Stability of the Long-Range Corrected Exchange-Correlation Functional and the Proca Procedural Functional in Time-Dependent Density-Functional Theory

Excitonic effects in the optical absorption spectra of solids can be described with time-dependent density-functional theory (TDDFT) in the linear-response regime, using a simple class of approximate, long-range corrected (LRC) exchange-correlation functionals. It was recently demonstrated that the LRC approximation can also be employed in real-time TDDFT to describe exciton dynamics. Here, we investigate the numerical stability of the time-dependent LRC approach using a two-dimensional model solid. It is found that the time-dependent Kohn-Sham equation with an LRC vector potential becomes more and more prone to instabilities for increasing exciton binding energies. The origin of these instabilities is traced back to time-averaged violations of the zero-force theorem, which leads to a simple and robust numerical stabilization scheme. This explains and justifies a recently proposed method by Dewhurst, J. K. [ <i>Phys. Rev. B</i> 2025, 111, L060302] to stabilize the LRC vector potential, known as the Proca procedural functional.