Generalized density functional theory framework for the non-linear density response of quantum many-body systems

TL;DR

This work develops a generalized DFT framework that directly relates free-energy functional derivatives to non-linear density response functions, enabling explicit expressions for linear, quadratic, and cubic responses, including the novel cubic response at the first harmonic $\chi_0^{(1,3)}(\mathbf{q})$ via mode coupling. It provides exact long-wavelength limits for the uniform electron gas (UEG) and connects higher-order responses to the third- and fourth-order derivatives of the non-interacting free-energy functional $F_s[n]$, offering stringent constraints for developing improved functionals. The authors benchmark these theoretical results against KS-DFT simulations and assess several non-interacting functionals (WTF, LKTF, XWMF) across temperatures from ground state to warm dense matter, revealing where common approximations succeed or fail in describing non-linear screening and harmonic coupling. The framework thus furnishes a systematic route to constrain and refine orbital-free and KS-DFT functionals for accurate non-linear electronic response, with implications for metals, semiconductors, and warm dense matter, and points toward extensions to time-dependent non-linear response and higher-order XC kernels.

Abstract

A density functional theory (DFT) framework is presented that links functional derivatives of free-energy functionals to non-linear static density response functions in quantum many-body systems. Within this framework, explicit expressions are derived for various higher-order response functions of systems that are homogeneous on average, including the first theoretical result for the cubic response at the first harmonic $χ_0^{(1,3)}(\vec{q})$. Specifically, our framework includes hitherto neglected mode-coupling effects that are important for the non-linear density response even in the presence of a single harmonic perturbation. We compare these predictions for $χ_0^{(1,3)}(\vec{q})$ to new Kohn-Sham DFT simulations, leading to excellent agreement between theory and numerical results. Exact analytical expressions are also obtained for the long-wavelength limits of the ideal quadratic and cubic response functions. Particular emphasis is placed on the connections between the third- and fourth-order functional derivatives of the non-interacting free-energy functional $F_s[n]$ and the ideal quadratic and cubic response functions of the uniform electron gas, respectively. These relations provide exact constraints that may prove useful for the future construction of improved approximations to $F_s[n]$, in particular for warm dense matter applications at finite temperatures. Here, we use this framework to assess several commonly employed approximations to $F_s[n]$ through orbital-free DFT simulations of the harmonically perturbed ideal electron gas. The results are compared with Kohn-Sham DFT calculations across temperatures ranging from the ground state to the warm dense regime. Additionally, we analyze in detail the temperature- and wavenumber-dependent non-monotonic behavior of the ideal quadratic and cubic response functions.

Figures (8)

• Figure 1: Linear static density response function of the ideal UEG at (a) $\theta=0.01$ and (b) $\theta=0.02$ with the density parameter set to $r_s=2$, and the wavenumber given in units of the Fermi wavenumber $q_F$. The solid line is the Lindhard function, while the dashed line corresponds to the analytic solution for the TF model. Symbols indicate data computed using OFDFT with an external harmonic perturbation and various non-interacting free energy functionals. Note that the results from the WTF and XWMF for $\chi_0^{(1)}(\mathbf{q})$ are almost identical.
• Figure 2: Quadratic static density response function of the ideal UEG at (a) $\theta=0.01$ and (b) $\theta=0.01$ with the density parameter $r_s=2$. The solid line represents the exact solution given by Eq. \ref{['eq:chi2_0']} and the dashed line shows the analytic solution using the TF model. Symbols correspond to the OFDFT calculations using the external harmonic perturbation and different approximations for the non-interacting free energy functional.
• Figure 3: Quadratic static density response function at second harmonic of the ideal UEG at different values of the degeneracy parameter with the density parameter set to $r_s=2$. The lines correspond to the exact solution given by Eq. \ref{['eq:chi2_0']}. Circles depict the results of the OFDFT calculations using the XWMF non-interacting free-energy functional.
• Figure 4: Cubic static density response function of the ideal UEG at (a) $\theta=0.01$ and (b) $\theta=0.01$ with the density parameter $r_s=2$. The solid line represents the exact solution given by Eq. \ref{['eq:Mikhailov3']} and the dashed line is the analytic solution using the TF model. Symbols correspond to the OFDFT calculations using the external harmonic perturbation and different approximations for the non-interacting free energy functional.
• Figure 5: Cubic static density response function at third harmonic of the ideal UEG at $\theta=2$ and $\theta=4$ with $r_s=2$. The lines correspond to the exact solution given by Eq. \ref{['eq:Mikhailov3']}. Symbols show the results of the OFDFT calculations using the XWMF and LKTF non-interacting free-energy functionals.