Great advances have been made during the past two decades in direct calculation of quantum transitions in complex molecular environments and intricate quantum dynamical processes involving open and/or driven environments. To this end, depending on the nature of physical observables and systems, transitions can be characterized by rates or may require more complete quantum description.

For the calculation of rates, Fermi’s golden rule (FGR) has been widely and successfully used for various molecular systems. However, in its applications to complex molecular systems, there are some ambiguities and issues requiring further refinement and development of FGR. In this talk, I will provide a short summary of our FGR-based theories of resonance energy transfer and charge transfer, and the energy gap law that can account for new quantum effects that were missing in previously established theories. Applications of some of these to light harvesting complexes and organic molecular aggregates are demonstrated as well.

For transitions that go beyond simple rate description, (quantum) master equation has been successful. In this talk, I will provide a general overview of these approaches and remaining issues to be addressed, and explain some of theoretical efforts we have been pursuing such as polaron transformed quantum master equation and quantum Fokker-Planck equation.

For driven quantum dynamical processes such as in quantum control and quantum sensing, accurate dynamics calculation of quantum systems driven by time dependent Hamiltonian is essential. However, efficient implementations of such calculations are in general challenging and may incur artifacts if not done correctly. Magnus expansion provides a formally superior starting point in this respect since any finite truncation approximation remains unitary. This talk presents simple and straightforward general quantum propagators based on the Magnus expansion we have recently developed, and their applications for driven quantum dynamics calculations.