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

Abstract

We perform ab initio quantum Monte Carlo (QMC) simulations of the warm dense uniform electron gas in the thermodynamic limit. By combining QMC data with linear response theory we are able to remove finite-size errors from the potential energy over the entire warm dense regime, overcoming the deficiencies of the existing finite-size corrections by Brown et al. [PRL 110, 146405 (2013)]. Extensive new QMC results for up to N = 1000 electrons enable us to compute the potential energy V and the exchange-correlation free energy Fxc of the macroscopic electron gas with an unprecedented accuracy of |∆V |/|V |, |∆Fxc|/|F |xc ∼ 10 −3 . A comparison of our new data to the recent parametrization of Fxc by Karasiev et al. [PRL 112, 076403 (2014)] reveals significant deviations to the latter.The uniform electron gas (UEG), consisting of electrons on a uniform neutralizing background, is one of the most important model systems in physics [1]. Besides being a simple model for metals, the UEG has been central to the development of linear response theory and more sophisticated perturbative treatments of solids, the formulation of the concepts of quasiparticles and elementary excitations, and the remarkable successes of density functional theory.The practical application of ground-state density functional theory in condensed matter physics, chemistry and materials science rests on a reliable parametrization of the exchange-correlation energy of the UEG [2], which in turn is based on accurate quantum Monte Carlo (QMC) simulation data [3]. However, the charged quantum matter in astrophysical systems such as planet cores and white dwarf atmospheres [4,5] is at temperatures way above the ground state, as are inertial confinement fusion targets [6][7][8], laser-excited solids [9], and pressure induced modifications of solids, such as insulator-metal transitions [10,11]. This unusual regime, in which strong ionic correlations coexist with electronic quantum effects and partial ionization, has been termed "warm dense matter" and is one of the most active frontiers in plasma physics and materials science.The warm dense regime is characterized by the existence of two comparable length scales and two comparable energy scales. The length scales are the mean interparticle distance,r, and the Bohr radius, a 0 ; the energy scales are the thermal energy, k B T , and the electronic Fermi energy, E F . The dimensionless parameters [12] r s =r/a 0 and Θ = k B T /E F are of order unity. Because Θ ∼ 1, the use of ground-state density functional theory is inappropriate and extensions to finite T are indispensible; these require accurate exchange-correlation functionals for finite temperatures [13][14][15][16][17]. Because neither r s nor Θ is small, there are no small parameters, and weakcoupling expansions beyond Hartree-Fock such as the Montroll-Ward (MW) and e 4 (e4) approximations [18,19], as well as linear response theory within the random- phase approximation (RPA) break down [20,21]. Finite-T Singwi-Tosi-Land-Sjölander (STLS) [22,23] local-fiel...

Ab initio results for the static structure factor of the warm dense electron gas

“…The solid black line corresponds to a spline combining STLS for small k with QMC elsewhere. Reproduced from Dornheim et al [20] with the permission of the authors…”

“…The main difference between these data sets is the different k-grid, while the functional form of the SSF is remarkably well converged with system size, see the inset. The light blue curve depicts the parabolic RPA expansion from Equation (23), which is of interest for finite-size corrections of the interaction energy, [20,21,36] but does not provide a sufficient description of the long-range correlations beyond the QMC data. The grey and red curves correspond to the full RPA and STLS results (see Section 2.5), respectively, and are in perfect agreement with each other and Equation (23) for small k, as expected.…”

“…where the RPA limit is recovered by setting G(q) = 0 in Equations (19) and (20). Unfortunately, the local field correction is not known in practice and one has to introduce an approximation.…”

“…This was realized by combining QMC data, which exactly incorporates all short-range exchange-correlation effects, but cannot capture the long-range effects due to the finite simulation cell, with the linear response theory, which is exact precisely in this limit. [18][19][20][21] The resulting accurate data for the UEG in the thermodynamic limit have subsequently been used to construct a complete parameterization of the exchange-correlation free energy with respect to temperature, density, and spin-polarization over the entire WDM regime. …”

Peculiarities of the momentum distribution functions of strongly correlated charged fermions

“…Let us mention new original approaches developed in [10][11][12][13]. The configuration path integral Monte Carlo (CPIMC) approach [10,11] for degenerate correlated fermions with arbitrary pair interactions at finite temperatures is based on representation of the N-particle density operator in a basis of (anti-)symmetrized N-particle states.…”