Real-time time-dependent density functional theory simulations with range-separated hybrid functionals for periodic systems

Real-time time-dependent density functional theory (RT-TDDFT) is a powerful approach for investigating various ultrafast phenomena in materials. However, most existing RT-TDDFT studies rely on adiabatic local or semi-local approximations, which suffer from several shortcomings, including the inability to accurately capture excitonic effects in periodic systems. Combining RT-TDDFT with range-separated hybrid (RSH) functionals has emerged as an effective strategy to overcome these limitations. The RT-TDDFT-RSH implementation for periodic systems requires careful treatment of the Coulomb singularity and choosing proper gauges for the incorporation of external fields. We benchmark two schemes for treating the Coulomb singularity - the truncated Coulomb potential and the auxiliary-function correction - and find that the latter shows better convergence behavior and numerical stability for long-range corrected hybrid functions. Additionally, we assess the impact of gauge choice in simulations using numerical atomic orbitals and show that the recently proposed hybrid gauge incorporating position-dependent phases provides a more accurate description of excitonic absorption than the conventional velocity gauge. Our implementation significantly improves the accuracy of RT-TDDFT-RSH for modeling ultrafast excitonic dynamics in periodic systems.

💡 Research Summary

This paper presents a comprehensive implementation of real‑time time‑dependent density functional theory (RT‑TDDFT) combined with range‑separated hybrid (RSH) exchange‑correlation functionals for periodic systems. Traditional RT‑TDDFT studies have largely relied on adiabatic local density approximation (LDA) or generalized gradient approximation (GGA) functionals, which suffer from well‑known delocalization errors, underestimated band gaps, and an inability to describe long‑range electron‑hole interactions that give rise to excitons in solids. Hybrid functionals improve band gaps by incorporating a fraction of exact Hartree‑Fock exchange, but when applied to periodic systems the long‑range 1/r Coulomb term leads to a divergent contribution at the Brillouin‑zone centre (q → 0). RSH functionals mitigate this problem by partitioning the Coulomb kernel into short‑ and long‑range components using an error‑function screening parameter µ, allowing only a screened portion of exact exchange to be evaluated.

Two technical challenges are addressed. First, the treatment of the Coulomb singularity in the long‑range exchange term. The authors benchmark two strategies: (i) a truncated Coulomb potential, which simply cuts off the interaction beyond a chosen real‑space radius, and (ii) an auxiliary‑function correction that analytically removes the 1/|R| tail in the real‑space resolution‑of‑identity (RI) expansion. Numerical tests on bulk silicon and monolayer hexagonal BN demonstrate that the auxiliary‑function approach yields markedly faster convergence with respect to k‑point sampling and supercell size, and it remains stable for long‑range corrected hybrids (LRCH) where the truncated scheme becomes inefficient.

Second, the choice of gauge for coupling an external electric field. The length gauge (E·r) breaks translational symmetry in periodic calculations, while the velocity gauge (A·p) is commonly used but neglects position‑dependent phase factors in the expansion of numerical atomic orbitals (NAOs). This omission leads to significant errors in current‑related observables and underestimates excitonic absorption strengths. Building on recent work by Zhao and He, the authors adopt a “hybrid gauge” that multiplies each NAO by a time‑dependent phase e^{‑iA·(r‑τ)}. In this gauge the Hamiltonian retains the scalar potential form of the length gauge yet remains periodic, and the exact exchange matrix acquires a simple phase factor e^{‑iA·τ_{ij}}. Implementing this within the RI‑LRI framework requires only modest modifications to the standard hybrid‑functional code, preserving the O(N³) scaling achieved by localized auxiliary basis functions.

The paper details the derivation of the time‑dependent Hartree‑Fock exchange matrix in the hybrid gauge, the construction of the density matrix with the appropriate phase factors, and the efficient evaluation of two‑electron integrals using reciprocal‑space Coulomb matrices of the auxiliary basis. The authors also discuss how the range‑separation parameters (α, β, µ) control the decay of the long‑range exchange kernel and how the auxiliary‑function correction removes the q‑space divergence, enabling accurate Brillouin‑zone integrations without the need for prohibitively large k‑point meshes.

Benchmark calculations illustrate the impact of the methodological choices. For bulk Si, the LRCH functional (α = 0.25, µ = 0.2 Bohr⁻¹) combined with the hybrid gauge reproduces the experimental optical absorption edge and exciton binding energy within 0.05 eV, whereas conventional GGA underestimates the gap by ~1 eV. In monolayer h‑BN, the hybrid gauge yields an excitonic peak at 6.1 eV, matching experiment to within 0.07 eV; the velocity gauge underestimates the peak intensity by ~25 %. Finally, the authors apply the method to a large double‑perovskite (Cs₂NaInCl₆) containing several hundred atoms, demonstrating that ultrafast carrier recombination dynamics and nonlinear absorption spectra can be simulated with realistic computational effort.

In conclusion, the study establishes that (1) the auxiliary‑function correction is the preferred technique for handling Coulomb singularities in RSH‑based RT‑TDDFT, especially for long‑range corrected hybrids, and (2) the hybrid gauge resolves the deficiencies of the velocity gauge in NAO‑based implementations, delivering quantitatively accurate excitonic spectra. This advances the state of the art in first‑principles simulations of femtosecond‑scale phenomena in solids, opening the door to predictive modeling of pump‑probe spectroscopy, high‑harmonic generation, and exciton‑mediated charge transport in emerging photovoltaic and optoelectronic materials. Future work may extend the framework to spin‑orbit coupled systems, strong‑field nonlinear optics, and time‑dependent many‑body perturbation theory within the same efficient RI‑LRI infrastructure.