Simulating electronic dynamics is essential for developing improved understanding of electronic processes in molecules and materials, including charge transfer and the evolution of excited states. Real-time timedependent density-functional theory (TDDFT) (and time-dependent Hartree-Fock theory, TDHF) methods can be utilized effectively in such cases, as a means of simulating electronic dynamics at relatively affordable computational costs. Real-time TDDFT has been used to study electron (and nuclear) dynamics in a myriad of cases: multinucleon transfer reactions through molecular and atomic collisions, molecules in oscillating electromagnetic fields of varying strengths, high-harmonic generation, resonant excitation dynamics (e.g. charge transfer, excitation-energy transfer, strong-field ionization, core excitations, plasmonic excitations ), perturbations in organic, biomolecules, chiral molecules, metallic systems, periodic systems, semiconductor materials, optical cavities, electronic stopping 32 etc. Runge and Gross proved that there exists a one-to-one mapping between the time-dependent density of a system and the external potential, justifying the use of TDDFT to simulate time-dependent electronic phenomena. This was further shown by van Leeuwen to be a special case of a more fundamental theorem, which states that the time-dependent density of a many-particle system can be reproduced by a unique external potential in another many-particle system with a differing two-particle interaction. 34 a) Electronic mail: cisborn@ucmerced.edu Although formally exact, an extremely common approximation made when using TDDFT in practice is the adiabatic approximation. One of the primary conditions within the formalism of TDDFT is that the evolving density depends on the initial state of the system and the history of its time-evolution. Within the Kohn-Sham formalism of non-interacting particles, this dependence is incorporated into the exchange-correlation (XC) potential of the time-dependent Kohn-Sham Hamiltonian. Assuming that this potential only depends on the instantaneous density, and not the density at previous points in time or the initial wave function or initial Kohn-Sham state, is the widely employed adiabatic approximation. This dependence on the initial state and the electron density at previous points in time is termed the "memory" of the potential and in practice incorporation of memory is rarely satisfied. The adiabatic approximation leads to qualitative errors in electron dynamics: incorrect charge transfer dynamics, peak shifting in time-resolved absorption spectra due to violation of the resonance condition, and incorrect electron dynamics when driven at resonant frequencies. The errors associated with the adiabatic approximation are often referred to as memory effects; they arise from the use of only the instantaneous density in the exchange-correlation potential. The TDHF method, which can be considered a special case within the generalized Kohn-Sham formalism of TDDFT, suffers from similar issues when used for real-time electron density propagation due to the dependence of the Fock operator on the electronic density, creating a similar set of nonlinear equations. It has been shown in one-dimensional model systems (such as the asymmetric Hubbard dimer and electron-Hydrogen atom scattering) as well as three-dimensional systems that adiabatic approximations made to the time-dependent exchange-correlation potential produce qualitative errors when used to simulate coherent electron dynamics, especially in cases involving charge transfer. In violation of one of the exact conditions that the TDKS XC potential must follow, some model systems and small molecules show frequency-shifts in peaks in their absorption spectra upon being resonantly driven from stationary states. The degree and direction of this shift is dependent on the amount of population driven from the ground state and the transition frequencies between the higher-lying states. Because larger systems will have a smaller relative change in the total electron density upon excitation of a single electron, the reference electron density input into the potential will be closer to that of the ground state compared to small model systems. It is unclear how such large systems would behave when driven resonantly starting in stationary states, one of the common applications for real-time TDDFT methods. Given the popularity of TDDFT methods for large-scale chemical applications, understanding and mitigating any associated systemic size-dependence (including errors due to memory effects) is an important undertaking.