Ab Initio Static Exchange–Correlation Kernel across Jacob’s Ladder without Functional Derivatives

Abstract

The electronic exchangecorrelation (XC) kernel constitutes a fundamental input for the estimation of a gamut of properties such as the dielectric characteristics, the thermal and electrical conductivity, or the response to an external perturbation. In this work, we present a formally exact methodology for the computation of the system specific static XC kernel exclusively within the framework of density functional theory (DFT) and without employing functional derivativesno external input apart from the usual XC-functional is required. We compare our new results with exact quantum Monte Carlo (QMC) data for the archetypical uniform electron gas model under both ambient and warm dense matter conditions. This gives us unprecedented insights into the performance of different XC functionals, and it has important implications for the development of new functionals that are designed for the application at extreme temperatures. In addition, we obtain new DFT results for the XC kernel of warm dense hydrogen as it occurs in fusion applications and astrophysical objects. The observed excellent agreement to the QMC reference data demonstrates that presented framework is capable to capture nontrivial effects such as XC-induced isotropy breaking in the density response of hydrogen at large wave numbers.

Non-empirical Mixing Coefficient for Hybrid XC Functionals from Analysis of the XC Kernel

“…The dashed red line illustrates the quadratic q dependence of K xc ( q ) based on PBE according to the compressibility sum rule. 45 It becomes exact in the limit of q → 0. The results for K xc ( q ) based on various LDA, GGA, meta-GGA, and hybrid XC functionals, including the data for the PBE, PBE0, PBE0–1/3, and AM05 functionals, were reported in refs ( 11 ) and ( 45 ).…”

“…45 By doing so, we circumvent the need to compute a second-order functional derivative of the XC functional and, thus, can calculate the static XC kernel across Jacob's ladder. 45 The key ingredient of our analysis is the exact quantum Monte Carlo data for the so-called local field correction (LFC) of the UEG, which is connected to the XC kernel as G(q) = −K xc (q)/v(q) and is commonly used in the quantum theory of the electron liquid. 34,49,50 The UEG is conveniently described by mean interparticle distance r s and the reduced temperature θ = T/T F , where r s is in units of the Bohr radius and T F is the Fermi temperature (energy).…”

“…To compute the static XC kernel K xc ( q ) = K xc ( q , ω = 0), we perturb the UEG with an external static harmonic perturbation V ext = 2 A cos­( q · r ). Then, static density response function χ­( q ) (defined by setting ω = 0 in eq ) is found from the difference δn ( r ) between the perturbed and unperturbed densities: − …”

“…This approach for computing χ­( q ) was used for the UEG by Dornheim et al , within the path integral Monte Carlo (PIMC) method. Recently, this method has been extended to the KS-DFT and has been illustrated for the UEG and warm dense hydrogen . By doing so, we circumvent the need to compute a second-order functional derivative of the XC functional and, thus, can calculate the static XC kernel across Jacob’s ladder .…”

“…The perturbation amplitude is set to A = 0.01. For a detailed discussion of the convergence with respect to the simulation parameters, see our recent papers. , …”