The generalized quantum master equation (GQME) provides a general and exact approach for simulating the reduced dynamics in open quantum systems where a quantum system is embedded in a quantum environment. Dynamics of open quantum systems is important in excitation energy, charge, and quantum coherence transfer as well as reactive photochemistry. The system is usually chosen to be the interested degrees of freedom such as the electronic states in light-harvesting molecules or tagged vibrational modes in a condensed-phase system. The environment is also called the bath, whose influence on the system has to be considered, and for instance can be described by the GQME formalisms using the projection operator technique. In this review, we provide a heuristic description of the development of two canonical forms of GQME, namely the time-convoluted Nakajima-Zwanzig form (NZ-GQME) and the time-convolutionless form (TCL-GQME). In the more popular NZ-GQME form, the memory kernel serves as the essential part that reflects the non-Markovian and non-perturbative effects, which gives formally exact dynamics of the reduced density matrix. We summarize several schemes to express the projection-based memory kernel of NZ-GQME in terms of projection-free time correlation function inputs that contain molecular information. In particular, the recently proposed modified GQME approach based on NZ-GQME partitions the Hamiltonian into a more general diagonal and off-diagonal parts. The projection-free inputs in the above-mentioned schemes expressed in terms of different system-dependent time correlation functions can be calculated via numerically exact or approximate dynamical methods. We hope this contribution would help lower the barrier of understanding the theoretical pillars for GQME-based quantum dynamics methods and also envisage that their combination with the quantum computing techniques will pave the way for solving complex problems related to quantum dynamics and quantum information that are currently intractable even with today's state-of-the-art classical supercomputers.

Generalized quantum master equation: A tutorial review and recent advances