Path-Integral Monte Carlo Simulation of the Warm Dense Homogeneous Electron Gas

Abstract

We perform calculations of the 3D finite-temperature homogeneous electron gas (HEG) in the warm-dense regime (rs ≡ (3/4πn) 1/3 a −1 B = 1.0−40.0 and Θ ≡ T /TF = 0.0625−8.0) using restricted path integral Monte Carlo (RPIMC). Precise energies, pair correlation functions, and structure factors are obtained. For all densities, we find a significant discrepancy between the ground state parameterized local density approximation (LDA) and our results around TF . These results can be used as a benchmark for improved functionals, as well as input for orbital-free DFT formulations. PACS numbers:The one-component plasma (OCP), a fundamental many body model, consists of a single species of charged particles immersed in a rigid neutralizing background. For electrons, the OCP is a model of simple metals and is often called the homogeneous electron gas (HEG), electron gas, or jellium. At zero-temperature, it is customary to define the natural length scale r s a B ≡ (3/4πn)and energy scale Ry = e 2 /2a B . When r s , the WignerSeitz radius, is small (high density) (r s → 0), the kinetic energy term dominates and the system becomes qualitatively similar to a non-interacting gas. At low density (r s → ∞), the potential energy dominates and the system is predicted to form a Wigner crystal [1]. In 3D at intermediate densities, a partially polarized state is predicted to emerge [2,3].Over the past few decades very accurate zerotemperature quantum Monte Carlo (QMC) calculations of the ground state HEG examined each of these phases [4,5]. In addition to determining phase boundaries, the results of these studies have proven invaluable in the rigorous parameterization of functionals in ground state density functional theory (DFT) [6].Recently there has been intense interest in extending the success of ground-state DFT to finite-temperature systems such as stellar, planetary interiors and other hot dense plasmas [7][8][9]. However, such attempts have met both fundamental and technical barriers when electrons have significant correlations.Some of the first Monte Carlo simulations explored the phases of the classical OCP [10]; note that its equation of state depends only on a single variable, the Coulomb coupling parameter Γ ≡ q 2 /(r s k B T ). First-order quantum mechanical effects have since been included [11,12]. However, the accuracy of these results quickly deteriorate as the temperature is lowered and quantum correlations play a greater role [13]. This breakdown is most apparent in the warm-dense regime where both Γ and the electron degeneracy parameter Θ ≡ T /T F are close to unity.Finite-temperature formulations of DFT have also met with challenges. There are two braod approaches to building finite-temperature functionals. In one approach, temperature effects are introduced by smearing the electronic density of states over a Fermi-Dirac distribution. As temperature increases, an ever-increasing number of molecular (Kohn-Sham) orbitals is required in order to evaluate the functional, making DFT calculations computationally intra...

Ab InitioQuantum Monte Carlo Simulation of the Warm Dense Electron Gas in the Thermodynamic Limit

“…Therefore, an accurate description of warm dense matter in general and of the warm dense UEG in particular can only be achieved using computational approaches, primarily quantum Monte-Carlo (QMC) methods which, however, are hampered by the fermion sign problem [28,29]. The pioneering QMC simulations of the warm dense UEG by Brown et al [30] eliminated the sign problem by invoking the (uncontrolled) fixed-node approximation [31], but were nevertheless restricted to small systems of N = 33 (spin-polarized) and N = 66 (unpolarized) elecarXiv:1607.08076v2 [physics.plasm-ph] 9 Sep 2016 trons and to moderate densities, r s ≥ 1. Recently, we were able to show [32][33][34] that accurate simulations of these systems are possible over a broad parameter range without any nodal restriction.…”

Accurate Homogeneous Electron Gas Exchange-Correlation Free Energy for Local Spin-Density Calculations

“…Observe that τ s is given both analytically and tabularly in the Supplementary Material for Ref. 1. From Eq.…”

“…The RPIMC data for the HEG [1] are the total kinetic T and potential (or interaction) U ee energies for given r s and t. The issues are which RPIMC data to use and how best to extract a broadly reliable f xc from that data.…”

“…In the absence of exact analytical solutions for its energetics, high-precision numerical results have been critical to insight. Recently published [1] restricted path integral Monte Carlo (RPIMC) data for the HEG over a wide range of temperatures and densities open the opportunity to obtain closed form expressions for HEG thermodynamics, in particular the exchange and correlation (XC) contributions. Such expressions extracted from Monte Carlo data are well-known for the zero-T HEG, where they have played a major role in understanding inhomogeneous electron-system behavior.…”

“…Thus, several F xc approximations have been proposed on the basis of various models; see Ref. 18 [1] and extracted a parametrization of the HEG XC free energy. That constitutes an LDA F xc .…”