Excited States of the Uniform Electron Gas

TL;DR

This work addresses the need for state-specific density functionals for electronic excited states by generalizing the uniform electron gas (UEG) to include a Fermi-surface gap $k_{ ext{F}\sigma}\Delta_{\sigma}$ and a local degree-of-excitation variable. The authors derive closed-form expressions for the gap-dependent reduced kinetic energies $t_{s\sigma}(\rho_{\sigma},\Delta_{\sigma})$ and exchange energies $\epsilon_{x\sigma}(\rho_{\sigma},\Delta_{\sigma})$, expressed as $t_{s\sigma}(\rho_{\sigma},\Delta_{\sigma}) = \Xi_s(\Delta_{\sigma}) C_F\rho_{\sigma}^{2/3}$ and $\epsilon_{x\sigma}(\rho_{\sigma},\Delta_{\sigma}) = \Xi_x(\Delta_{\sigma}) C_x\rho_{\sigma}^{1/3}$, with gap-dependent factors $\Xi_s$ and $\Xi_x$. The method also identifies a gap-dependent leading term in the correlation energy in the high-density limit, $\epsilon^{(2d)}(\Delta) \sim \lambda_0(\Delta)\ln r_s$, with a six-term decomposition $\lambda_0(\Delta) = (1/\pi^2)\sum_{k=1}^6 \lambda_0^{(k)}$ and limits $\lambda_0(0) = \lambda_0 = (1-\ln 2)/\pi^2$ and $\lambda_0(1) \approx 0.00578826$. A density-matching condition yields $\kappa_{\sigma}$, ensuring identical spin densities between ground and excited states, and the framework recovers the ground state as $\Delta_{\sigma}\to 0$. Overall, the paper provides a rigorous basis for constructing state-specific (semi)local functionals by embedding excitation information into the UEG, paving the way for excited-state DFT beyond ensemble approaches.

Abstract

The uniform electron gas (UEG) is a cornerstone of density-functional theory (DFT) and the foundation of the local-density approximation (LDA), one of the most successful approximations in DFT. In this work, we extend the concept of UEG by introducing excited-state UEGs, systems characterized by a gap at the Fermi surface created by the excitation of electrons near the Fermi level. We report closed-form expressions of the reduced kinetic and exchange energies of these excited-state UEGs as functions of the density and the gap. Additionally, we derive the leading term of the correlation energy in the high-density limit. By incorporating an additional variable representing the degree of excitation into the UEG paradigm, the present work introduces a new framework for constructing local and semi-local state-specific functionals for excited states.

Figures (4)

• Figure 1: Schematic representation of ground- and excited-state UEGs. In a ground-state UEG (left), all electronic levels are filled from $k=0$ to the Fermi level at $k = k_{\text{F}\uparrow}$ for the spin-up electrons and $k = k_{\text{F}\downarrow}$ for the spin-down electrons. In an excited-state UEG, a gap of magnitude $k_{\text{F}\sigma}\Delta_{\sigma}$ opens at the Fermi level for each spin manifold with $0 \le \Delta_{\sigma} \le 1$. The electrons in the energy levels from $k_{\text{F}\sigma}(1-\Delta_{\sigma})$ to $k_{\text{F}\sigma}$ are excited to occupy the energy levels from $k_{\text{F}\sigma}$ to $k_{\text{F}\sigma}(1+\kappa_{\sigma}\Delta_{\sigma})$. The factor $0 \le \kappa_{\sigma} \le 1$ is determined such that the spin-$\sigma$ density of the ground- and excited-state UEGs are identical. The red lines indicate regions where the infrared catastrophe may occur due to the vanishing gap between occupied and unoccupied states.
• Figure 2: Left: $\kappa_{\sigma}$ as a function of $\Delta_{\sigma}$, as given by Eq. \ref{['eq:kap_vs_sig']}. Right: $\Xi_\text{s}$ and $\Xi_\text{x}$ as functions of $\Delta_{\sigma}$, as given by Eqs. \ref{['eq:Xis_vs_sig']} and \ref{['eq:Xix_vs_sig']}, respectively.
• Figure 3: $\rho_1$ as a function of $r_{12}$, as given by Eqs. \ref{['eq:rho1']}, for $k_{\text{F}\sigma} = 10$ and various values of $\Delta_{\sigma}$.
• Figure 4: $\Lambda_0$ as a function of $\Delta$, as given by Eq. \ref{['eq:lam0_vs_Del']}.