Mixed Stochastic-Deterministic Density Functional Theoretic Decomposition of Kubo-Greenwood Conductivities in the Projector Augmented Wave Formalism

TL;DR

The paper tackles the challenge of predicting charge transport in warm dense matter by fusing mixed stochastic-deterministic DFT (mDFT) with the Kubo–Greenwood framework and real-time TDDFT. It develops a PAW-compatible, mixed-resolution approach that decomposes Onsager coefficients $\mathscr{L}_{mn}$ into deterministic, stochastic, and mixed orbital contributions, enabling accurate conductivity spectra with substantially smaller deterministic subspaces. The authors demonstrate that mDFT reproduces KS conductivities while offering orbital-resolved insights across CH, Be, and CH/Be, and they validate real-time TDDFT results against KG calculations within uncertainties. This work provides a scalable, physically transparent framework for transport in extreme states of matter, with implications for astrophysical plasmas and inertial confinement fusion modeling.

Abstract

Pairing the accuracy of Kohn-Sham density-functional framework with the efficiency of a stochastic algorithmic approach, mixed stochastic-deterministic Density Functional Theory (mDFT) achieves a favorable computational scaling with system sizes and electronic temperatures. We employ the recently developed mDFT formalism to investigate the dynamic charge-transport properties of systems in the warm dense matter regime. The optical conductivity spectra are computed for single- and multi- component mixtures of carbon, hydrogen, and beryllium using two complementary approaches: Kubo-Greenwood in the mDFT picture and real-time Time-Dependent mDFT. We further devise a decomposition of the Onsager coefficients leading up to the Kubo-Greenwood spectra to exhibit contributions from the deterministic, stochastic, and mixed electronic state transitions at different incident photon energies.

Figures (6)

• Figure 1: Optical conductivity spectra as a function of the photon energy obtained using the Kubo-Greenwood formalism within KS-DFT (yellow dashdotted line) and mDFT (blue dashed line). Electrical and Thermal Conductivity (a, b): CH mixture at $(\rho, T)= (0.9 \text{ g/cm}^3, 7.8 \text{ eV})$, (c, d): Be at $(\rho, T)= (1.84 \text{ g/cm}^3, 4.4 \text{ eV})$ and, (e, f): CH/Be ternary mixture at $(\rho, T) = (1.37 \text{g/cm}^3, 5.0 \text{ eV})$, respectively. A Drude-model fit (green solid line) to the mDFT conductivity is included as a reference.
• Figure 2: CH mixture at $(\rho, T)= (0.9 \text{ g/cm}^3, 7.8 \text{ eV})$: (a) Occupied density of states (DOS) obtained with KS-DFT and mDFT. The pink and orange-shaded regions indicate the contributions of deterministic KS and stochastic orbitals, respectively, to the mixed DOS. (b)-(d) Frequency-dependent Onsager coefficients $L_{mn}(\omega)$ decomposed into transitions among Kohn-Sham ($N_\psi$), stochastic ($N_\chi$), and mixed deterministic-stochastic ($N_\psi, N_\chi$) orbitals. The full $L_{mn}(\omega)$ shown in black dashed lines comprises transitions amongst all orbitals.
• Figure 3: Be at $(\rho, T)= (1.84 \text{ g/cm}^3, 4.4 \text{ eV})$: (a) Occupied density of states (DOS) obtained with KS-DFT and mDFT. The pink and orange-shaded regions indicate the contributions of deterministic KS and stochastic orbitals respectively to the mixed DOS. (b)-(d) Frequency-dependent Onsager coefficients $L_{mn}(\omega)$ decomposed into transitions among Kohn-Sham ($N_\psi$), stochastic ($N_\chi$), and mixed deterministic-stochastic ($N_\psi, N_\chi$) orbitals. The full $L_{mn}(\omega)$ shown in black dashed lines comprises transitions amongst all orbitals.
• Figure 4: CH/Be at $(\rho, T)= (1.37 \text{ g/cm}^3, 5.0 \text{ eV})$: (a) Occupied density of states (DOS) obtained with KS-DFT and mDFT. The pink and orange-shaded regions indicate the contributions of deterministic KS and stochastic orbitals respectively to the mixed DOS. (b)-(d) Frequency-dependent Onsager coefficients $L_{mn}(\omega)$ decomposed into transitions among Kohn-Sham ($N_\psi$), stochastic ($N_\chi$), and mixed deterministic-stochastic ($N_\psi, N_\chi$) orbitals. The full $L_{mn}(\omega)$ shown in black dashed lines comprises transitions amongst all orbitals.
• Figure 5: Comparison of electrical AC conductivity spectra obtained from Kubo-Greenwood + Kohn-Sham DFT and real-time Time-Dependent mixed DFT--based Ehrenfest dynamics for: (a) CH $(1:1)$ mixture, (b) Be and, (c) CH/Be ternary mixture. The cyan-shaded region shows the uncertainty from mixed DFT.