Development of Real-Time TDDFT Program with <b>k</b>-Point Sampling and DFT + <i>U</i> in a Gaussian and Plane Waves Framework

We developed a <b>k</b>-point sampling real-time TDDFT (RT-TDDFT) program within the Gaussian and plane waves (GPW) framework of the CP2K software suite. In addition to standard real-time propagation of time-dependent Kohn-Sham orbitals, we make use of symmetry-based <b>k</b>-point reduction and <b>k</b>-point parallelization schemes so that our RT-TDDFT program in the GPW framework is feasible for practical large-scale calculations. We also implemented DFT + <i>U</i> as a relevant extension for real-time simulations of systems with strong electron correlations. In particular, we extended the "tensorial" subspace representation approach for DFT + <i>U</i>, following the formulation in [Chai, Z., et al. <i>J. Chem. Theory Comput.</i>, <b>2024,</b> <i>20,</i> 8984], to <b>k</b>-point sampling RT-TDDFT. Our extension, which is, to our knowledge, the first application of the "tensorial" subspace representation approach to <b>k</b>-point sampling RT-TDDFT, is found to be robust and efficient with small additional costs owing to the locality of Gaussian basis functions, indicating that it is a promising approach to RT-TDDFT + <i>U</i> for solids. We show details of our implementation in CP2K and the results of our benchmark calculations.