From the Fermi Golden Rule to Open Quantum Systems: Basic Concepts on Non-radiative Rates

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