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Ninety years ago, Wigner derived the leading order expansion term in ℏ 2 for the tunneling rate through a symmetric barrier. His derivation included two contributions: one came from the parabolic barrier, but a second term involved the fourth-order derivative of the potential at the barrier top. He left us with a challenge, which is answered in this paper, to derive the same but for an asymmetric barrier. A crucial element of the derivation is obtaining the ℏ 2 expansion term for the projection operator, which appears in the flux-side expression for the rate. It is also reassuring that an analytical calculation of semiclassical transition state theory (TST) reproduces the anharmonic corrections to the leading order of ℏ 2 . The efficacy of the resulting expression is demonstrated for an Eckart barrier, leading to the conclusion that especially when considering heavy atom tunneling, one should use the expansion derived in this paper, rather than the parabolic barrier approximation. The rate expression derived here reveals how the classical TST limit is approached as a function of ℏ and, thus, provides critical insights to understand the validity of popular approximate theories, such as the classical Wigner, centroid molecular dynamics, and ring polymer molecular dynamics methods. 

Molecular aggregates with long-range excitonic couplings have drastically different photophysical properties compared to their monomer counterparts. From Kasha's model for one-dimensional systems, positive or negative excitonic couplings lead to blue or red-shifted optical spectra with respect to the monomers, labeled H-and J-aggregates, respectively. The overall excitonic couplings in higher dimensional systems are much more complicated and cannot be simply classified from their spectral shifts alone. Here, we provide a unified classification for extended 2D aggregates using temperature dependent peak shifts, thermal broadening, and quantum yields. We discuss the examples of six 2D aggregates with J-like absorption spectra but quite drastic changes in quantum yields and superradiance. We find the origin of the differences is, in fact, a different excitonic band structure where the bright state is lower energy than the monomer but still away from the band edge. We call this an “I-aggregate.” Our results provide a description of the complex excitonic behaviors that cannot be explained solely on Kasha's model. Furthermore, such properties can be tuned with the packing geometries within the aggregates providing supramolecular pathways for controlling them. This will allow for precise optimizations of aggregate properties in their applications across the areas of optoelectronics, photonics, excitonic energy transfer, and shortwave infrared technologies.