The flow chart, Fig. S1, shows how to drive a self-consistent solution for TDKS equation. Quantum Espresso performs the step (A) (B) (C) (H) (I) (K). It’s generally required for constructing \(\mathcal{H_k}(t_2)\) to guess an initial guess of \(\rho^{in} (\mathbf{G},t_2 )\) before TD evolution. The new charge density \(\rho^{new} (\mathbf{G},t_2 )\) can be calculated by the Eqs (10)-(19). The self-consistent calculation will stop until \(\rho^{new} (\mathbf{G},t_2 )\) is consistent with \(\rho^{in} (\mathbf{G},t_2 )\).
