Bright Frenkel Excitons in Molecular Crystals: A Survey

We computed the optical properties of a large set of molecular crystals (∼2200 structures) composed of molecules whose lowest excited states are strongly coupled and generate wide excitonic bands. Such bands are classified in terms of their dimensionality (1-, 2-, and 3-dimensional), the position of the optically allowed state in relation with the excitonic density of states, and the presence of Davydov splitting. The survey confirms that one-dimensional aggregates are rare in molecular crystals highlighting the need to go beyond the simple low-dimensional models. Furthermore, this large set of data is used to search for technologically interesting and less common properties. For instance, we considered the largest excitonic bandwidth that is achievable within known molecular crystals and identified materials with strong super-radiant states. Finally, we explored the possibility that strong excitonic coupling can be used to generate emissive states in the near-infrared region in materials formed by molecules with bright visible absorption and we could identify the maximum allowable red shift in this material class. These insights with the associated searchable database provide practical guidelines for designing materials with interesting optical properties.


A. Excitonic coupling versus dipole-dipole interaction for 0D structures
Our results indicated that 4.3% of all materials have |J1|< 0.05 eV and they are characterized by extremely narrow bands. Since we have selected molecules with bright excited states, this can only happen where the molecules are distant or the dipoles are perpendicular to each other. As shown in Figure S1 (left panel), for about 95% of the cases the dipole-dipole interaction JDD is also very small, however, in 5% of the structures JDD is larger than 0.1 eV while J1 remains below 0.05 eV that is due to the fact that in these structures the higher multipole terms are dominating. We have labelled these structures as 0D excitons as they have very small excitonic bandwidth and retain most of the molecular characteristics and are excluded from further analysis. The molecular diagram of materials depicting larger dipole-dipole interactions are shown in the right panel.

B. Excitonic coupling versus intermolecular distances
As stated in the main manuscript, our results indicate that increasing values of molecular volume and the length of side chains negatively affect excitonic band widths, because of the larger intermolecular distances. In Figure S2, the relation between the largest excitonic coupling and the associated intermolecular distance is shown. As can be seen, there is a relatively large rank correlation of strength −0.65 between the two parameters reflecting the fact that the closely packed molecular pairs are expected to yield larger values of excitonic couplings. Figure S2. A scatter plot reflecting the relation between the computed excitonic coupling and the associated intermolecular distance.

C. Excitonic bandwidth with an expanded cut-off 30 Å
As explained in the main manuscript, in these calculations, we have considered all molecular pairs in van der Waals contact and also those with mass centers closer than 10 Å. As was shown in Figure S2, with the considered criteria, pairs distanced up to 20 Å are included in our calculations. This implies that the majority of "important" excitonic couplings have been considered within the defined cut-off. However, to investigate this in further detail, we have computed the excitonic bandwidth of 100 structures with a relaxed cut-off 30 Å finding a correlation with strength +0.97 ( Figure S3) between the two set of computed bandwidths W1 (with cut-off 10 Å) and W2 (with cut-off 30 Å) confirming the validity of the initially defined threshold. In the main manuscript, the excitonic bandwidths are obtained with a dielectric constant set to 1. As shown in Figure S4, considering an averaged dielectric constant 2.95, leads to median bandwidth 0.32 eV and an interquartile range 0.27 eV which are in excellent agreement with the values reported in Refs. [1][2][3][4][5] .
Furthermore, the attained maximum bandwidth 1.16 eV (considering the top 5% of the data) is in conformity with the values reported e.g. in Refs. 6-8 . Figure S4. The distribution of excitonic bandwidth scaled by an averaged dielectric constant 2.95. The variation range of among the materials with 1-, 2-, and 3D bands is shown in the inset.

E. Impact of dielectric anisotropy on the excitonic bandwidth
It is clear that when performing a large-scale screening, one cannot afford the use of the most accurate possible methods. A survey of current literature reveals that the majority of current studies do not consider the impact of dielectric constant, e.g. Refs. 9-11 , where the number of considered crystals are significantly fewer than the one studied in this work. On the other hand, the range of dielectric anisotropy for different materials is shown to be broad. It was instructive to collect this information in Table S1, where the range of dielectric anisotropy reported in the literature is given. It has to be noted that the refractive index when measured with frequency that is absorbed by the sample develops a special character which depends on the exciton and that cannot be precisely counted as anisotropy. Accordingly, the values reported in this table are for non-resonant frequencies. The mean value of the difference between dielectric constant in perpendicular directions, collected from 12 different experimental reports on 9 different crystals, is ∆ϵ = 1.32 which can be used as a guide to evaluate the error incurred in assuming isotropic dielectric response as this work and others, e.g. Refs. 12-15 , have done. Studies focusing on a more limited number of cases would benefit from the inclusion of such corrections as shown e.g. in Refs. 16,17 .

F. Distribution of difference between molecules' and solids' emission energies
The distribution of difference in molecular and solid emission energies (in absolute value) is represented in Figure S5 showing a median value 0.11 eV and a maximum (considering the top 5%) 0.56 eV (with extreme maximum being 1.42 eV) which is in accordance with majority of values reported in the literature, e.g. Refs. [30][31][32][33][34] . Figure S5. The distribution of energy difference between molecular and solid emission energies in absolute value.

G. Comparison with experimental measurements
A comparison with experimental spectra is provided in Figure S6 for Pentacene and Rubrene crystals (isotropic absorption). Computed absorption spectra are scaled in intensity and shifted in energy by −0.20 and −0.28 eV. The computed spectra are consistent with the experiment (the energy splitting between the electronic peaks is slightly underestimated with respect to the experiment) but, overall, this comparison is not sufficiently stringent to validate our results because we have not included vibronic effects. We have therefore compared the excitonic coupling computed in this work with those extracted from other theoretical work that by including vibronic coupling have successfully reproduced experimental spectra or other observables. We report such comparison in Table S2 and the results are fully satisfactory. Experimental and the calculated spectra are normalized and aligned by an energy shift of −0.2 eV and −0.28 eV in the left and right panels, respectively. For Pentacene the experimental spectrum is digitalized from Ref. 35 which is based on analysis of Ref. 36 . The dataset used for experimental spectrum of Rubrene is based on Ref. 37 . Table S2. The values of excitonic couplings computed in this work compared and also those reported in literature.  Lumin. 2005, 112 (1-4), 308-311.