We study transport of electronic excitations (Frenkel excitons) along molecular aggregate addressing complex behaviour, which emerges beyond the limits of certain common model assumptions. We focus on the vibronic regime of transport, when quantum diffusion of Frenkel excitons is modulated by bath of underdamped vibrations. Harmonic approximation to electronic potential surfaces (linear coupling regime) allows for cost effective treatment of vibrational modulation of electronic transitions using cumulant approaches, so it is often used far beyond its formal validity. Similarly, harmonic approximation stays in the heart of normal mode quantum chemical analysis commonly used to ab initio parametrizations of the transport models.
In the present communication we surpass the limits of harmonic analysis and analyse subexponential and power law vibrational relaxation appearing outside linear coupling regime. This dynamics is probed by optical spectroscopies. In particular we supplement absorption and fluorescence spectra by femtosecond time-resolved electronic spectroscopy (such as transient absorption, or two-dimensional spectroscopy (2D)), an advanced tool for the investigation of electron-vibrational coupling.
We summarize signatures of cubic anharmonicity  and quadratic coupling  in absorption/fluorescence and 2D spectra, nad turn to analysis of excitation transport between perylenes in orthogonally arranged dyad. Vibrations are responsible for fluctuations from orthogonal equilibrium geometries registered by fast excitonic transport observed between chromophores in transient grating measurements. However, the ab initio Quantum Chemistry calculations show large discrepancies between couplings obtained by normal mode analysis of vibrations and by Molecular Dynamics sampling of molecular geometries. This can be partly assigned to failure of the harmonic approximation; rather very complex potential surfaces are suggested to drive the random trajectories by the evidence of non-Gaussian statistics of couplings for MD sampled geometries .