Hierarchical Equations of Motion for Quantum Chemical Dynamics: Recent Methodology Developments and Applications

ConspectusQuantum effects are critical to understanding many chemical dynamical processes in condensed phases, where interactions between molecules and their environment are usually strong and non-Markovian. In this Account, we review recent progress from our group in development and application of the hierarchical equations of motion (HEOM) method, highlighting its ability to address some challenging problems in quantum chemical dynamics.In the HEOM method, the bath degrees of freedom are represented using effective modes from exponential decomposition of the bath correlation function. Complex spectral densities and low temperature simulations often require a larger number of modes, making the simulations very expensive. Recent advances, such as the barycentric spectral decomposition (BSD) technique, can significantly reduce the number of effective modes, allowing to handle complex spectral densities and enabling simulations at very low temperatures, including near-zero temperature dynamics.Another key improvement in the computational efficiency is the use of tensor network methods like matrix product states and hierarchical tensor networks. These techniques allow for efficient HEOM propagation with thousands of effective modes, crucial for simulating large molecular systems interacting with multiple baths. This combination enables simulations of excitation energy transfer (EET) in systems like the Fenna-Matthews-Olson (FMO) complex and even larger systems with experimentally determined spectral densities.The versatility of the HEOM method is demonstrated through applications to a wide range of chemical dynamics problems. Simulations of EET and related ultrafast spectroscopy are first briefly covered. Applications of the HEOM to quantum tunneling effects in proton transfer reactions are then presented. Early works have studied the non-Kramers dependence of the rate constant as a function of bath friction due to deep tunneling and revealed vibrationally nonadiabatic dynamics within the so-called nontraditional view of proton transfer reactions. A recent work on the large kinetic isotope effects in soybean lipoxygenase also indicated that many quantum correction approximations to classical transition-state theory may fall short in describing deep tunneling effects.Charge transport and separation dynamics in organic semiconductors are another area where the HEOM method has been instrumental. We first demonstrate that the HEOM provides a unified description of both band-like and thermally assisted charge carrier transport in organic materials. The effect of non-nearest neighbor transitions is then investigated by combining generalized master equations with exact memory kernels. The HEOM method also enables simulation of charge separation in organic photovoltaics (OPVs) and reveals how factors such as external electric fields, entropy, and charge delocalization influence the charge separation barrier and dynamics.Moreover, HEOM has been applied to investigate hydrogen atom scattering on the Au(111) surface and vibrational energy relaxation at molecule-metal interfaces. These studies provide deeper insights into how electron-hole pair excitations and temporary charge transfer states influence the nuclear motion, offering a new framework for simulating nonadiabatic dynamics on metal surfaces.In summary, the HEOM method has developed into a robust tool for simulating quantum effects in condensed phases. Future developments in algorithm efficiency and computational power will likely expand its applicability to even more complex systems.