Enhancing the Efficiency of Time-Dependent Density Functional Theory Calculations of Dynamic Response Properties

TL;DR

The paper addresses the high computational cost of TDDFT for modeling dynamic response properties in XRTS under extreme conditions. It introduces an η-convergence test in the imaginary-time domain combined with a constrained noise-attenuation workflow that leverages the one-to-one mapping between the dynamic structure factor $S(\mathbf{q},\omega)$ and the imaginary-time correlation function $F(\mathbf{q},\tau)$ via a two-sided Laplace transform. The approach is demonstrated on solid-density hydrogen and isochorically heated aluminum, achieving up to an order-of-magnitude speedup while producing unbiased, noise-filtered $S_p(\mathbf{q},\omega)$ that retain key spectral features, with results validated against benchmark data such as PIMC. This method provides a broadly applicable, efficient toolkit for ab initio XRTS modeling and for deriving related dynamic dielectric properties through fluctuation-dissipation and Kramers-Kronig relations, significantly aiding experimental interpretation and model benchmarking under ambient and extreme conditions.

Abstract

X-ray Thomson scattering (XRTS) constitutes an essential technique for diagnosing material properties under extreme conditions, such as high pressures and intense laser heating. Time-dependent density functional theory (TDDFT) is one of the most accurate available ab initio methods for modeling XRTS spectra, as well as a host of other dynamic material properties. However, strong thermal excitations, along with the need to account for variations in temperature and density as well as the finite size of the detector significantly increase the computational cost of TDDFT simulations compared to ambient conditions. In this work, we present a broadly applicable method for optimizing and enhancing the efficiency of TDDFT calculations. Our approach is based on a one-to-one mapping between the dynamic structure factor and the imaginary time density--density correlation function, which naturally emerges in Feynman's path integral formulation of quantum many-body theory. Specifically, we combine rigorous convergence tests in the imaginary time domain with a constraints-based noise attenuation technique to improve the efficiency of TDDFT modeling without the introduction of any significant bias. As a result, we can report a speed-up by up to an order of magnitude, thus potentially saving millions of CPU hours for modeling a single XRTS measurement of matter under extreme conditions.

Figures (10)

• Figure 1: TDDFT results for (a) the DSF $S(\mathbf{q},\omega)$ and (b) the ITCF $F(\mathbf{q},\tau)$ of solid density hydrogen with $T=4.8 ~{\rm eV}$ and $\rho=0.08 ~{\rm g/cc}$ at $q=0.946~\textup{~\AA}^{-1}$. Results are shown for different Lorentzian smearing parameters $\eta$. The TDDFT results were averaged using 20 different ion configurations generated using molecular dynamics simulations (see Sec. \ref{['s:dft_info']} for details).
• Figure 2: Panel (a): dependence of the ITCF minimum, $F(\mathbf{q}, \tau=\beta/2)$, and the area under the ITCF, $\int_0^{\beta} F(\mathbf{q}, \tau) \, {\mathrm{d}}\tau$, on the Lorentzian smearing parameter $\eta$. Panel (b): dependence of the minimum of the shifted ITCF (defined in Eq.(\ref{['eq:tilde_ITCF']})), $\widetilde{F}(\mathbf{q}, \tau=\beta/2)$, and the area under the shifted ITCF, $\int_0^{\beta} \widetilde{F}(\mathbf{q}, \tau) \, {\mathrm{d}}\tau$, on the Lorentzian smearing parameter $\eta$. The results are presented for solid density hydrogen with $T=4.8 ~{\rm eV}$ and $\rho=0.08 ~{\rm g/cc}$ at $q=0.946~\textup{~\AA}^{-1}$.
• Figure 3: (a) Derivative of the reference DSF, $\partial S(q,\omega)/\partial \omega$, with $\eta=0.1~{\rm eV}$ and at the same parameters as shown in Fig. \ref{['fig:ilust_h']}. Panel (b) shows the power spectral density of $\partial S(q,\omega)/\partial \omega$ with the Fourier frequencies of the dominant narrowband noise oscillations marked on the x-axis. Panel (c) displays the grid search over multiple window-size-polynomial-degree-combinations (color map) and the optimum window size for each polynomial degree (black symbols). In panel (d) we compare the reference DSF with $\eta=0.1~{\rm eV}$ with the data obtained using the SG filer with different polynomial degrees and corresponding to them optimal window sizes as depicted in panel (a).
• Figure 4: Absolute value of the residual ITCF $F_n(q,\tau)$ [(a)], the magnitude of the mean $\bar{F}_n(q)$ [(b)], and the RMS $\sigma_n(q)$ [(c)] at different values of the polynomial degree and corresponding optimal window length. The results are for solid density hydrogen with $T=4.8 ~{\rm eV}$ and $\rho=0.08 ~{\rm g/cc}$ at $q=0.946~\textup{~\AA}^{-1}$.
• Figure 5: Workflow for applying the $\eta$-convergence test in the imaginary time domain combined with the constraints-based noise attenuation technique. For the first step, the generation of the DSFs with various $\eta$ is realized at the post-processing phase of the TDDFT simulation and requires negligible computational time.