositions of two chromophores that maximize the rate constant of the SF process starts with a search for the largest local maxima of jTRPj2 in the 6-D space. Afterward, we will discard unphysical maxima and possibly also those for which the ρ(E) term is unfavorable. Locating the maxima of jTRPj2 is an arduous task that requires a systematic search of that part of the 6-D space in which the partners A and B are close to each other. Techniques such as the genetic algorithm91 that have been developed for such searches do not guarantee that all the local extrema will be found. Our preferred procedure is to combine preselection of extrema on a 6-D grid with subsequent gradient optimization starting at the preselected points. To create a relatively sparse grid in a 6-D space, with ˚ steps for translations and 10° for rotations, one has to evaluate jTRPj2 at 0.2 A 107–109 points or even more, depending on the size of chromophores. A systematic search for maxima on the jTRPj2 surface is therefore presently limited to the use of simple formulas such as (42)–(44). These do not provide any information about intermolecular repulsions and the energetic accessibility of the geometries at which the maxima of jTRPj2 are located. By itself, the function jTRPj2 typically shows local maxima at geometries at which the HOMO and LUMO overlap strongly but the molecules interpenetrate to a ridiculous extent. This is illustrated in Fig. 4, which shows a perspective ˚, view of a plot of jTRPj2 as a function of Tx and Ty at Tz ¼ 3 A Rx ¼ Ry ¼ Rz ¼ 0, and thus displays the results for a two-dimensional subspace of the total six-dimensional space. The values of jTRPj2 are quite large, ˚ the molecules are pressed closer together than they would since at Tz ¼ 3 A ever come under ordinary conditions, making the intermolecular overlap integrals fairly large. The value at the maxima drops as Tz is increased and grows as it is decreased. It reaches a peak at Tz ¼ 0, a completely unphysical situation with the molecules interleaved in the same plane. This unphysical maximum could however be viewed as the parent of the various realizable maxima that can be arrived at by moving the two molecules further apart along one or another direction.

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Fig. 4 jTRPj2 in units of eV2 for two parallel ethylene molecules separated by 3.0 Å as a function of slip along in-plane directions x and y. The maxima are located at x ¼  1.0 and y ¼ 0.0 Å.

The inaccessible maxima of jTRPj2 thus provide insight into the origin of maxima that can actually be accessed, and our present task is to find the latter. For this purpose, we use a “search function,” which combines information about jTRPj2 with information about the part of our 6-D space that is excluded when atoms in the two partner molecules are modeled as hard spheres. The search function is defined as F ¼ αEREP 2  jTRP j2

(45)

where EREP is a repulsion term and α is a weighting coefficient. The function F has no real physical significance and only helps us to find structures at which jTRPj2 is large, yet the two partners are not unrealistically close. We have tested the use of several van der Waals potentials for EREP and in the end concluded that for our purpose a hard sphere potential seems to be the best and use X    EREP2 ¼ 10106 μν exp ςdμν = rμ vdW + rν vdW , (46) where the summation runs over all atoms of molecule A (μ) and molecule B (ν), ς ¼ 244.0, dμν is the distance between atoms μ and ν, and rμ vdW is the

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Fig. 5 The search function F2 in units of eV2 for two parallel ethylene molecules separated by 3.0 Å as a function of slip along in-plane directions x and y.

van der Waals radius92 of atom μ. The results are quite independent of the choice of the weighting coefficient α, and we set α ¼ 1. Fig. 5 shows the value of the search function F in the same two˚ , Rx ¼ Ry ¼ Rz ¼ 0) that was used dimensional subspace Tx,Ty (Tz ¼ 3 A for Fig. 4. Now, the unphysical region of interpenetrated molecules is excluded and the four minima of the search function lie at its circumference. As Tz is increased, the excluded region of the two-dimensional subspace shrinks and ultimately disappears, and the four minima of F coalesce pairwise into two, identical with those of jTRPj2. This is a typical result; the optimal geometries that can be realistically accessed surround a much higher maximum that cannot be accessed because of intermolecular repulsions. Clearly, increased hydrostatic pressure would in general be favorable for reaching higher values of jTRPj2. Using the search function F(Tx, Ty, Tz, Rx, Ry, Rz) and the simple approximation (43) for the evaluation of jTRPj2, numerical determination of the location of minima on a 6-D grid of 109 points takes several hours of CPU time on a modern Intel processor for a small molecule such as ethylene and several days for a large one, such as 1,3-diphenylisobenzofuran.

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Subsequent refinement of the best dimer geometries is performed by an optimization procedure based on numerical evaluation of gradients and the Hessian.

2.5 State Energies in the 6-D Space of Rigid Dimer Geometries Up to this point, we have ignored the ρ(E) term in Eqs. (43) and (44). When it is approximated similarly as in Marcus theory of CT, its effect on the expected rate of SF is reflected in two primary energy-related terms, the reorganization energy λ and the exoergicity of the SF process, ΔESF. While λ can be reasonably considered independent of the mutual disposition of the partners A and B, the exoergicity ΔESF cannot. For some chromophores, such as pentacene, the resulting variation of the SF rate may be of little practical consequence since E(S1) is sufficiently larger than 2E(T1) that SF is exothermic for any realistic mutual disposition of the chromophores A and B and will always prevail over competing decay channels. For chromophores in which E(S1) and 2E(T1) are less favorable, such as tetracene, the variation of ΔESF as a function of the mutual chromophore disposition may be critically important for the outcome of the competition between SF and other decay processes. In this section, we use the HOMO/LUMO model and the same approximations that were applied in the search for the local maxima of jTRPj2 in Section 2.4 to evaluate trends in the possibly detrimental effect of chromophore interaction on ΔESF. The results can then be used to eliminate some of the otherwise favorable local maxima of jTRPj2 from consideration. To estimate roughly the reduction of the exoergicity ΔESF of the conversion of the lowest excited singlet energy in a dimer into the biexciton 1(TT) relative to expectations based on the properties of the monomeric chromophore, we start with expressions given in Appendix for the elements of the Hamiltonian matrix (14). After neglect of intermolecular overlap and introduction of the ZDO approximation, the Hamiltonian matrix for the dimer system becomes

ð47Þ where

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    ^ AB UA UB EðUA UB Þ ¼ EðUA Þ + E ðUB Þ + UA UB H

(48)

^ AB describes the interaction between partners A and B. and H The elements of the Fock operator F^ in the HOMO/LUMO basis are already familiar from Section 2.4 and (hAlAjhBlB) is the Coulomb interaction between the HOMO–LUMO transition densities on chromophores A and B. In direct analogy to the procedure used in Section 2.2 to derive expression (15) for the matrix element TRP, we assume that the singlet excited states localized on one of the partners A and B and the biexciton state are described by mixtures of electron configurations dominated by S1S0, S0S1, or 1(T1T1), respectively, but also containing small admixtures of the higher energy configurations 1D+D and 1DD+. Next, the contribution of the charge-separated configurations is approximated by first-order perturbation theory and terms containing products of two small numbers are neglected. As before, we now assume that the partners A and B are the same chemical species, the S1S0, S0S1, and 1(T1T1) configurations have the same energy (i.e., SF would be isoergic in the absence of interactions between partners A and B), and the 1D+D and 1DD+ states are also degenerate and higher in energy by ΔE than S1S0 or S0S1. Strictly speaking, in this approximation, even if A and B are the same chemical species, at most geometries E(S1S0) and E(S0S1) will differ slightly, by 2[(lAlA|hBhB)  (hAhA|lBlB)], while 1D+D and 1DD+ will differ by half as much, (lAlA|hBhB)  (hAhA|lBlB). Also E(T1T1) and E(S1S0) will differ slightly, by (lAlA|lBlB) + (hAhA|lBlB)  (lAlA|hBhB)  (hAhA|hBhB). We neglect these differences presently, but they could be easily taken into account if necessary. Note that in alternant hydrocarbons such as tetracene, the charge distributions hAhA and lAlA are the same, as are hBhB and lBlB, and (lAlA|hBhB)  (hAhA|lBlB) vanishes. The resulting approximations for the (positive) Davydov splitting ΔEDS between the in-phase S1+ ¼ ðS1 S0 + S0 S1 Þ=21=2 and the out-of-phase S1 ¼ ðS1 S0  S0 S1 Þ=21=2 combination of the localized singlet excited states, for the endoergicity ΔESF  , and for the matrix elements TRP  for SF from S1  to the biexciton TT are ΔEDS ¼ jE ðS1 + Þ  E ðS1  Þj          ¼ 4 ðhA lA jhB lB Þ + lA F^lB hA F^hB =ΔE 

(49)

   ΔE ¼ E ð TT Þ  E SF h      S1 ¼ 2ðhA lAjhB lB Þ        i 2 + lA F^lB  hA F^hB  ð3=2Þ j lA F^hB 2 + hA F^lB j2 =ΔE (50)

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  ^ ^ B >  < lA jFjh ^ B> TRP  ¼ ¼ 31=2 < hA jFjl   ^ B >  < lA jFjl ^ B > =2ΔE < hA jFjh

(51)

Introduction of a minimum valence basis set of natural atomic orbitals (NAOs) and conversion of Fock matrix elements between AOs on different partners into resonance integrals followed by the use of the Mulliken approximation for conversion to overlap integrals similarly as in Section 2.3.3 yields the formulas     (52) ΔEDS ¼ 4 ðhA lA jhB lB Þ + k2 Σ μν clμ clν Sμν ðΣ κλ chκ chλ Sκλ Þ=ΔE  n   2 ΔESF  ¼  2ðhA lA jhB lB Þ + k2 Σ μν clμ clν  chμ chν Sμν  (53) 2  2 o ð3=2Þ Σ μν clμ chν Sμν  + Σ μν chμ clν Sμν  =ΔE  ^ > TRP  ¼ < S1 jHjTT
    1=2 2 ¼ 3 k Σμν chμ chν  clμ clν Sμν ½Σκλ ðchκ clλ  clκ chλ ÞSκλ  =2ΔE

(54) where μ and κ are located on partner A, and ν and λ are on partner B. If we now approximate the ρ(E) term in the Fermi Golden rule using Marcus theory, the SF rates from the S1 + and S1  combinations of the localized excited singlet states are h  i  2 2   W SF ¼ ð2π=ℏÞTRP   ð4πλkB T Þ1=2 exp  ΔESF  + λ =4πλkB T (55) If the Davydov splitting ΔEDS is large enough, only the more stable combination of locally excited singlet states is populated significantly and needs to be considered in the evaluation of SF rate. If ΔEDS is small, both combinations will be populated according to Boltzmann statistics and will contribute to SF. Then, the overall rate will be
 W ðSFÞ ¼ 1  ½ exp ðΔEDS =kB T Þ + 11 W ðSF + Þ (56)
 + ½ exp ðΔEDS =kB T Þ + 11 W ðSF Þ It is seen that excitation delocalization and the resulting Davydov splitting make SF less exoergic or more endoergic and in that sense usually are detrimental to SF. Although the formulas (53) and (54) are only

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approximate, they are likely to reflect trends correctly and will permit a rejection of some of the optimal dimer geometries obtained in Section 2.4. Finally, for the biexciton binding energy, in this approximation we obtain E(T + T) E(1TT) ¼ (3/2)k2(jΣ μνclμchνSμνj2 + jΣ μνchμclνSμνj2)/ΔE. In this case the entropy change associated with the dissociation into free triplets will not be negligible, as was assumed for processes treated in the rest of our derivations.

3. APPLICATIONS 3.1 Two Ethylene Molecules Our first illustration of the simplified HOMO/LUMO model is its application to the simplest π-electron chromophore, ethylene. Although the energies of its S0, T1, and S1 states fulfil the requirements for isoergic SF, it is obviously not a practical molecule for solar cells. We take advantage of its simplicity to derive intuitive understanding and also use it for method testing. Each ethylene molecule has two identical natural atomic orbitals, one on each atom. We have expanded the NAOs in terms of a contracted Gaussian basis set, taken from Pople’s 6-311+G basis set.93,94 The expansion coefficients of NAO in terms of contracted Gaussians were calculated by Weinhold’s NBO analysis95,96 using the SCF wave function of ethylene. Construction of the orbitals HOMO and LUMO is simple, as they are L€ owdin orthonormalized in-phase and out-of-phase combinations of 2pπ NAOs on each atom, respectively. For the evaluation of jTRPj we used ΔE ¼ 1 eV. For preselection of best structures in the 6-D space, we used a grid size of ˚ for translations and 10˚ for rotations and performed calculations at a 0.2 A total of 7.12  108 dimer geometries. We found about 13  103 local minima of the search function F. These were optimized, sorted, and duplicates (mostly due to symmetry) were removed. Only 43 structures remained, and only a dozen of these have jTRPj values larger than 0.01% of the largest jTRPj value found. The six best structures are shown in Fig. 7, along with their jTRPj2 values in units of eV2. They all obey the simple rule1 deduced from inspection of formula (43): to maximize jTRPj2, position the ethylene molecules A and B in such a way that one of the NAOs of A overlaps both NAOs of B, and the other NAO of A has as little overlap with the NAOs of B as possible. The resulting local maxima of jTRPj2 are relatively narrow peaks with a

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width of 1–1.3 A˚ at half height and if this result is general, it will easily account for the sensitivity of the SF rate to the details of molecular packing. The qualitative rule applies to all non-polar chromophores if one (i) replaces the two NAOs of ethylene by two semilocalized LMOs of a general chromophore, defined as the sum (LMO+) and the difference (LMO) of its HOMO and LUMO, with LMO+ located mostly on one side and LMO mostly on the other side of the nodal surface that is introduced upon going from HOMO to LUMO, and (ii) looks for geometries in which LMO+ on partner A overlaps both LMO+ and LMO of partner B, while LMO on partner A has as little overlap as possible with both LMO+ and LMO of partner B. Polarity introduces complications and even a perfectly stacked pair of polarized ethylenes has a non-vanishing jTRPj if it is arranged head-to-tail. In Table 1 we show the results obtained for the effects of intermolecular interaction at the six ethylene dimer geometries shown in Fig. 6. They are the excitonic coupling matrix element (hAlAjhBlB) and, with the choices ΔE ¼ 1 and 2 eV, the Davydov splitting ΔEDS and the singlet fission electronic matrix element TRP and endoergicity ΔESF starting at either one of the two exciton states (a positive value of ΔESF implies a process that is more endoergic than would be expected from the properties of isolated molecules). The dependence of the magnitude of excitonic splitting on the relative orientation of the HOMO–LUMO transition moments in the two molecules is apparent and it is clear that trends are not sensitive to the details of the calculation. It is seen that structure 1, which has the largest jTRPj2 value, also has the largest Davydov splitting and suffers the most from the endoergicity induced by intermolecular interactions. The next best structure 2, with a somewhat smaller jTRPj2 value, leads to no Davydov splitting and actually gains a little exoergicity from intermolecular interactions. It would most likely be the best dimer geometry choice if one wished to perform SF on an ethylene dimer.

3.2 Two Clipped Cibalackrot Molecules Our second illustration of the simplified HOMO/LUMO model is its application to a much larger chromophore, diindolo[3,2,1-de;3ʹ,2ʹ,1ʹ-ij] [1,5]naphthyridine-6,13-dione (Fig. 7). Its structure is related to indigo and to another industrial dye, cibalackrot, which differs only by the presence of a phenyl substituent on each of the carbon atoms adjacent to a carbonyl.

Table 1 Energetic Effects of Intermolecular Interaction in Ethylene Dimer (eV)a Structure 1 2 3

4

5

6

(hAlA|hBlB)

0.11

0

0

0.01

0.06

0

ΔEDS

1.58, 1.01

0, 0

0, 0

0.49, 0.27

0.69, 0.47

0, 0

ΔESF+

0.42, 0.32

0.05, 0.02

0.25, 0.12

0.42, 0.22

0.17, 0.15

0.04, 0.02

ΔESF

1.16, 0.69

0.05, 0.02

0.25, 0.12

0.08, 0.05

0.52, 0.32

0.04, 0.02

|TRP+|

0.13, 0.07

0.46, 0.23

0.14, 0.07

0.04, 0.02

0.08, 0.04

0.05, 0.03

|TRP|

0.86, 0.43

0.14, 0.07

0.08, 0.04

0.04, 0.02

0.05, 0.03

a

0.46, 0.23 



For the first entry in a column, ΔE = ΔE( D D ) = ΔE( D D ) = 1 eV; for the second, ΔE = 2 eV. 1

+

1

+

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Fig. 6 The six highest local maxima of jTRPj2 for a pair of ethylene molecules. The four NAOs are each colored differently, with a different shade in each of their two lobes. Projections of the two molecules onto the interior walls of the front octant are also shown. jTRPj2 values are given in eV2.

The geometry of this molecule was optimized with the B3LYP/6-311 +G* method and its frontier orbitals, HOMO and LUMO, were expressed in the basis of NAOs of π-symmetry by performing a HF/6-311+G calculation followed by Weinhold’s NBO analysis with the Gaussian program suite in a single run. Each NAO was expressed as fixed combination of four pz contracted Gaussian functions, identical to the basis set used in the SCF

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Fig. 7 Chemical structure of diindolo[3,2,1-de;3ʹ,2ʹ,1ʹ-ij][1,5]naphthyridine-6,13-dione.

procedure. The NAOs on each molecule were L€ owdin orthogonalized and used as a representation of HOMO and LUMO. For the evaluation of jTRPj we used ΔE ¼ 1 eV. We calculated the values of the search function F defined in Eq. (45) on ˚ a 6-D grid of 3.13 x 108 points with steps Δ (20° for rotations and 0.75 A for translations). We used them for preselection of physically realizable dimer structures with the largest jTRPj and found 58  103 structures for which F(xi  Δ) > F(xi) < F(xi + Δ) in all six dimensions. Subsequently, all 10,000 structures of local minima of F for which jFj exceeded 0.01% of its value in the deepest minimum were optimized starting from the preselected structures. We then removed all duplicates (either random, or resulting from symmetry of the system). Finally, we obtained 2500 structures of dimers with search function values in the range from 0.3 to 0.3  1014 eV2. Positive search function values reflect situations in which repulsion exceeds jTRPj2 and are of no interest. They were eliminated, leaving 1800 structures. At most of these, jTRPj is very small. Elimination of points with jTRPj smaller than 0.01% of the maximum jTRPj value found completed the search and provided about 350 dimer structures. Only a few of them are really significant and about a dozen have a jTRPj value that exceeds 10% of the value less than one order of magnitude lower than the best structure (i.e., a difference of two orders in expected rate constant). The first six best structures are shown in Fig. 8. They are all approximately slip-stacked and mostly also rotated to some degree. The jTRPj2 values in eV2 are shown. The first structure among those that do not contain two nearly coplanar stacked chromophores is 250th on the list of optimized structures and has a jTRPj value of 15 meV. In Table 2, we show the results obtained for the excitonic coupling matrix element (hAlAjhBlB) evaluated in the point charge approximation, Davydov splitting ΔEDS, the endoergicity ΔESF  , and the electronic matrix element TRP  for SF from the two exciton states at the six dimer geometries

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Fig. 8 The first six best structures of diindolo[3,2,1-de;3ʹ,2ʹ,1ʹ-ij][1,5]naphthyridine-6,13dione dimers in side and top projections and their jTRPj2 values in eV2, numbered 1–6 from left to right in the order of decreasing jTRPj2 value.

Table 2 Energetic Effects of Intermolecular Interaction in a Dimer of the Heterocycle of Fig. 7 (eV)a Structure 1 2 3 4 5 6

(hAlA| hBlB)

0.00

ΔEDS

3.38, 1.70 0.56, 0.28

0.36, 0.14 0.14, 0.06 1.15, 0.57 1.14, 0.57

ΔESF+

3.13, 1.57 0.16, 0.08

0.02, 0.01

0.03, 0.01

ΔESF

0.25, 0.13

0.34, 0.15

0.11, 0.05 0.97, 0.48 0.95, 0.47

|TRP+|

0.93, 0.46 0.45, 0.23

0.60, 0.30 0.38, 0.19 0.00, 0.00 0.00, 0.00

|TRP|

0.00, 0.00 0.17, 0.08

0.00, 0.00 0.15, 0.07 0.51, 0.25 0.51, 0.25

a

0.00

0.39, 0.19

0.02

0.00

0.00

0.18, 0.09

0.00

0.19, 0.10

For the first entry in a column, ΔE = ΔE(1D+D) = ΔE(1DD+) = 1 eV; for the second, ΔE = 2 eV.

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shown in Fig. 8 (a positive value of ΔESF  implies a process endoergic relative to expectations based on properties of the monomer). As in Table 1, results are shown for two choices of ΔE, 1 and 2 eV. The excitonic coupling is very small when evaluated as the electrostatic interaction of transition charge densities. When approximated as a dipole–dipole interaction, it is about four times smaller than was the case for two ethylenes in Table 1, because the transition dipole is only about half of that in ethylene. The Davydov splitting is totally dominated by the term mediated by the charge-transfer states and seems unrealistically large in this approximation, but the trend within the six structures shown is probably reliable. Similarly as in the case of ethylene, structure 1, which has the largest jTRPj2 value, also has by far the largest Davydov splitting and as a result suffers the most from the endoergicity induced by intermolecular interactions. It would be a very good choice for H-type excimer formation, but a very poor choice for SF. Structures 5 and 6 are somewhat better, but still very unfavorable energetically. In the present approximation, in structures 2 and 4, intermolecular interactions also make the formation of the biexciton less favorable. However, in structure 3 we find an ideal combination. It has one of the largest jTRP+j2 values and at the same time, its energy remains essentially unaffected by intermolecular interactions. It is thus expected to be the best dimer geometry choice for SF in a dimer of this heterocycle.

4. OUTLOOK The ability to search through a six-dimensional space of dimer geometries in pursuit of approximate positions of local extrema of the SF electronic matrix element (jTRPj), the Davydov splitting (EDS), the endoergicity of SF (ΔESF), and the biexciton binding energy offers interesting possibilities for design of structures optimized for SF, either by crystal engineering of monomers or by synthesis and crystallization of covalent dimers. We plan to release a computer program that will allow any interested party to perform such a search for a chromophore of interest. First, however, it will be necessary to build confidence in results obtained by the ruthlessly simplified procedure outlined here. The agreement of the results of the full and the simplified HOMO/LUMO model is encouraging, but obtaining a comparison with trends in both experimental data and high-level computational results is critically important. Our hope is that the structures of the local extrema obtained by the

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simple and fast procedure will indicate the approximate locations in the six-dimensional search space in which focused searches by high-level methods should be performed. We are acutely aware of the inadequacy of all treatments that are limited to the treatment of dimers. If testing results are encouraging and provide some assurance that the simple method is valuable, one can imagine extending it to more general treatments of larger aggregates or single crystals.

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Biosciences, and Geosciences, under award number DE-SC0007004. Work in Prague was supported by the Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic (RVO:  grant 15-19143S. We are grateful to Alexandr Zaykov and Petr 61388963) and GACR Felkel for assistance with computations and computer graphics, respectively.

APPENDIX The complete expressions for the matrix elements of the 6  6 Hamiltonian matrix of the HOMO/LUMO model without neglect of intermolecular overlap are listed below. ^ 0S0i: hS0S0jHjS     ^ S0 S0 ¼NS0S0 2 f4ShAhB 3 ½FhAhB  ðhA hA jhA hB Þ  ðhA hB jhB hB Þ S0 S0 H +½2FhAhA + 2ðhA hA jhA hA Þ + 6ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  2FhBhB + 2ðhB hB jhB hB ÞShAhB 2  4FhAhB ShAhB + 2FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhB hB jhB hB Þ + 2FhBhB g

^ 0S1i: hS0S0jHjS     ^ S0 S1 ¼NS0S0 NS0S1 ð21=2 f½FhAlB  ðhA hA jhA lB Þ  2ðhA lB jhB hB Þ S0 S0  H + ðhA hB jhB lB ÞShAhB 3 + f3ShAlB ½FhAhB  ðhA hA jhA hB Þ ðhA hB jhB hB Þ + FhBlB  ðhA hA jhB lB Þ  ðhB hB jhB lB Þ  2ðhA hB jhA lB ÞgShAhB 2 + f½FhAhA  ðhA hA jhA hA Þ 3ðhA hA jhB hB Þ  ðhA hB jhA hB Þ + FhBhB  ðhB hB jhB hB Þ ShAlB + 3ðhA hB jhB lB Þ  ðhA lB jhB hB Þ + FhAlB gShAhB + FhAhB ShAlB  FhBlB gÞ

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^ 1S0i: hS0S0jHjS     ^ S1 S0 ¼NS0S0 NS1S0 21=2 ð½FlAhB  2ðhA hA jlA hB Þ  ðlA hB jhB hB Þ S0 S0 H + ðhA lA jhA hB ÞShAhB 3 + f3SlAhB ½FhAhB  ðhA hA jhA hB Þ  ðhA hB jhB hB Þ  FhAlA + ðhA hA jhA lA Þ + ðhA lA jhB hB Þ + 2ðhA hB jlA hB ÞgShAhB 2  f½FhAhA  ðhA hA jhA hA Þ  3ðhA hA jhB hB Þ  ðhA hB jhA hB Þ + FhBhB  ðhB hB jhB hB Þ  SlAhB  ðhA hA jlA hB Þ + 3ðhA lA jhA hB Þ + FlAhB gShAhB  FhAhB SlAhB + FhAlA Þ ^ +Di: hS0S0jHjD   +   ^ D D ¼NS0S0 N + 21=2 ð½fFhBlB  2 ðhA hA jhB lB Þ  ðhB hB jhB lB Þ S0 S0 H + ðhA hB jhA lB ÞgShAhB 3 + fShAlB ½FhBhB  ðhB hB jhB hB Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ  FhAlB + ðhA hA jhA lB Þ + ðhA lB jhB hB Þ + 2ðhA hB jhB lB ÞgShAhB 2 + f½2FhAhB + 2ðhA hA jhA hB ÞShAlB + ðhA hA jhB lB Þ  3ðhA hB jhA lB Þ  FhBlB gShAhB + ½FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhB hB jhB hB Þ + 2FhBhB ShAlB + FhAlB Þ

^ D+i: hS0S0jHjD    +  ^ D D ¼ NS0S0 N + 21=2 f½FhAlA  ðhA hA jhA lA Þ  2ðhA lA jhB hB Þ S0 S0  H + ðhA hB jlA hB ÞShAhB 3 + fSlAhB ½FhAhA + ðhA hA jhA hA Þ + 2ðhA hA jhB hB Þ  ðhA hB jhA hB Þ  FlAhB + ðhA hA jlA hB Þ + ðlA hB jhB hB Þ + 2ðhA lA jhA hB ÞgShAhB 2  f½2FhAhB  2ðhA hB jhB hB ÞSlAhB + 3ðhA hB jlA hB Þ  ðhA lA jhB hB Þ + FhAlA gShAhB + ½2FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + FhBhB  ðhB hB jhB hB ÞSlAhB + FlAhB g

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^ 1T1i: hS0S0jHjT 

   ^ T1 T1 ¼NS0S0 NT1T1 31=2 ðf½FhAhA  ðhA hA jhA hA Þ  3ðhA hA jhB hB Þ S0 S0 H + 2ðhA hB jhA hB Þ + FhBhB  ðhB hB jhB hB ÞSlAhB + ½ðhA lA jhB hB Þ  FhAlA + ðhA hA jhA lA Þ  ðhA hB jlA hB ÞShAhB  ðhA hA jlA hB Þ + FlAhB gShAlB + f½ðhA hA jhB lB Þ  FhBlB + ðhB hB jhB lB Þ  ðhA hB jhA lB ÞShAhB  ðhA lB jhB hB Þ + FhAlB gSlAhB + ShAhB 2 ðhA lA jhB lB Þ + ½ðhA lA jhA lB Þ  ðlA hB jhB lB Þ  ShAhB + ðhA lB jlA hB ÞÞ

^ 1S0i: hS1S0jHjS 

   ^ S1 S0 ¼NS1S0 2 ðf½8ðhA hA jlA hB Þ + 4ðhA lA jhA hB Þ S1 S0 H  4ðlA hB jhB hB Þ + 4FlAhB SlAhB + 2ðhA hA jhB hB Þ + 2ðhA hA jlA lA Þ  ðhA hB jhA hB Þ  ðhA lA jhA lA Þ + ðhB hB jhB hB Þ + ðlA hB jlA hB Þ + ðlA lA jhB hB Þ  FhBhB  FlAlA gShAhB 2 + f½4ðhA hA jhA hB Þ  4ðhA hB jhB hB Þ + 4FhAhB SlAhB 2 + ½6ðhA hB jlA hB Þ + 2ðhA lA jhB hB Þ  2FhAlA + 2ðhA hA jhA lA Þ  SlAhB + 2ðhA hA jhA hB Þ  2ðhA hB jlA lA Þ  2FhAhB  2ðhA lA jlA hB ÞShAhB +½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ  FhBhB + ðhB hB jhB hB ÞSlAhB 2 + ½2ðhA hA jlA hB Þ  4ðhA lA jhA hB Þ  2FlAhB SlAhB + FhAhA  4ðhA hA jhB hB Þ  ðhA hA jlA lA Þ + 2ðhA hB jhA hB Þ + 2ðhA lA jhA lA Þ  ðhB hB jhB hB Þ + 2FhBhB + FlAlA  ðhA hA jhA hA ÞgÞ

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^ 0S1i: hS1S0jHjS     ^ S0 S1 ¼NS1S0 NS0S1 ðfShAhB 3 ½FlAlB  2ðhA hA jlA lB Þ  2ðlA lB jhB hB Þ S1 S0  H + ðlA hB jhB lB Þ + ðhA lA jhA lB Þ + fShAlB ½FlAhB  2ðhA hA jlA hB Þ  ðlA hB jhB hB Þ + ðhA lA jhA hB Þ +SlAhB ½FhAlB  ðhA hA jhA lB Þ  2ðhA lB jhB hB Þ + ðhA hB jhB lB Þ +3SlAlB ½FhAhB  ðhA hA jhA hB Þ  ðhA hB jhB hB Þ + 3ðhA hB jlA lB Þ  ðhA lA jhB lB Þ + ðhA lB jlA hB Þg  ShAhB 2 + f½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ + ðhA hB jhA hB Þ  FhBhB + ðhB hB jhB hB ÞSlAlB + f2SlAhB + ½FhAhB  ðhA hA jhA hB Þ  ðhA hB jhB hB Þ  FhAlA + ðhA hA jhA lA Þ + ðhA lA jhB hB Þ + ðhA hB jlA hB ÞgShAlB + ½ðhA hA jhB lB Þ + ðhA hB jhA lB Þ  FhBlB + ðhB hB jhB lB ÞSlAhB + ðhA hA jlA lB Þ  2ðhA lA jhA lB Þ  2ðlA hB jhB lB Þ + ðlA lB jhB hB Þ  FlAlB Þg  ShAhB  FhAhB SlAlB + ½SlAhB ðhA hB jhA hB Þ  ðhA lA jhA hB Þ  ShAlB  SlAhB ðhA hB jhB lB Þ  ðhA hB jlA lB Þ + 2 ðhA lA jhB lB ÞÞ

^ +Di: hS1S0jHjD 

  +  ^ D D ¼NS1S0 N + ðf½4ðhA hA jhB lB Þ + 2ðhA hB jhA lB Þ  2ðhB hB jhB lB Þ S1 S0 H + 2FhBlB ÞSlAhB + ð2ðhA hA jhB hB Þ  ðhA hB jhA hB Þ + ðhB hB jhB hB Þ  FhBhB SlAlB + 2ðhA hA jlA lB Þ  ðhA lA jhA lB Þ + ðlA hB jhB lB Þ + ðlA lB jhB hB Þ  FlAlB gShAhB 2 + ðf½2ðhA hA jhB hB Þ  ðhA hB jhA hB Þ + ðhB hB jhB hB Þ  FhBhB ShAlB + ðhA hA jhA lB Þ + 3ðhA hB jhB lB Þ + ðhA lB jhB hB Þ  FhAlB gSlAhB + ½2FhAhB + 2ðhA hA jhA hB ÞSlAlB + ½FlAhB + 2ðhA hA jlA hB Þ  ðhA lA jhA hB ÞShAlB  2ðhA hB jlA lB Þ  ðhA lA jhB lB Þ  ðhA lB jlA hB ÞÞShAhB + f½FhAhB + ðhA hA jhA hB Þ  ShAlB + ðhA hA jhB lB Þ  2ðhA hB jhA lB Þ  FhBlB gSlAhB + ½FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhB hB jhB hB Þ + 2FhBhB SlAlB +½FhAlA  ðhA hA jhA lA ÞShAlB  ðhA hA jlA lB Þ + 2ðhA lA jhA lB Þ + FlAlB Þ

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^ D+i: hS1S0jHjD 

   + ^ D D ¼NS1S0 N + fShAhB 3 ½FlAlA  2ðhA hA jlA lA Þ  2ðlA lA jhB hB Þ S1 S0 H + ðhA lA jhA lA Þ + ðlA hB jlA hB Þ + ½3ðhA hB jlA lA Þ + 3FhAhB  3ðhA hA jhA hB Þ  3ðhA hB jhB hB ÞShAhB 2 + f½FhAhA + 2ðhA hA jhB hB Þ  ðhA hB jhA hB Þ + ðhA hA jhA hA ÞSlAhB 2 + ½2ðhA hA jlA hB Þ  2FlAhB + 2ðlA hB jhB hB Þ  2ðhA lA jhA hB Þ  SlAhB  FhAhA + 3ðhA hA jhB hB Þ + ðhA hA jlA lA Þ + ðhA hB jhA hB Þ  2ðhA lA jhA lA Þ + ðhB hB jhB hB Þ  2ðlA hB jlA hB Þ + ðlA lA jhB hB Þ  FhBhB  FlAlA + ðhA hA jhA hA ÞgShAhB + ½FhAhB + ðhA hB jhB hB ÞSlAhB 2 +½2ðhA lA jhB hB Þ + 2FhAlA ÞSlAhB  ðhA hB jlA lA Þ  FhAhB + 2ðhA lA jlA hB Þg

^ 1T1i: hS1S0jHjT     ^ T1 T1 ¼NS1S0 NT1T1 ð3=2Þ1=2 f½ðlA lA jhB lB Þ + FhBlB  2ðhA hA jhB lB Þ S1 S0 H  ðhB hB jhB lB Þ + ðhA hB jhA lB ÞShAhB 2 + f½ðlA lA jhB hB Þ  FhBhB + ðhB hB jhB hB Þ + 2ðhA hA jhB hB Þ  ðhA hB jhB hB Þ  FlAlA + 2ðhA hA jlA lA Þ  ðhA lA jhA lA Þ  ðlA hB jlA hB ÞShAlB + ½ðhA lA jhB hB Þ  FhAlA + ðhA hA jhA lA Þ  hA hB jlA hB ÞSlAlB + ðhA hB jhB lB Þ  ðhA lA jlA lB Þ + ðhA lB jhB hB Þ  ðhA lB jlA lA Þ  FhAlB + ðhA hA jhA lB ÞgShAhB + ½ðhA hA jhB lB Þ  FhBlB + ðhB hB jhB lB Þ  ðhA hB jhA lB ÞSlAhB 2 + f½ðhA lA jhB hB Þ + FhAlA  ðhA hA jhA lA Þ + ðhA hB jlA hB ÞShAlB + ½FhAhA  ðhA hA jhA hA Þ  3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + FhBhB  ðhB hB jhB hB ÞSlAlB  ðhA hA jlA lB Þ + 2ðhA lA jhA lB Þ  ðlA hB jhB lB Þ  ðlA lB jhB hB Þ + FlAlB gSlAhB + ½FhAhB + ðhA hA jhA hB Þ  ðhA hB jlA lA Þ + ðhA lA jlA hB ÞShAlB + ½ðhA hA jlA hB Þ + FlAhB SlAlB  ðhA hB jhA lB Þ + ðlA hB jlA lB Þg

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^ 0S1i: hS0S1jHjS     ^ S0 S1 ¼ NS0S1 2 ðf4 ShAlB ½FhAlB  ðhA hA jhA lB Þ  2ðhA lB jhB hB Þ S0 S1 H + ðhA hB jhB lB Þ  FhAhA + ðhA hA jhA hA Þ + 2ðhA hA jhB hB Þ  ðhA hB jhA hB Þ  FlBlB + ðhA hA jlB lB Þ + 2ðhB hB jlB lB Þ + ðhA lB jhA lB Þ  ðhB lB jhB lB ÞgShAhB 2 + f4ShAlB 2 ½FhAhB  ðhA hA jhA hB Þ  ðhA hB jhB hB Þ + ½2ðhA hA jhB lB Þ + 6ðhA hB jhA lB Þ  2FhBlB + 2ðhB hB jhB lB ÞShAlB  2ðhA hB jlB lB Þ  2FhAhB + 2ðhA hB jhB hB Þ  2ðhA lB jhB lB ÞgShAhB + ½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ  FhBhB + ðhB hB jhB hB ÞShAlB 2 + ½4ðhA hB jhB lB Þ + 2ðhA lB jhB hB Þ  2FhAlB ShAlB + 2FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhB hB jlB lB Þ + 2ðhB lB jhB lB Þ + FhBhB  ðhB hB jhB hB Þ + FlBlB Þ

^ +Di: hS0S1jHjD   +   ^ D D ¼ NS0S1 N + fShAhB 3 ½FlBlB  2ðhA hA jlB lB Þ  2ðhB hB jlB lB Þ S0 S1 H + ðhA lB jhA lB Þ + ðhB lB jhB lB Þ + ½3FhAhB  3ðhA hA jhA hB Þ  3ðhA hB jhB hB Þ + 3ðhA hB jlB lB ÞShAhB 2 + f½FhBhB  ðhB hB jhB hB Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ  ShAlB 2 + ½2ðhA lB jhB hB Þ  2FhAlB + 2ðhA hA jhA lB Þ  2ðhA hB jhB lB ÞShAlB  FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + ðhA hA jlB lB Þ  2ðhA lB jhA lB Þ + ðhB hB jlB lB Þ  2ðhB lB jhB lB Þ  FhBhB + ðhB hB jhB hB Þ  FlBlB gShAhB + ½FhAhB + ðhA hA jhA hB ÞShAlB 2 + ½2ðhA hA jhB lB Þ + 2FhBlB ShAlB  ðhA hB jlB lB Þ + 2 ðhA lB jhB lB Þ  FhAhB g

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^ D+i: hS0S1jHjD 

   + ^ D D ¼NS0S1 N + ðf2ShAlB ½FhAlA  ðhA hA jhA lA Þ S0 S1 H  2ðhA lA jhB hB + hA hB jlA hB Þ  SlAlB ½FhAhA  ðhA hA jhA hA Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ  FlAlB + ðhA hA jlA lB Þ + 2ðlA lB jhB hB Þ  ðlA hB jhB lB Þ + ðhA lA jhA lB gShAhB 2 + ðfSlAhB ½FhAhA  ðhA hA jhA hA Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ  FlAhB + ðhA hA jlA hB Þ + ðlA hB jhB hB Þ + 3ðhA lA jhA hB ÞgShAlB + ½FhAlB + 2ðhA lB jhB hB Þ  ðhA hB jhB lB ÞSlAhB + ½2FhAhB + 2ðhA hB jhB hB ÞSlAlB  2ðhA hB jlA lB Þ  ðhA lA jhB lB Þ  ðhA lB jlA hB ÞÞShAhB + f½FhAhB + ðhA hB jhB hB ÞSlAhB  2ðhA hB jlA hB Þ + ðhA lA jhB hB Þ  FhAlA gShAlB + ½FhBlB  ðhB hB jhB lB ÞSlAhB + ½2FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + FhBhB  ðhB hB jhB hB ÞSlAlB + 2ðlA hB jhB lB Þ  ðlA lB jhB hB Þ + FlAlB Þ

^ 1T1i: hS0S1jHjT     ^ T1 T1 ¼NS0S1 NT1T1 ð3=2Þ1=2 f½FhAlA  ðhA hA jhA lA Þ  2ðhA lA jhB hB Þ S0 S1 H + ðhA hB jlA hB Þ + ðhA lA jlB lB ÞShAhB 2  f½FhAhA  ðhA hA jhA hA Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ  ðhA hA jlB lB Þ + FlBlB  2ðhB hB jlB lB Þ + ðhA lB jhA lB Þ + ðhB lB jhB lB ÞSlAhB + ½ðhA hA jhB lB Þ + FhBlB  ðhB hB jhB lB Þ + ðhA hB jhA lB ÞSlAlB  ðhA hA jlA hB Þ  ðhA lA jhA hB Þ + ðlA hB jlB lB Þ + ðlA lB jhB lB Þ + FlAhB  ðlA hB jhB hB ÞgShAhB + ½ðhA lA jhB hB Þ  FhAlA + ðhA hA jhA lA Þ  ðhA hB jlA hB ÞShAlB 2  f½ðhA hA jhB lB Þ  FhBlB + ðhB hB jhB lB Þ  ðhA hB jhA lB ÞSlAhB + ½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ  2ðhA hB jhA hB Þ  FhBhB + ðhB hB jhB hB ÞSlAlB + ðhA hA jlA lB Þ + ðhA lA jhA lB Þ  2ðlA hB jhB lB Þ + ðlA lB jhB hB Þ  FlAlB gShAlB  ½FhAhB  ðhA hB jhB hB Þ + ðhA hB jlB lB Þ  ðhA lB jhB lB ÞSlAhB +½ðhA lB jhB hB Þ + FhAlB SlAlB  ðhA hB jlA hB Þ + ðhA lB jlA lB Þg

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^ +D-i: hD+D-jHjD  +   +  ^ D D ¼ N + 2 f½ðhB hB jlB lB Þ + ðhB lB jhB lB Þ  FhBhB + ðhB hB jhB hB Þ D D H + 2ðhA hA jhB hB Þ  ðhA hB jhA hB Þ  FlBlB + 2ðhA hA jlB lB Þ  ðhA lB jhA lB ÞShAhB 2 + f½2FhBlB + 4 ðhA hA jhB lB Þ  2ðhA hB jhA lB ÞShAlB  2ðhA hB jlB lB Þ  2ðhA lB jhB lB Þ  2FhAhB + 2ðhA hA jhA hB ÞgShAhB + ½ðhB hB jhB hB Þ + 2FhBhB  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB ÞShAlB 2 + ½2FhAlB  2ðhA hA jhA lB ÞShAlB + FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhA hA jlB lB Þ + 2ðhA lB jhA lB Þ  ðhB hB jhB hB Þ + 2FhBhB + FlBlB g

^ D+i: hD+D-jHjD  +    + ^ D D ¼ N + N + fShAhB 3 ½FlAlB  2ðhA hA jlA lB Þ  2ðlA lB jhB hB Þ D D H + ðlA hB jhB lB Þ + ðhA lA jhA lB Þ + fShAlB ½FlAhB  2ðhA hA jlA hB Þ  ðlA hB jhB hB Þ + ðhA lA jhA hB Þ  SlAhB ½FhAlB  ðhA hA jhA lB Þ  2ðhA lB jhB hB Þ + ðhA hB jhB lB Þ + 3SlAlB ½FhAhB  ðhA hA jhA hB Þ  ðhA hB jhB hB Þ + 3 ðhA hB jlA lB Þ + ðhA lA jhB lB Þ  ðhA lB jlA hB Þg  ShAhB 2 + ðf2SlAhB ½FhAhB  ðhA hA jhA hB Þ  hA hB jhB hB Þ  FhAlA + ðhA hA jhA lA Þ + ðhA lA jhB hB Þ  3ðhA hB jlA hB gShAlB + ½ðhA hA jhB lB Þ  3ðhA hB jhA lB Þ  FhBlB + ðhB hB jhB lB ÞSlAhB + ½FhAhA + ðhA hA jhA hA Þ + 3ðhA hA jhB hB Þ + ðhA hB jhA hB Þ  FhBhB + ðhB hB jhB hB ÞSlAlB + ðhA hA jlA lB Þ  2ðhA lA jhA lB Þ  2ðlA hB jhB lB Þ + ðlA lB jhB hB Þ  FlAlB ÞShAhB + f½2FhAhA  2ðhA hA jhA hA Þ  6ðhA hA jhB hB Þ + 3ðhA hB jhA hB Þ + 2FhBhB  2ðhB hB jhB hB ÞSlAhB  2ðhA hA jlA hB Þ + ðhA lA jhA hB Þ + 2FlAhB ÞgShAlB + ½ðhA hB jhB lB Þ  2ðhA lB jhB hB Þ + 2FhAlB SlAhB  FhAhB SlAlB  ðhA hB jlA lB Þ + 2ðhA lB jlA hB Þg

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^ 1T1i: hD+DjHjT 

   ^ T1 T1 ¼N + NT1T1 ð3=2Þ1=2 ðfSlAlB ½FhBlB  2ðhA hA jhB lB Þ  ðhB hB jhB lB Þ D + D H + ðhA hB jhA lB Þ + ðlA lB jhB lB ÞgShAhB 2 + ðf½2ðhA hA jhB lB Þ  ðhA hB jhA lB Þ + ðhB hB jhB lB Þ  FhBlB SlAhB + ½2ðhA hA jhB hB Þ  ðhA hB jhA hB Þ + ðhB hB jhB hB Þ  FhBhB SlAlB + 2ðhA hA jlA lB Þ  ðhA lA jhA lB Þ  2ðlA hB jhB lB Þ + ðlA lB jhB hB Þ  FlAlB gShAlB  ðhA lB jhB lB ÞSlAhB + ½ðhA hB jhB lB Þ + ðhA lB jhB hB Þ  FhAlB + ðhA hA jhA lB ÞSlAlB + ðhA hA jhA lA Þ  ðhA hB jlA hB Þ  FhAlA + ðhA lA jhB hB Þ  ðhA lA jlB lB Þ  ðhA lB jlA lB ÞÞShAhB + fSlAhB ½FhBhB  ðhB hB jhB hB Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + FlAhB  2ðhA hA jlA hB Þ + ðhA lA jhA hB ÞgShAlB 2 + f½ðhA hB jhB lB Þ  2ðhA lB jhB hB Þ + 2FhAlB  2ðhA hA jhA lB ÞSlAhB + ½FhAhB + ðhA hA jhA hB ÞSlAlB  ðhA hB jlA lB Þ + 2ðhA lB jlA hB ÞgShAlB + ½FhAhA  ðhA hA jhA hA Þ  3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhA hA jlB lB Þ + 2ðhA lB jhA lB Þ  ðhB hB jlB lB Þ + FhBhB  ðhB hB jhB hB Þ + FlBlB SlAhB  SlAlB ðhA hB jhA lB Þ  ðhA hA jlA hB Þ + ðlA hB jlB lB Þ + FlAhB gÞ

^ D+i: hDD+jHjD   +   + ^ D D ¼N + 2 f½FhAhA + ðhA hA jhA hA Þ + 2ðhA hA jhB hB Þ D D H  ðhA hB jhA hB Þ + ðhA hA jlA lA Þ + ðhA lA jhA lA Þ  FlAlA + 2ðlA lA jhB hB Þ  ðlA hB jlA hB ÞShAhB 2 f½2FhAlA + 4ðhA lA jhB hB Þ  2ðhA hB jlA hB ÞSlAhB  2ðhA hB jlA lA Þ  2ðhA lA jlA hB Þ  2FhAhB + 2ðhA hB jhB hB ÞgShAhB + ½2FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB ÞSlAhB 2 + ½2 + FlAhB  2ðlA hB jhB hB Þ  SlAhB + 2FhAhA  ðhA hA jhA hA Þ  4ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ + 2ðlA hB jlA hB Þ  ðlA lA jhB hB Þ + FhBhB  ðhB hB jhB hB Þ + FlAlA g

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^ 1T1i: hDD+jHjT   +   ^ T1 T1 ¼ N + NT1T1 ð3=2Þ1=2 ½ðSlAlB fFlAlB ½FhAlA  ðhA hA jhA lA Þ D D H  2ðhA lA jhB hB Þ + ðhA hB jlA hB Þ + ðhA lA jlA lB ÞgShAhB 2  ðfShAlB ½FhAlA  ðhA hA jhA lA Þ  2ðhA lA jhB hB Þ + ðhA hB jlA hB Þ + SlAlB ½FhAhA  ðhA hA jhA hA Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + FlAlB  ðhA hA jlA lB Þ  2ðlA lB jhB hB Þ + ðlA hB jhB lB Þ + 2 ðhA lA jhA lB ÞgSlAhB + ShAlB ðhA lA jlA hB Þ + ½ðhA hA jlA hB Þ  ðhA lA jhA hB Þ + FlAhB  ðlA hB jhB hB ÞSlAlB  ðhA hA jhB lB Þ + ðlA hB jlA lB Þ + ðlA lA jhB lB Þ + FhBlB  ðhB hB jhB lB Þ + ðhA hB jhA lB ÞÞ  ShAhB  fShAlB ½FhAhA  ðhA hA jhA hA Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + FhAlB  2ðhA lB jhB hB Þ + ðhA hB jhB lB Þg  SlAhB 2 + f½2ðhA hA jlA hB Þ  ðhA lA jhA hB Þ  2FlAhB + 2ðlA hB jhB hB ÞShAlB  ½FhAhB  ðhA hB jhB hB ÞSlAlB + ðhA hB jlA lB Þ  2ðhA lB jlA hB ÞgSlAhB + ½FhAhA  ðhA hA jhA hA Þ  3ðhA hA jhB hB Þ + 2ðhA hB jhA hB Þ  ðhA hA jlA lA Þ + 2ðlA hB jlA hB Þ  ðlA lA jhB hB Þ + FhBhB  ðhB hB jhB hB Þ + FlAlA ShAlB + SlAlB ðhA hB jlA hB Þ + ðhA lB jhB hB Þ  ðhA lB jlA lA Þ  FhAlB 

^ 1T1i: hT1T1jHjT 

   ^ T1 T1 ¼ NT1T1 2 ð1=2Þðf½ 8ðhA hA jlA lB Þ + 4ðhA lA jhA lB Þ + 4ðlA hB jhB lB Þ  8ðlA lB jhB hB Þ T1 T1 H + 4FlAlB SlAlB + FlBlB  2ðhA hA jlA lA Þ  2ðhA hA jlB lB Þ + ðhA lA jhA lA Þ + ðhA lB jhA lB Þ  2ðhB hB jlB lB Þ + ðhB lB jhB lB Þ + ðlA hB jlA hB Þ  2ðlA lA jhB hB Þ + ðlA lA jlB lB Þ + 2ðlA lB jlA lB Þ + FlAlA gShAhB 2 + ðf½8 ðhA hA jlA lB Þ  4ðhA lA jhA lB Þ  4ðlA hB jhB lB Þ + 8ðlA lB jhB hB Þ  4ðFlAlB ÞSlAhB + ½8ðhA hA jlA hB Þ  4ðhA lA jhA hB Þ + 4ðlA hB jhB hB Þ  4FlAhB SlAlB + 4ðhA hA jhB lB Þ  2ðhA hB jhA lB Þ + 2ðhB hB jhB lB Þ  4ðlA hB jlA lB Þ  2ðlA lA jhB lB Þ  2FhBlB gShAlB + f½4ðhA hA jhA lB Þ  4ðhA hB jhB lB Þ  4FhAlB + 8ðhA lB jhB hB ÞSlAlB + 2ðhA hA jhA lA Þ  2ðhA hB jlA hB Þ  2FhAlA + 4ðhA lA jhB hB Þ  2ðhA lA jlB lB Þ  4ðhA lB jlA lB ÞgSlAhB + ½4ðhA hA jhA hB Þ + 4FhAhB  4ðhA hB jhB hB ÞSlAlB 2 +½8ðhA hB jlA lB Þ + 2ðhA lA jhB lB Þ  4ðhA lB jlA hB ÞSlAlB  2ðhA hA jhA hB Þ + 2FhAhB  2ðhA hB jhB hB Þ + 2ðhA hB jlA lA Þ + 2ðhA hB jlB lB Þ  2ðhA lA jlA hB Þ  2ðhA lB jhB lB ÞÞShAhB + f½8ðhA hA jlA hB Þ + 4ðhA lA jhA hB Þ  4ðlA hB jhB hB Þ + 4FlAhB SlAhB  2ðhA hA jhB hB Þ  2ðhA hA jlA lA Þ + ðhA hB jhA hB Þ + ðhA lA jhA lA Þ  ðhB hB jhB hB Þ + FhBhB + 3ðlA hB jlA hB Þ  ðlA lA jhB hB Þ + FlAlA gShAlB 2 + f½4ðhA hA jhA lB Þ + 4ðhA hB jhB lB Þ + 4FhAlB  8ðhA lB jhB hB ÞSlAhB 2 + f½4ðhA hA jhA hB Þ  4FhAhB + 4ðhA hB jhB hB ÞSlAlB  4ðhA hB jlA lB Þ + 2ðhA lA jhB lB Þ + 8ðhA lB jlA hB ÞgSlAhB + ½6ðhA hB jlA hB Þ  2FhAlA + 2ðhA hA jhA lA Þ + 2ðhA lA jhB hB ÞSlAlB  2ðhA hA jhA lB Þ  2ðhA lA jlA lB Þ + 2FhAlB  2ðhA lB jhB hB Þ + 2ðhA lB jlA lA ÞgShAlB + ½FhAhA  ðhA hA jhA hA Þ  2ðhA hA jhB hB Þ + ðhA hB jhA hB Þ + FlBlB  ðhA hA jlB lB Þ  2ðhB hB jlB lB Þ + 3ðhA lB jhA lB Þ + ðhB lB jhB lB ÞSlAhB 2 + f½2ðhA hA jhB lB Þ  6ðhA hB jhA lB Þ  2FhBlB + 2ðhB hB jhB lB ÞSlAlB  2ðhA hA jlA hB Þ + 2ðlA hB jlB lB Þ  2ðlA lB jhB lB Þ + 2FlAhB  2ðlA hB jhB hB ÞgSlAhB + ½FhAhA  ðhA hA jhA hA Þ  3ðhA hA jhB hB Þ + 4ðhA hB jhA hB Þ + FhBhB  ðhB hB jhB hB ÞSlAlB 2 + ½2ðhA hA jlA lB Þ  2ðlA lB jhB hB Þ + 2FlAlB SlAlB + 2FhAhA + 2FlBlB  2ðhA hA jhA hA Þ  6ðhA hA jhB hB Þ  2ðhA hA jlA lA Þ  2ðhA hA jlB lB Þ + 5ðhA hB jhA hB Þ + 3ðhA lB jhA lB Þ  2ðhB hB jhB hB Þ  2ðhB hB jlB lB Þ + 2FhBhB + 3ðlA hB jlA hB Þ  2ðlA lA jhB hB Þ + 2ðlA lA jlB lB Þ + ðlA lB jlA lB Þ + 2FlAlA Þ

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