Communication Cite This: J. Am. Chem. Soc. 2017, 139, 15572-15575
pubs.acs.org/JACS
Packing Guidelines for Optimizing Singlet Fission Matrix Elements in Noncovalent Dimers Eric A. Buchanan† and Josef Michl*,†,‡ †
Department of Chemistry, University of Colorado, Boulder, Colorado 80309-0215, United States Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Flemingovo nám. 2, 16610 Prague 6, Czech Republic
‡
perturbative treatment of their mixing with locally excited configurations that produces the |S1S0⟩, |S0S1⟩ (or |S±⟩ = |S1S0⟩ ± |S 0S1⟩), and |T1T1 ⟩ states; (iii) in expressions for Hamiltonian matrix elements, intermolecular overlap is negligible; (iv) zero differential overlap approximation is acceptable; (v) frontier MOs are described in an orthonormal minimum basis set of natural atomic orbitals (AOs); (vi) Mulliken approximation is used for the matrix elements Fμv between an AO μ on partner A and AO v on partner B: Fμv = kSμv, where Sμv is the overlap of AOs μ and v. F is the Fock operator for the ground state and includes interactions with all electrons. For MOs x on A and y on B, it follows that Fxy = ⟨xA| F|yB⟩ = kSxy in a pair of hydrocarbons. Since k is a function of the average electronegativity of μ and v,32,33 for π systems containing heteroatoms of different electronegativities, this relation is only approximate, but it is employed nevertheless. The guideline stated that one of the 2pz AOs of ethylene A should overlap with both AOs of ethylene B, while the other AO of ethylene A should overlap as little as possible with the AOs of ethylene B. Its validity has been confirmed by inspection of the first few dozen most important computed local maxima of |TA| for a pair of ethylene molecules.31 To maximize the actual SF rate, the effect of intermolecular interactions on state energies and on the biexciton binding energy also needs to be considered.31,34,35 The expressions provided by the simplified model 31 allow their rapid approximate evaluation. There has been much interest in numerical optimizations of the electronic matrix element for SF for dimeric chromophores.2,29,31,36−40 In the past, such efforts were constrained to an examination of a limited number of geometries in some subspace of the total six-dimensional space of relative locations of two rigid bodies. Recently, the simplified frontier orbital model has been used for complete searches for all significant local maxima within the full space.31 This requires an evaluation of |T| at 108−109 dimer geometries but can be readily done even for large π-electron systems when the simplified theory is used. In spite of this advance, it appears desirable to formulate in words a general qualitative guideline for finding dimer geometries with a large electronic matrix element |T| for general chromophores. Such a guideline for the maximization of |T| is proposed presently. It only requires the knowledge of the intermolecular overlap of the AOs of the partners and the shape of their frontier orbitals.
ABSTRACT: A simplified version of the frontier orbital model for a noncovalent dimer is used to derive guidelines for dimer geometries that maximize the square of the electronic matrix element for singlet fission. The use of the guidelines requires only the knowledge of the highest occupied and lowest unoccupied orbital of the monomer and the overlaps of the atomic orbitals on partner A with those on partner B. inglet fission (SF)1,2 is a photophysical process that converts a singlet excitation into two triplet excitations and offers an opportunity3 to circumvent the Shockley− Queisser limit4 on the theoretical efficiency of solar cells. Unfortunately, it proceeds efficiently in only very few materials, most if not all of which are not suitable for practical use. This motivates a search for additional candidates, in terms of both molecular structure and molecular packing. A simple guideline was formulated for chromophore choice a decade ago:5 In order to make SF isoergic or even better,6−8 slightly exoergic, use a biradicaloid electronic structure that is intermediate between a perfect biradical and an ordinary closed-ground-shell molecule. In the language of the reformulated9,10 classical ‘3 × 3 model’ of biradicals,11,12 such structures are reached from a perfect biradical by applying just enough but not too much covalent perturbation. More recently, this guideline has been elaborated13−16 and it has been used in searches for new chromophores.5,16−20 In contrast, although it has been long recognized that chromophore packing is very important for SF,1,2,21−28 simple rules for optimal packing of a general chromophore are not available beyond the general understanding that intermolecular overlap is necessary and the vague feeling that slip-stacked dimer geometries are favorable.1 Only for ethylene, the simplest but very impractical π-electron chromophore, has a packing guideline been deduced29 for maximizing |TA|, where TA is the electronic matrix element for the conversion of singlet excitation on partner A (|S1S0⟩; the first symbol indicates the state of partner A and the second the state of partner B) into the singlet biexciton |T1T1⟩.1,2 It used the frontier orbital model1,2,6,7,30,31 whose active space is limited to the highest occupied (hA, hB) and lowest unoccupied (lA, lB) molecular orbitals on partners A and B. It relied on a series of additional simplifications:2,29,31 (i) in isolated molecules, state energies satisfy E(S1) − E(T1) = E(T1) − E(S0); (ii) energies E(S1) + ΔE of charge transfer configurations (one electron transferred from one to the other partner) are high enough to justify a
S
© 2017 American Chemical Society
Received: July 28, 2017 Published: October 17, 2017 15572
DOI: 10.1021/jacs.7b07963 J. Am. Chem. Soc. 2017, 139, 15572−15575
Communication
Journal of the American Chemical Society We define general orbitals r and s on each partner as mixtures of h and l:
T A = [(3/2)1/2 /ΔE](FppFqp − FpqFqq)
(13)
r = h cos α + l sin α
(1)
T B = [(3/2)1/2 /ΔE](FppFpq − FqpFqq)
(14)
s = h sin α − l cos α
(2)
T +(π /4, π /4) = T +
where the range of α is 0 [r ≡ r(0) = h, s ≡ s(0) = −l] to π/2 [r ≡ r(π/2) = l, s ≡ s(π/2) = h]. The evenly mixed orbitals (α = π/4) are referred to as p ≡ r(π/4) = 2−1/2(h + l) and q ≡ s(π/ 4) = 2−1/2(h − l). The general initial singlets in the SF process in a dimer A + B are the excitonic states +
|S (ω) = |S1S0 cos ω + |S0S1 sin ω
(3)
|S−(ω) = |S1S0 sin ω − |S0S1 cos ω
(4)
= [(3)1/2 /2ΔE](Fpp − Fqq)(Fpq + Fqp) (15)
T −(α , π /4) = T − = [(3)1/2 /2ΔE](Fpp + Fqq)(Fqp − Fpq) (16) −
Note that the expressions for T have the same form not only for α = 0 and ± π/4 but also for any value of α. Neglecting the variation of ΔE with geometry, we express |T| in terms of the overlaps between MOs on partner A and those on partner B. The results obtained using the canonical MOs h and l or the transformed MOs p and q have the same form:
where ω ranges from −π/4 to π/4. Often only the lower energy excitonic state will play a significant role, but when the difference E(±) of the energies of the |S+⟩ and |S−⟩ states is small, populations in both may contribute to the SF process. The results of the simple model31 can be rewritten as eqs 5−8.
|T A| ≈ const × |ShhShl − SlhSll| = const × |SppSqp − SpqSqq| (17)
|T B| ≈ const × |ShhSlh − ShlSll| = const × |SppSpq − SqpSqq|
+ A S+(α , ω)|Hint|TT 1 1 ≈ T (α , ω) = T (α) cos ω
+ T B(α) sin ω
(18)
(5)
|T +| ≈ const′ × |(Shh − Sll)(Shl + Slh)|
A − S−(α , ω)|Hint|TT 1 1 ≈ T (α , ω) = T (α) sin ω
− T B(α)cos ω
= const′ × |(Spp − Sqq)(Spq + Sqp)|
(6)
|T −| ≈ const′ × |(Shh + Sll)(Shl − Slh)|
T A(α) = [(3/2)1/2 /2ΔE]{(cos 4α)(Fss − Frr )(Frs + Fsr )
= const′ × |(Spp + Sqq)(Sqp − Spq)|
+ [(sin 4α)/2][(Frr − Fss)2 − (Frs + Fsr )2 ] + (Frr + Fss)(Fsr − Frs)}
+ [(sin 4α)/2][(Frr − Fss)2 − (Frs + Fsr )2 ] (8)
where Hint is the interaction Hamiltonian in the diabatic picture, and tan 2ω = 2 ⟨S1S0|Hint|S0S1⟩/[E(S1S0) − E(S0S1)]. When E(±) = 0, the populations of |S+⟩ and |S−⟩ are equal, and ω can be chosen arbitrarily. The sum |T+(α,ω)|2 + |T−(α,ω)|2 is independent of ω and equals |TA(α)|2 + |TB(α)|2. These expressions for the electronic matrix elements T simplify when orbitals r and s are the canonical orbitals h and l (α = 0 or π/2) or their sum and difference p and q [α = π/4, p = 2−1/2(h + l), q = 2−1/2(h − l)], and when the initial excitation is either fully localized on partner A (TA, ω = 0) (or B: TB, ω = π/2) or evenly distributed on A and B (T+ and T−, ω = ±π/4). For α = 0, the expressions become T A = [(3/2)1/2 /ΔE](FhhFhl − FlhFll)
(9)
T B = [(3/2)1/2 /ΔE](FhhFlh − FhlFll)
(10)
(20)
Expressions 17−20 are simple enough that they permit a formulation of qualitative guidance to optimal dimer geometries. For our purposes Hückel MOs are adequate, but MOs obtained by more advanced methods can be used. The choice between MOs h and l or MOs p and q is a matter of taste; the same guidelines apply. We find it easier to visualize overlaps of orbitals that are as localized within A and B as possible. In a molecule that has a plane of symmetry perpendicular to the direction of the h → l transition moment and hence MOs not polarized but evenly delocalized along this direction, we choose to work with p and q, which are partly localized on one and the other side of the symmetry plane. For ethylene, H2CCH2, these are the two 2pz AOs, and indeed, eqs 17 and 18 revert to those already published.29 In a polyacene, p is mostly localized along one and q mostly along the other long rim (Figure 1).
(7)
T B(α) = [(3/2)1/2 /2ΔE]{(cos 4α)(Fss − Frr )(Frs + Fsr ) − (Frr + Fss)(Fsr − Frs)}
(19)
Figure 1. Anthracene as an example. (Left) Canonical orbitals h and l and their sum and difference, p and q. (Right) Top and side views of a slip-stacked dimer. |TA| is large since both qA and pA have significant overlap with qB (red and blue double-headed arrows), but only pA overlaps pB, whereas qA does not. |T−| is large because pA has significant overlap with both qB and pB (blue and green arrows), but qA overlaps only qB (red arrow) and is too far for significant overlap with p B.
T +(0, π /4) = T + = [(3)1/2 /2ΔE](Fhh − Fll)(Fhl + Flh) (11)
T −(α , π /4) = T − = [(3)1/2 /2ΔE](Fhh + Fll)(Fhl − Flh) (12)
For α = π/4, the expressions become 15573
DOI: 10.1021/jacs.7b07963 J. Am. Chem. Soc. 2017, 139, 15572−15575
Journal of the American Chemical Society
■
ACKNOWLEDGMENTS This work was supported by the U.S. DoE BES, Division of Chemical Sciences, Biosciences, and Geosciences (DESC0007004). Work in Prague was supported by the Institute of Organic Chemistry and Biochemistry (RVO: 61388963) and GAČ R (15-19143S).
In a molecule that has a strong donor character at one end of the h → l transition moment direction and strong acceptor character at the other end, and hence has h and l strongly polarized in opposite sense along this direction, we choose to work with orbitals h and l, which are largely localized on the two ends of the molecule. For instance, in aminoborane, H2BNH2, in the first approximation h is the 2pz AO on N and l is the 2pz on B. In a polyacene carrying donor moieties along one rim and acceptors along the other, the situation would be similar. We assume that orbital phases can be chosen in a way that makes products of overlaps such as SppSqp all positive (suprafacial interaction). When this is not the case (antarafacial interaction), the guidelines for optimizing |TA| and |TB| change in an obvious way. Inspection of eqs 17 and 18 leads to the conclusion that |TA| and |TB| will be large when overlaps are large and the overlap products SppSqp and SpqSqq (or SppSpq and SqpSqq) have very different magnitudes. This requires one of the orbitals on partner A, say, pA, to overlap both orbitals pB and qB on partner B, and the other orbital on partner A (qA) to have as little overlap with orbitals on partner B as possible. Figure 1 provides an illustration (slip-stacked anthracenes). The rules for optimizing |T+| or |T−| are derived similarly from eqs 19 and 20. Again assuming that the overlaps are positive, |T+| will be maximized when the overlaps Spq and Sqp are large and either Spp or Sqq is large but the other is small, and |T−| will be maximized when Spp and Sqq are large and either Spq or Sqp is large, but not both. If the overlaps cannot be all positive, the rule changes in an obvious fashion (Figure 1). Polyacene crystals, the champions of singlet fission,1,2 satisfy the rules for large matrix elements perfectly. The herringbone arrangement of neighbor molecules in their crystals is similar to that in Figure 1, except that, in addition to a slip along the short axis, one of the molecules is also tilted relative to the other, and this weakens the overlap indicated by the double-headed green arrow on the right-hand side of Figure 1. Finally, note that eqs 17−20 show that if partners A and B both possess a symmetry operation relative to which the h → l transition moment is antisymmetric and this operation is preserved at the dimer geometry, all four elements T will vanish by symmetry alone (e.g., in a perfectly stacked polyacene, this symmetry element would be reflection in a plane containing the long axes of both partners). This result is valid generally and follows from the antisymmetry of the initial state (one h → l excitation) and the symmetry of the final state (two such excitations). The addition of a quantum of an antisymmetric vibration to either the initial electronic state or the final electronic state will remove the forbiddenness. Calculations in which such vibrations were included indeed produced SF rates different from zero even when the equilibrium geometry was perfectly stacked.41,42 In summary, simple guidelines are now available for optimal molecular packing of a dimer when maximizing the electronic matrix element for singlet fission.
■
Communication
■
REFERENCES
(1) Smith, M. B.; Michl, J. Chem. Rev. 2010, 110, 6891−6936. (2) Smith, M. B.; Michl, J. Annu. Rev. Phys. Chem. 2013, 64, 361−386. (3) Hanna, M. C.; Nozik, A. J. J. Appl. Phys. 2006, 100, 074510. (4) Shockley, W.; Queisser, H. J. J. Appl. Phys. 1961, 32, 510−519. (5) Paci, I.; Johnson, J. C.; Chen, X.; Rana, G.; Popović, D.; David, D. E.; Nozik, A. J.; Ratner, M. A.; Michl, J. J. Am. Chem. Soc. 2006, 128, 16546−16553. (6) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. J. Chem. Phys. 2013, 138, 114102. (7) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. J. Chem. Phys. 2013, 138, 114103. (8) Busby, E.; Berkelbach, T. C.; Kumar, B.; Chernikov, A.; Zhong, Y.; Hlaing, H.; Zhu, X. Y.; Heinz, T. F.; Hybertsen, M. S.; Sfeir, M. Y.; Reichman, D. R. J. Am. Chem. Soc. 2014, 136, 10654−10660. (9) Bonačić-Koutecký, V.; Koutecký, J.; Michl, J. Angew. Chem., Int. Ed. Engl. 1987, 26, 170−189. (10) Michl, J. J. Am. Chem. Soc. 1996, 118, 3568−3579. (11) Salem, L.; Rowland, C. Angew. Chem., Int. Ed. Engl. 1972, 11, 92−111. (12) Michl, J. Mol. Photochem. 1972, 4, 257−286. (13) Minami, T.; Nakano, M. J. Phys. Chem. Lett. 2012, 3, 145−150. (14) Minami, T.; Ito, S.; Nakano, M. J. Phys. Chem. Lett. 2013, 4, 2133−2137. (15) Nakano, M. Chem. Rec. 2017, 17, 27−62. (16) Wen, J.; Havlas, Z.; Michl, J. J. Am. Chem. Soc. 2015, 137, 165− 172. (17) Johnson, J. C.; Nozik, A. J.; Michl, J. J. Am. Chem. Soc. 2010, 132, 16302−16303. (18) Akdag, A.; Havlas, Z.; Michl, J. J. Am. Chem. Soc. 2012, 134, 14624−14631. (19) Minami, T.; Ito, S.; Nakano, M. J. Phys. Chem. Lett. 2012, 3, 2719−2723. (20) Zeng, T.; Ananth, N.; Hoffmann, R. J. Am. Chem. Soc. 2014, 136, 12638−12647. (21) Albrecht, W. G.; Michel-Beyerle, M. E.; Yakhot, V. Chem. Phys. 1978, 35, 193−200. (22) Dillon, R. J.; Piland, G. B.; Bardeen, C. J. J. Am. Chem. Soc. 2013, 135, 17278−17281. (23) Ryerson, J. L.; Schrauben, J. N.; Ferguson, A. J.; Sahoo, S. C.; Naumov, P.; Havlas, Z.; Michl, J.; Nozik, A. J.; Johnson, J. C. J. Phys. Chem. C 2014, 118, 12121−12132. (24) Arias, D. H.; Ryerson, J. L.; Cook, J. D.; Damrauer, N. H.; Johnson, J. C. Chem. Sci. 2016, 7, 1185−1191. (25) Roberts, S. T.; McAnally, R. E.; Mastron, J. N.; Webber, D. H.; Whited, M. T.; Brutchey, R. L.; Thompson, M. E.; Bradforth, S. E. J. Am. Chem. Soc. 2012, 134, 6388−6400. (26) Dron, P. I.; Michl, J.; Johnson, J. C. J. Phys. Chem. A 2017, DOI: 10.1021/acs.jpca.7b07362. (27) Yost, S. R.; Lee, J.; Wilson, M. W. B.; Wu, T.; McMahon, D. P.; Parkhurst, R. R.; Thompson, N. J.; Congreve, D. N.; Rao, A.; Johnson, K.; Sfeir, M. Y. Nat. Chem. 2014, 6, 492−497. (28) Eaton, S. W.; Miller, S. A.; Margulies, E. A.; Shoer, L. E.; Schaller, R. D.; Wasielewski, M. R. J. Phys. Chem. A 2015, 119, 4151− 4161. (29) Havlas, Z.; Michl, J. Isr. J. Chem. 2016, 56, 96−106. (30) Král, K. Czech. J. Phys. 1972, 22, 566−571. (31) Buchanan, E. A.; Havlas, Z.; Michl, J. In Advances in Quantum Chemistry: Ratner Vol., Vol. 75; Sabin, J. R., Brändas, E. J., Eds.; Elsevier: Cambridge, MA, 2017; pp 175−227. (32) Wolfsberg, M.; Helmholz, L. J. Chem. Phys. 1952, 20, 837−843.
AUTHOR INFORMATION
Corresponding Author
*
[email protected] ORCID
Josef Michl: 0000-0002-4707-8230 Notes
The authors declare no competing financial interest. 15574
DOI: 10.1021/jacs.7b07963 J. Am. Chem. Soc. 2017, 139, 15572−15575
Communication
Journal of the American Chemical Society (33) Hoffmann, R. J. Chem. Phys. 1963, 39, 1397−1412. (34) Kolomeisky, A. B.; Feng, Y.; Krylov, A. I. J. Phys. Chem. C 2014, 118, 5188−5195. (35) Ito, S.; Nagami, T.; Nakano, M. Phys. Chem. Chem. Phys. 2017, 19, 5737−5745. (36) Wang, L.; Olivier, Y.; Prezhdo, O. V.; Beljonne, D. J. Phys. Chem. Lett. 2014, 5, 3345−3353. (37) Mirjani, F.; Renaud, N.; Gorczak, N.; Grozema, F. C. J. Phys. Chem. C 2014, 118, 14192−14199. (38) Renaud, N.; Sherratt, P. A.; Ratner, M. A. J. Phys. Chem. Lett. 2013, 4, 1065−1069. (39) Ito, S.; Nagami, T.; Nakano, M. J. Phys. Chem. A 2016, 120, 6236−6241. (40) Wu, Y.; Liu, K.; Liu, H.; Zhang, Y.; Zhang, H.; Yao, J.; Fu, H. J. Phys. Chem. Lett. 2014, 5, 3451−3455. (41) Matsika, S.; Feng, X.; Luzanov, A. V.; Krylov, A. I. J. Phys. Chem. A 2014, 118, 11943−11955. (42) Tamura, H.; Huix-Rotllant, M.; Burghardt, I.; Olivier, Y.; Beljonne, D. Phys. Rev. Lett. 2015, 115, 107401.
15575
DOI: 10.1021/jacs.7b07963 J. Am. Chem. Soc. 2017, 139, 15572−15575
