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As introduced above, H- and J-aggregation has been identified for various chromophores. The intermolecular interactions in aggregates depend strongly on the orientation of the molecular transition dipole moments (TDM). As shown in Figure 1.9,

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the TDM-TDM interaction leads to splitting of excited states, each higher or lower than the original monomeric excited state. This is often referred to as exciton splitting. According to Kasha’s Exciton model,105 the “face-to-face” dipole arrangement leads to an allowed transition from the ground state to the higher excited state (while the transition to the lower excited state is forbidden), resulting in a hypsochromically shifted absorbance peak. After excitation, the electrons in the higher excited state quickly relax to the lower excited state, from where the radiative transition to the ground state is suppressed. This type of aggregate is called an H-aggregate. When the molecules (and hence the TDM) are described as packing in a “head-to-tail” geometry, the transition from ground state to lower excited states is allowed, resulting in a bathochromically shifted absorbance peak and an enhanced emission rate. This class of aggregates is called the J-aggregates. As Kasha stated in his paper, this “head-to-tail” J-aggregate is more likely to be formed when the long geometrical molecular axis packed parallelly while the transition dipole moment is along the molecular short axis. The optically allowed state as often referred to as the “bright” state, and the forbidden state as the “dark” state. For slip stacking geometry as shown in the inset of Figure 1.9, the ordering of the bright and dark states is dependent on 𝜃, an angle between the transition dipole moment and the line of the molecular centers. When 𝜃 = 54.70_{, the bright and dark states are degenerate, and the exciton splitting is zero} (the optical transition is independent of intermolecular distance and the strength of the interaction coupling).

The exciton model has been proven to be very successful in rationalizing the absorbance and emission behavior of the molecular aggregates. The model links the intermolecular Coulombic interactions to the molecular (and transition dipole moment)

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geometry, which has profound influence on design and synthesis of macromolecular assemblies with specific photophysical properties. However, there are some limitations associated with this model. For example, the energy levels are subjected to the gas-to- crystal shift due to the changing electric field of the environments (i.e. solvents, surrounding materials, etc.). This can disturb the interpretation of spectral shift of aggregates. In addition, the vibronic peaks (i.e. 0-𝑣 transitions, 𝑣 = 1,2,3 …) for organic materials can spread over 0.4 - 0.7 eV in the absorbance and emission spectra. These broad vibronic bands can further complicate the spectral behaviors of the aggregates.

Figure 1.9 Energy level diagram for the Exciton model with ideal aggregates. The molecules are symbolized by the oval shapes with the double arrow representing the transition dipole moment. The intermolecular interaction in the dimer causes the splitting of the LUMO level. For the J-aggregate with the transition dipole moments aligned, the lower state is optically allowed, while the higher state is forbidden. On the other hand, the transition from the ground state to the higher state is allowed for the H- aggregate with a parallel transition dipole moment arrangement. For slip stacking geometry shown in the inset, the allowed transition depends on the slip angle, 𝜃.

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Figure 1.10 Energy level diagram of the Exciton model in ideal H- and J-aggregates with consideration of vibrational states. (a) and (b) represent the situations when the coulombic coupling is weak compared to the vibrational energy, 𝜔0, while (c) and (d) correspond to the strong coupling regime. For weakly coupled aggregates, the original vibrational states split into many sub-states where the ones on the top (bottom) of the package |𝐴_𝑛⟩ are optically allowed for H-(J-) aggregate. When the coupling is strong, the splitting in vibrational states leads to a continuous distribution of vibrational states and the optically allowed state is located on the top (bottom) of the band for the H- (J-) aggregate. This figure is taken from reference 54.

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In order to properly understand the photophysical properties of the aggregates, Spano has expanded Kasha’s model to account for the effects of intermolecular coupling, vibronic coupling and disorder in crystals on an equal footing.61,106 A detailed energy level diagram is drawn in Figure 1.10. Specifically, the absorbance line shape consists of all the transitions from the ground state (with no vibrational excitations) to the vibronically excited states based on the Franck-Condon principle. Each transition peak can be expressed as 0-0, 0-1, 0-2 … transitions, with the first and second number denoting the vibrational excitation in the ground and the excited state respectively. When the Coulombic coupling is weak as compared to the energy of a vibrational quanta (𝜔0 = 1400 cm-1 for typical organic conjugated molecules), the vibrational states of aggregates are split into many sub- states within an energy package (|𝐴₁⟩, |𝐴2⟩, etc. as shown in Figure 1.10). The optically allowed state is located on the top (bottom) of each vibrational package for H- (J-) aggregate. In addition, Spano has showen that the first two vibronic peak intensities are dependent on the Coulombic coupling strength, 𝑉,

𝐼𝐴1 𝐼𝐴2 = (1 − 0.96_𝜔𝑉 0) 2 𝜆2_{(1 + 0.29} 𝑉 𝜔₀) 2 (1.16)

where 𝑉 is the Coulombic coupling term and 𝜆2 _{is the Huang-Rhys factor. Therefore, the} ratio of first two vibronic peaks increases when 𝑉 < 0 (corresponding to H-aggregate) but decreases when 𝑉 > 0 (corresponding to J-aggregate).

When the coupling strength is large as compared to 𝜔0, the split in vibrational states is significant such that the vibrational levels are spread over the entire exciton band, while the bright state is located on the top (bottom) of the band for the H- (J-) aggregate. In this

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case, the absorbance spectra would feature a single peak that is significantly blue- or red- shifted from the original monomeric peaks.

This ratio rule provides a more reliable method to identify the type of aggregates based on spectral line shape. As discussed above, the spectral shift might be due to the nonresonant intermolecular interactions (e.g. gas-to-crystal shift), while the vibronic peak ratio is less affected.