Ab InitioThermodynamic Results for the Degenerate Electron Gas at Finite Temperature

Abstract

The uniform electron gas (UEG) at finite temperature is of key relevance for many applications in dense plasmas, warm dense matter, laser excited solids and much more. Accurate thermodynamic data for the UEG are an essential ingredient for many-body theories, in particular, density functional theory. Recently, first-principle restricted path integral Monte Carlo results became available which, however, due to the fermion sign problem, had to be restricted to moderate degeneracy, i.e. low to moderate densities with rs =r/aB 1. Here we present novel first-principle configuration PIMC results for electrons for rs ≤ 1. We also present quantum statistical data within the e 4 -approximation that are in good agreement with the simulations at small to moderate rs. [11,12]. Besides, the electron component is of crucial importance for understanding the properties of atoms, molecules and existing and novel materials. The most successful approach has been density functional theory (DFT)-combined with an approximation for the exchange-correlation potential. Its success is based on the availability of accurate zero temperature data for the UEG which is obtained from analytically known limiting cases combined with first-principle quantum Monte Carlo data [13].In recent years more and more applications have emerged where the electrons are highly excited, e.g. by compression of the material or by electromagnetic radiation (see above), which require to go beyond zero temperature DFT. This has led to an urgent need for accurate thermodynamic data of the UEG at finite temperature. One known limiting case is the highly degenerate ideal Fermi gas (IFG), and perturbation theory results around the IFG, starting with the Hartree-Fock and first order correlation corrections (Montroll-Ward) [14,15] [27]. It is well known that fermionic PIMC simulation in continuous space suffer from the fermion sign problem (FSP) which is known to be NP hard [28]. This means, with increasing quantum degeneracy, i.e. increasing parameter χ = nλ 3 DB , which is the product of density and thermal DeBroglie wave length, λ, the simulations suffer an exponential loss of accuracy. RPIMC formally avoids the FSP by an additional assumption on the nodes of the density matrix, however, it also cannot access high densities [29], r s < 1 [r s =r/a B , wherer is the mean interparticle distance, n −1 = 4πr 3 /3 and a B the Bohr radius]. Also, the quality of the simulations around r s = 1, at low temperatures Θ = k B T /E F ≤ 0.125 [E F is the Fermi energy] is unknown. However, this leaves out the high-density range that is of high importance, e.g. for deuterium-tritium implosions at NIF where mass densities of 400 gcm −3 (up to 1596 gcm −3 ) have recently been reported [9] (are expected along the implosion path [8]), corresponding to r s ≈ 0.24 (r s = 0.15), see Fig. 1.The authors of Ref.[27] also performed DPIMC simulations which confirmed that, for Θ < 0.5 and r s 4, these simulations are practically not possible, see Fig. 1. We also mention independent recent DPIMC ...

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

“…However, to nevertheless obtain accurate QMC results at WDM conditions, we have introduced two novel QMC methods that are efficient at complementary parameter regimes. The CPIMC method [12,13] is formulated in anti-symmetric Fock-space and can be interpreted as a Monte Carlo simulation of the exact, infinite perturbation expansion around the ideal (non-interacting) system. Therefore, it excels at strong degeneracy and high density, but becomes inefficient towards strong coupling.…”

“…[10,11] It was only recently that the combination of two novel methods (configuration path integral Monte Carlo [CPIMC] [12,13] ) and permutation blocking path integral Monte Carlo [PB-PIMC] [14,15] ) that are available at complementary parameter ranges allowed to conduct the first unbiased simulation of the UEG. At first, these efforts were limited to a finite number of electrons N in a finite simulation cell of volume V. [16,17] In practice, however, one is interested in the thermodynamic limit, which is given by the limit of an infinite number of particles at fixed density (or, equivalently, fixed r s ).…”

Ab initioquantum Monte Carlo simulations of the uniform electron gas without fixed nodes: The unpolarized case

“…[22]) we have combined two novel complementary approaches: our configuration path integral Monte Carlo (CPIMC) method [23][24][25] excels at high to medium density and arbitrary temperature, while our permutation blocking path integral Monte Carlo (PB-PIMC) approach [26,27] significantly extends standard fermionic PIMC [28,29] towards lower temperature and higher density. Surprisingly, it has been found that existing RPIMC results are inaccurate even at high temperatures.…”

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

“…An independently developed third approach, density matrix QMC [40,41], confirmed the excellent quality of these results. The only significant errors remaining are finite-size effects [37,[42][43][44][45][46], which arise from the difference between the small systems simulated and the infinite [thermodynamic limit (TDL)] system of interest.…”

“…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. Our approach combines two independent methods, configuration path-integral Monte Carlo (CPIMC) [35][36][37] and permutation blocking PIMC [38,39], which allow for accurate simulations at high (r s 1) and moderate densities (r s 1 and θ 0.5), respectively. An independently developed third approach, density matrix QMC [40,41], confirmed the excellent quality of these results.…”