Real-time time-dependent electronic structure theory is one of the most promising methods for investigating time-dependent molecular responses and electronic dynamics. Since its first modern use in the 1990s, it has been used to study a wide variety of spectroscopic properties and electronic responses to intense external electromagnetic fields, complex environments, and open quantum systems. It has also been used to study molecular conductance, excited state dynamics, ionization, and nonlinear optical effects. Real-time techniques describe non-perturbative responses of molecules, allowing for studies that go above and beyond the more traditional energy-or frequency-domain-based response theories. Recent progress in signal analysis, accurate treatment of environmental responses, relativistic Hamiltonians, and even quantized electromagnetic fields have opened up new avenues of research in time-dependent molecular response. After discussing the history of real-time methods, we explore some of the necessary mathematical theory behind the methods, and then survey a wide (yet incomplete) variety of applications for real-time methods. We then present some brief remarks on the future of real-time timedependent electronic structure theory.

Single-reference techniques based on coupled-cluster (CC) theory, in the forms of linear response (LR) or equation of motion (EOM), are highly accurate and widely used approaches for modeling valence absorption spectra. Unfortunately, these equations with singles and doubles (LR-CCSD and EOM-CCSD) scale as O(N⁶), which may be prohibitively expensive for the study of high-energy excited states using a conventional eigensolver. In this paper, we present an energy-specific non-Hermitian eigensolver that is able to obtain high-energy excited states (e.g., XAS K-edge spectrum) at low computational cost. In addition, we also introduce an improved trial vector for iteratively solving the EOM-CCSD equation with a focus on high-energy eigenstates. The energy-specific EOM-CCSD approach and its low-scaling alternatives are applied to calculations of carbon, nitrogen, oxygen, and sulfur K-edge excitations. The results are compared to other implementations of CCSD for excited states, energy-specific linear response time-dependent density functional theory (TDDFT), and experimental results with multiple statistical metrics are presented and evaluated.

We present a linear response formalism for the description of the electronic excitations of a noncollinear reference defined via Kohn-Sham spin density functional methods. A set of auxiliary variables, defined using the density and noncollinear magnetization density vector, allows the generalization of spin density functional kernels commonly used in collinear DFT to noncollinear cases, including local density, GGA, meta-GGA and hybrid functionals. Working equations and derivations of functional second derivatives with respect to the noncollinear density, required in the linear response noncollinear TDDFT formalism, are presented in this work. This formalism takes all components of the spin magnetization into account independent of the type of reference state (open or closed shell). As a result, the method introduced here is able to afford a nonzero local xc torque on the spin magnetization while still satisfying the zero-torque theorem globally. The formalism is applied to a few test cases using the variational exact-two-component reference including spin-orbit coupling to illustrate the capabilities of the method.

We report the development of a real time propagation method for solving the time-dependent relativistic exact two-component density functional theory equations (RT-X2C-TDDFT). The method is fundamentally non-perturbative and may be employed to study nonlinear responses for heavy elements which require a relativistic Hamiltonian. We apply the method to several group 12 atoms as well as heavy-element hydrides, comparing with the extensive theoretical and experimental studies on this system, which demonstrates the correctness of our approach. Because the exact two-component Hamiltonian contains spin-orbit operators, the method is able to describe the non-zero transition moment of otherwise spin-forbidden processes in non-relativistic theory. Furthermore, the two-component approach is more cost effective than the full four-component approach, with similar accuracy. The RT-X2C-TDDFT will be useful in future studies of systems containing heavy elements interacting with strong external fields.

Bioinspired constructs consisting of benzimidazole-phenol moieties bearing N-phenylimines as proton-accepting substituents have been designed to mimic the H-bond network associated with the TyrZ-His190 redox relay in photosystem II. These compounds provide a platform to theoretically and experimentally explore and expand proton-coupled electron transfer (PCET) processes. The models feature H-bonds between the phenol and the nitrogen at the 3-position of the benzimidazole and between the 1H-benzimidazole proton and the imine nitrogen. Protonation of the benzimidazole and the imine can be unambiguously detected by infrared spectroelectrochemistry (IRSEC) upon oxidation of the phenol. DFT calculations and IRSEC results demonstrate that with sufficiently strong electron-donating groups at the para-position of the N-phenylimine group (e.g.,-OCH3 substitution), proton transfer to the imine is exergonic upon phenol oxidation, leading to a oneelectron, two-proton (E2PT) product with the imidazole acting as a proton relay. When transfer of the second proton is not sufficiently exergonic (e.g.,-CN substitution), a one-electron, one-proton transfer (EPT) product is dominant. Thus, the extent of proton translocation along the H-bond network, either ~1.6 Å or ~6.4 Å, can be controlled through imine substitution. Moreover, the H-bond strength between the benzimidazole NH and the imine nitrogen, which is a function of their relative pKa values, and the redox potential of the phenoxyl radical/phenol couple are linearly correlated with the Hammett constants of the substituents. In all cases, a high potential (~1 V vs SCE) is observed for the phenoxyl radical/phenol couple. Designing and tuning redox-coupled proton wires is important for understanding bioenergetics and developing novel artificial photosynthetic systems.