From the Fermi\nGolden Rule to Open Quantum Systems:\nBasic Concepts on Non-radiative Rates

The accurate estimate of nonradiative rates is a compelling\nendeavor\nto reliably address the fate of excited states in molecular functional\nmaterials, an issue of fundamental relevance and of enormous interest\nfor applications in organic optoelectronic devices and in the development\nof innovative materials for quantum technologies. Addressing nonradiative\nrates is a tricky task as it unavoidably requires to abandon the familiar\nrealm of closed quantum systems. The popular Fermi golden rule, and\nmore specifically its state-to-state version, applies to closed quantum\nsystems, and only describes non-radiative transitions between degenerate\nstates. Several strategies have been proposed to <i>open</i> the state-to-state Fermi golden rule. Here, we address the most\npopular strategy, based on the generating function approach, that,\nallowing to easily deal with a large number of vibrational degrees\nof freedom, is implemented in several popular computational codes.\nSpecifically, non-radiative rates are calculated via a Fourier transform\nof the Fermi golden rule from the energy to the time domain to obtain\nthe generating function, whose back-Fourier transform defines the\nrate-spectra, i.e., the non-radiative rates as a function of the adiabatic\ngap. The back-Fourier transform is unavoidably calculated on a finite\ntime-window, leading to broadened rate spectra. Working with a finite\ntime-observation window implicitly corresponds to a damping of the\nsignal, hence <i>opening</i> the closed quantum system of\nthe state-to-state Fermi Golden rule. Exploiting the simplest model\nfor non-radiative rates, i.e., a displaced oscillator model, we compare\nresults obtained via the generating function approach with those obtained\nvia a genuine open quantum system approach, the Redfield-Bloch approach.\nTo facilitate the comparison, we introduce a hybrid approach, damping\nthe oscillating terms in the generating function as to mimic the relaxation\ndynamics of coherences calculated in the Redfield-Bloch model. The\nrole of (quasi-)static disorder is briefly addressed. The main message\nof this work is that a strictly <i>ab initio</i> estimate\nof non-radiative rates is not possible, due to the need to introduce\nin a phenomenological way of the interaction between the system and\nthe environment.