Comment pubs.acs.org/JPCC
Reply to “Comment on ‘Magnetic Field Effects on Singlet Fission and Fluorescence Decay Dynamics in Amorphous Rubrene’” Geoffrey B. Piland and Christopher J. Bardeen* Department of Chemistry University of California, Riverside, California 92521, United States
J. Phys. Chem. C 2013, 117 (3), 1224−1236. DOI: 10.1021/jp309286v J. Phys. Chem. C 2016, 120. DOI: 10.1021/acs.jpcc.6b04934 involve performing individual kinetic calculations for each INTRODUCTION
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orientation and then summing up all these time-dependent contributions to obtain a total NS1(t) signal. It turns out that both approaches result in very similar dynamics for both amorphous rubrene and polycrystalline tetracene. We use the latter approach for the simulations in this paper, since it is physically more reasonable to assume that the triplet dynamics occur within single crystal domains, and that the total fluorescence is given by a sum over signals emanating from all these domains.” The fact that we did not observe a difference between the two approaches in our calculations, while TH do, most likely stems from the incorrect calculation of orientations described in point 1 above. C.3. Form of the Intermolecular Interaction Term. The exact form of the interaction Hamiltonian between molecules A and B is unknown. We agree with TH that a dipole−dipole interaction is generally regarded as the most likely form,2,5 and in an earlier paper6 we did use a dipole−dipole Hamiltonian given by Benk and Sixl7 to analyze our quantum beat data. However, the isotropic exchange interaction term (eq (13) of TH) is much simpler and gave the same qualitative behavior at high fields, so we used that expression for the calculations in PBKB.
In the accompanying comment, Tapping and Huang (TH) raise several issues with our 2013 paper “Magnetic Field Effects on Singlet Fission and Fluorescence Decay Dynamics in Amorphous Rubrene” by Piland, Burdett, Kurunthu, and Bardeen1 (hereafter referred to as PBKB). The experimental results and the Merrifield kinetic analysis in that paper are correct, but we agree with TH that the analysis of the magnetic field on the triplet-pair eigenstates is not correct. To clarify some of the issues in PBKB, we divide our response to their Comment into two parts: Conceptual and Typographical Errors in our original paper that we correct here.
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CONCEPTUAL C.1. Orientation Effects on Triplet-Pair States. The most important point of TH is that we neglected to rotate the individual molecular axes to correctly derive the triplet-pair zero-field states. We agree with the analysis of TH, and indeed the necessity of such a coordinate transformation was pointed out on p. 506 in ref 2 albeit suggesting a different coordinate transformation. From our survey of the literature, Lendi et al. were the first to perform a full analysis of how randomly aligned molecular pairs interact with magnetic fields.3 It should be pointed out that the incorrect approach we used has also been used by subsequent workers.4 We agree with TH that differences between random and ordered molecules should be more obvious at lower fields. C.2. Spatial Averaging over Triplet Pair Orientations. We agree with TH that the correct way to calculate the total fluorescence signal is to calculate the signal for each molecular pair individually and then sum the total. We pointed this out in a later paper (Chem. Phys. Lett. 2013, 585, p 7): “In order to use the |ClS|2 values as inputs for the kinetic calculations, we have two choices. In ref 1 we simply summed over the energyordered |ClS|2 values for all orientations in order to get a set of nine averaged |ClS|2 values, which were used as inputs for a single kinetic calculation. A more rigorous calculation would
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TYPOGRAPHICAL ERRORS IN PBKB T.1. Zero-Field Hamiltonian Typo. Equation 3 in TH is correct, while eq 3 in PBKB incorrectly has a + instead of a − 1 2 sign in front of the 3 S ̂ term due to a typographical error. We did use the correct Hamiltonian for the calculations, which explains why the eigenvalues in the Supporting Information Hamiltonian matrix were correct, except for the typo described below in point 2. T.2. Hamiltonian Matrix in PBKB Supporting Information. The Hamiltonian matrix in PBKB’s Supporting Information contains several typographical errors. Here we give both the original and corrected versions.
Received: July 13, 2016 Revised: September 27, 2016
© XXXX American Chemical Society
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DOI: 10.1021/acs.jpcc.6b07021 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Comment
The Journal of Physical Chemistry C
Original Hamiltonian matrix: ⎡ ⎛D ⎞ −X ⎢ 2⎜ − E⎟ − igβhzB − igβhyB − igβhzA ⎝ ⎠ ⎢ 3 ⎢ 2D ⎢ igβhzB X − igβhxB − igβhzA 3 ⎢ ⎢ B 0 0 igβhxB D − E ⎢ igβhy ⎢ 2D ⎢ igβh A 0 X − igβhzB z ⎢ 3 Ĥ = ⎢⎢ ⎛D ⎞ 0 igβhzA igβhzB 2⎜ + E⎟ −X ⎢ ⎝3 ⎠ ⎢ A B B ⎢ 0 0 igβhz igβhy igβhx ⎢ ⎢ igβh A 0 0 X igβhxA y ⎢ ⎢ 0 0 0 igβhyA igβhxA ⎢ ⎢ ⎢ −X 0 0 igβhyA −X ⎣
⎤ −X ⎥ ⎥ ⎥ 0 0 0 ⎥ − igβhyA ⎥ A A⎥ 0 X − igβhz − igβhy ⎥ ⎥ 0 0 ⎥ − igβhyB − igβhxA ⎥ ⎥ 0 − igβxB − igβhxA − X ⎥ ⎥ ⎥ A⎥ 0 D+E X − igβhx ⎥ 0 D − E − igβhzB − igβhyB ⎥ ⎥ ⎥ X igβhzB D + E − igβhxB ⎥ 4D ⎥⎥ igβhxA igβhyB igβhxB 3 ⎦ 0
− igβhyA
0
Corrected Hamiltonian matrix: ⎡ ⎛D ⎞ 0 0 −X − igβhyA ⎢ 2⎜ − E⎟ − igβhzB − igβhyB − igβhzA ⎝ ⎠ 3 ⎢ ⎢ 2D ⎢ igβhzB X 0 0 − igβhxB − igβhzA − igβhyA 3 ⎢ ⎢ D ⎢ igβhyB igβhxB − − E 0 0 X 0 − igβhzA 3 ⎢ ⎢ 2D ⎢ igβhzA 0 0 X − igβhzB − igβhyB − igβhxA 3 ⎢ ⎢ ⎛D ⎞ Ĥ = ⎢ − X 0 0 igβhzA igβhzB 2⎜ + E⎟ − igβxB − igβhxA ⎝3 ⎠ ⎢ ⎢ D ⎢ 0 0 0 igβhzA igβhyB igβhxB X − +E ⎢ 3 ⎢ D ⎢ igβh A 0 0 0 X igβhxA − − E − igβhzB y ⎢ 3 ⎢ D ⎢ 0 0 0 igβhyA igβhxA X igβhzB − + E ⎢ 3 ⎢ A A B ⎢ −X 0 0 igβhy igβhx igβhy igβhxB −X ⎣
T.3. Zero-Field Parameters D and E Should Be Those of Isolated Molecules. We agree with the TH position that the molecular D and E values should be used for rubrene. However, the only zero-field splitting parameters we could find for rubrene were the D* and E* values for crystalline rubrene which were the same as those of tetracene.8 Our calculations were insensitive to the magnitude of the zero-field splitting, but this was likely an artifact of our incorrect orientational averaging. T.4. Correct Value for gβ. TH are correct that J gβ = 2∗9.274 × 10−24 T = 0.934 cm−1 T−1, and this value
⎤ −X ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ − igβhyA ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ −X ⎥ ⎥ ⎥ − igβhxA ⎥ ⎥ ⎥ − igβhyB ⎥ ⎥ ⎥ − igβhxB ⎥ ⎥ ⎥ 4D ⎥ − 3 ⎦
was quoted in the text of our paper. However, when the calculations were done, a value of gβ 9.274 × 10−24JT −1 = 2∗ = 1.76 × 1011 Hz T−1 ℏ 1.055 × 10−34Js = 5.87 cm−1 T−1
was used in PBKB, leading to a value for gβ that was a factor of 2π too large. B
DOI: 10.1021/acs.jpcc.6b07021 J. Phys. Chem. C XXXX, XXX, XXX−XXX
Comment
The Journal of Physical Chemistry C
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors acknowledge financial support from the National Science Foundation grant CHE-1152677. REFERENCES
(1) Piland, G. B.; Burdett, J. J.; Kurunthu, D.; Bardeen, C. J. Magnetic field effects on singlet fission and fluorescence decay dynamics in amorphous rubrene. J. Phys. Chem. C 2013, 117, 1224−1236. (2) Swenberg, C. E.; Geacintov, N. E. Exciton Interactions in Organic Solids. In Organic Molecular Photophysics; Birks, J. B., Ed.; Wiley & Sons: London, 1973. (3) Lendi, K.; Berber, P.; Labhart, H. Influence of a Magnetic Field on Delayed Fluorescence of Aromatic Hydrocarbons in Solution. Chem. Phys. 1976, 18, 449−468. (4) Yokoyama, K.; Wakikawa, Y.; Miura, T.; Fujimori, J.; Ito, F.; Ikoma, T. Solvent Viscosity Effect on Triplet-Triplet Pair in Triplet Fusion. J. Phys. Chem. B 2015, 119, 15901−15908. (5) Wang, R.; Zhang, C.; Zhang, B.; Liu, Y.; Wang, X.; Xiao, M. Magnetic Dipolar Interaction between Correlated Triplets Created by Singlet Fission in Tetracene Crystals. Nat. Commun. 2015, 6, 8602. (6) Burdett, J. J.; Bardeen, C. J. Quantum beats in crystalline tetracene delayed fluorescence due to triplet pair coherences produced by direct singlet fission. J. Am. Chem. Soc. 2012, 134, 8597−8607. (7) Benk, H.; Sixl, H. Theory of two coupled triplet states. Application to bicarbene structures. Mol. Phys. 1981, 42, 779−801. (8) Lesin, V. I.; Pristupa, A. I.; Frankevich, E. L. Magnetic resonance spectrum of triplet exciton pairs in polycrystalline layers of rubrene with nonequivalent locations of the molecules in the unit cells, as detected from the quantum yield. Opt. Spect. 1981, 51, 477−480.
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DOI: 10.1021/acs.jpcc.6b07021 J. Phys. Chem. C XXXX, XXX, XXX−XXX
