Real-time time-dependent density functional theory for high-energy density physics

TL;DR

This paper addresses predicting electronic response properties of high-energy density (HED) systems, including $S(\mathbf{q},\omega)$, conductivity, and stopping power, using real-time TDDFT. It provides a practical framework and tutorial for computing DSF, optical properties, and stopping power within the real-time TDDFT formalism, emphasizing Hermitian perturbations and current-density formulations. Key contributions include a detailed discussion of linear response via $\tilde{\chi}_{nn}$, the use of TD-CDFT for optical properties, and approaches to beyond-linear regimes via stopping power, together with practical considerations like initialization, pseudopotentials, time stepping, and finite-size effects. The work guides future applications and development of TDDFT in HED science, emphasizing validation, uncertainty quantification, and computational acceleration to extend regimes such as degenerate plasmas and core-electron dynamics.

Abstract

Electronic response properties of high-energy density (HED) systems influence planetary structure, drive evolution of fusion targets, and underpin diagnostics in laboratory astrophysics. Real-time time-dependent density functional theory (TDDFT) offers a versatile modeling framework capable of accurately predicting the dynamic response of HED materials -- including free-free, bound-free, and bound-bound contributions without requiring ad hoc state partitioning; capturing both collective and non-collective behavior; and applicable within the linear-response regime and beyond. We review the theoretical formalism of real-time TDDFT as applied to HED systems, provide a practical tutorial for computing relevant response properties (dynamic structure factors, conductivity, and stopping power), and comment on avenues for further development of this powerful computational method in service of HED science.

Figures (7)

• Figure 1: Exemplary pair of TDDFT density response calculations for a particular choice of $\mathbf{q}$. In the top and bottom panels, the perturbing potential takes the form of Eq. \ref{['eq:cosVpert']} and \ref{['eq:sinVpert']}, respectively, where the temporal envelope $I_0 f(t)$ is shown in red. The corresponding real and imaginary parts of the density response computed through real-time evolution of Eq. \ref{['eq:tdks']} are shown in teal and purple, respectively.
• Figure 2: Exemplary post-processing steps to compute the DSF from the time-dependent density-response data shown in Fig. \ref{['fig:dsf_example_t']}. First, the perturbing potential and total density response $\delta n^\mathrm{T}(\mathbf{q},t) = \delta n^\mathrm{C}(\mathbf{q},t) + i\delta n^\mathrm{S}(\mathbf{q},t)$ are transformed into the frequency domain (top two panels) after scaling by a window function $\Delta(t)$ in the latter case. Then, the density-density response function and corresponding DSF are evaluated according to Eqs. \ref{['eq:sincos_response2']} and \ref{['eq:dsf']}.
• Figure 3: Robustness of DSF predictions across three different types of perturbation envelope $f(t)$. Panels (a) and (b) show the Gaussian (solid green), square (dashed pink), and sinusoidal (dotted orange) pulses in the time and frequency domain, respectively, where points indicate the discrete time steps. Panel (c) shows excellent agreement among the resulting DSFs.
• Figure 4: Propagation of initial-state errors in real-time TDDFT calculations of linear-response functions. Panels (a) and (b) show the real part of the density response $\delta n(\mathbf{q},t)$ normalized by the perturbation intensity $I_0$, while panels (c) and (d) show the resulting dynamic structure factor. Fully converged results (solid black) are compared to cases where the initial states $\phi_j(\mathbf{r},t=0)$ were less converged (dashed blue and dashed-dotted purple). When $I_0$ is too weak compared to the initial-state errors (dashed-dotted purple), fictitious dynamics (dotted red) pollute the density response in (a) and distort the resulting DSF in (c). For sufficiently small initial-state errors, subtracting the fictitious dynamics from the computed response can recover accurate results as shown in (b) and (d).
• Figure 5: Low-$q$ response functions predicted using real-time TDDFT for solid-density aluminum at a temperature of 1eV. The solid curves were computed through the density-response formalism of Section \ref{['sec:dsf']}, while the dashed curve at $\mathbf{q}=0$ was computed through the current-response formalism of Section \ref{['sec:cond']}.