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Abstract Time-dependent density functional theory continues to draw a large number of users in a wide range of fields exploring myriad applications involving electronic spectra and dynamics. Although in principle exact, the predictivity of the calculations is limited by the available approximations for the exchange-correlation functional. In particular, it is known that the exact exchange-correlation functional has memory-dependence, but in practise adiabatic approximations are used which ignore this. Here we review the development of non-adiabatic functional approximations, their impact on calculations, and challenges in developing practical and accurate memory-dependent functionals for general purposes. 

We extend the DeePMD neural network architecture to predict electronic structure properties necessary to perform non-adiabatic dynamics simulations. While learning the excited state energies and forces follows a straightforward extension of the DeePMD approach for ground-state energies and forces, how to learn the map between the non-adiabatic coupling vectors (NACV) and the local chemical environment descriptors of DeePMD is less trivial. Most implementations of machine-learning-based non-adiabatic dynamics inherently approximate the NACVs, with an underlying assumption that the energy-difference-scaled NACVs are conservative fields. We overcome this approximation, implementing the method recently introduced by Richardson [J. Chem. Phys. 158, 011102 (2023)], which learns the symmetric dyad of the energy-difference-scaled NACV. The efficiency and accuracy of our neural network architecture are demonstrated through the example of the methaniminium cation CH2NH2+. 

Advances in coherent light sources and development of pump–probe techniques in recent decades have opened the way to study electronic motion in its natural time scale. When an ultrashort laser pulse interacts with a molecular target, a coherent superposition of electronic states is created and the triggered electron dynamics is coupled to the nuclear motion. A natural and computationally efficient choice to simulate this correlated dynamics is a trajectory-based method where the quantum-mechanical electronic evolution is coupled to a classical-like nuclear dynamics. These methods must approximate the initial correlated electron–nuclear state by associating an initial electronic wavefunction to each classical trajectory in the ensemble. Different possibilities exist that reproduce the initial populations of the exact molecular wavefunction when represented in a basis. We show that different choices yield different dynamics and explore the effect of this choice in Ehrenfest, surface hopping, and exact-factorization-based coupled-trajectory schemes in a one-dimensional two-electronic-state model system that can be solved numerically exactly. This work aims to clarify the problems that standard trajectory-based techniques might have when a coherent superposition of electronic states is created to initialize the dynamics, to discuss what properties and observables are affected by different choices of electronic initial conditions and to point out the importance of quantum-momentum-induced electronic transitions in coupled-trajectory schemes. 

Although useful to extract excitation energies of states of double-excitation character in time-dependent density functional theory that are missing in the adiabatic approximation, the frequency-dependent kernel derived earlier [Maitra et al., J. Chem. Phys. 120, 5932 (2004)] was not designed to yield oscillator strengths. These are required to fully determine linear absorption spectra, and they also impact excited-to-excited-state couplings that appear in dynamics simulations and other quadratic response properties. Here, we derive a modified non-adiabatic kernel that yields both accurate excitation energies and oscillator strengths for these states. We demonstrate its performance on a model two-electron system, the Be atom, and on excited-state transition dipoles in the LiH molecule at stretched bond-lengths, in all cases producing significant improvements over the traditional approximations. 

Coupled electron–ion dynamicsviathe exact factorization approach gives improved mixed quantum classical methods due to trajectory-coupling.