Hybrid Gauge Approach for Accurate Real-Time TDDFT Simulations with Numerical Atomic Orbitals

Ultrafast real-time dynamics are critical for understanding a broad range of physical processes. Real-time time-dependent density functional theory (rt-TDDFT) has emerged as a powerful computational tool for simulating these dynamics, offering insight into ultrafast processes and light-matter interactions. In periodic systems, the velocity gauge is essential because it preserves the system's periodicity under an external electric field. Numerical atomic orbitals (NAOs) are widely employed in rt-TDDFT codes due to their efficiency and localized nature. However, directly applying the velocity gauge within the NAO basis set neglects the position-dependent phase variations within atomic orbitals induced by the vector potential, leading to significant computational errors - particularly in current calculations. To resolve this issue, we develop a hybrid gauge that incorporates both the electric field and the vector potential, preserving the essential phase information in atomic orbitals and thereby eliminating these errors. Our benchmark results demonstrate that the hybrid gauge fully resolves the issues encountered with the velocity gauge in NAO-based calculations, providing accurate and reliable results. This algorithm offers a robust framework for future studies on ultrafast dynamics in periodic systems using NAO bases.