Path Integral Monte Carlo Research Articles

Dynamic exchange correlation effects in the strongly coupled electron liquid

The first quasiexact path integral Monte Carlo (PIMC) investigation of the dynamic local field correction G̃(q,zl;rs,Θ) in the imaginary Matsubara frequency domain is reported, focusing on the unpolarized uniform electron gas in the warm dense matter and strongly coupled regimes. The impact of dynamic exchange–correlation effects on the static structure factor is investigated. A straightforward explanation is provided for previously reported spurious effects in the so-called [Dornheim , ], where the frequency dependence of the local field correction is neglected. The comprehensive analysis of the asymptotic Matsubara order behavior, the density dependence, and the temperature dependence hints at the intriguing possibility of constructing an analytical four-parameter representation of G̃(q,zl;rs,Θ) valid for a substantial part of the uniform electron gas phase diagram, which would constitute key input for thermal density functional theory simulations. Published by the American Physical Society 2024

On accurate calculations of equilibrium H/D fractionations at super-cold conditions (≤200 K) I: Full Partition Function Ratio (FPFR) vs. Path Integral Monte Carlo (PIMC)

Accurate estimations of Hydrogen/Deuterium (H/D) isotope effects have been a long-standing problem since the beginning of stable isotope geochemistry, primarily due to their strong nuclear quantum effects (NQEs). Currently, the D/H ratio plays a key role in deciphering the origin and evolution of water ice and organics in the solar system. Additionally, the Big-bang origin of Deuterium necessitates that the observed D/H variations be linked to isotope exchange reactions, such as HD + H2O ↔ H2 + HDO. Therefore, an accurate benchmark dataset describing the equilibrium H/D isotope fractionations at low temperatures (≤ 200 K) is essential for interpreting rapidly accumulating astrochemical observation data. In this study, we implemented and compared two methods developed for evaluating the NQEs of H/D isotopes: 1) the Full Partition Function Ratio (FPFR) method and 2) the Path Integral Monte Carlo (PIMC) method. Both were assessed for their accuracy and efficiency in estimating equilibrium isotope effects (EIEs) using 10 common interstellar small molecules (H2, HF, HCl, HCN, HNC, H2O, H2S, NH3, HCHO, and CH4) as examples. Most of them are calculated at high precision for the first time. For the FPFR method, several higher-order corrections to the Urey-Bigeleisen-Mayer model were considered, including anharmonicity, quantum mechanical rotation with nuclear-spin statistics, and corrections to the Born-Oppenheimer approximation. For the PIMC method, four different actions related to the thermal density matrix are used. These include the primitive and three fourth-order actions (Takahashi and Imada, Suzuki-Chin and Chin approximations), based on the direct scaled-coordinates transformation. Three main conclusions are obtained: 1) Under the framework of Born-Oppenheimer approximation, the PIMC method holds the same and even slightly better accuracy (more comprehensive descriptions of quantum anharmonicities) in estimating EIEs for H/D isotopes as the FPFR method. However, both methods suffer from the neglection of the DBOC when using the Born-Oppenheimer potential energy surfaces. For the PIMC method, additional corrections for particle distinguishability must be considered when nuclear spin statistics exhibit its effects at low temperatures (e.g., < 160 K for H2). 2) Among the four actions, three fourth-order actions show better convergence and efficiency (2.2 to 26.5 times speedups at 50 K) than the primitive action, making them superior candidates for PIMC simulations at super-cold (≤ 200 K) conditions. 3) A combination of accurate potential energy surface with the PIMC method serves as an alternative way for theoretical estimations of H/D isotope effects.

Ab initio density response and local field factor of warm dense hydrogen

We present quasi-exact ab initio path integral Monte Carlo (PIMC) results for the partial static density responses and local field factors of hydrogen in the warm dense matter regime, from solid density conditions to the strongly compressed case. The full dynamic treatment of electrons and protons on the same footing allows us to rigorously quantify both electronic and ionic exchange–correlation effects in the system, and to compare the results with those of earlier incomplete models such as the archetypal uniform electron gas or electrons in a fixed ion snapshot potential that do not take into account the interplay between the two constituents. The full electronic density response is highly sensitive to electronic localization around the ions, and our results constitute unambiguous predictions for upcoming X-ray Thomson scattering experiments with hydrogen jets and fusion plasmas. All PIMC results are made freely available and can be used directly for a gamut of applications, including inertial confinement fusion calculations and the modeling of dense astrophysical objects. Moreover, they constitute invaluable benchmark data for approximate but computationally less demanding approaches such as density functional theory or PIMC within the fixed-node approximation.

Simple proof that there is no sign problem in path integral Monte Carlo simulations of fermions in one dimension.

It is widely known that there is no sign problem in path integral Monte Carlo (PIMC) simulations of fermions in one dimension. As far as the author is aware, there is no direct proof of this in the literature. This work shows that the sign of the N-fermion antisymmetric free propagator is given by the product of all possible pairs of particle separations, or relative displacements. For a nonvanishing closed-loop product of such propagators, as required by PIMC, all relative displacements from adjacent propagators are paired into perfect squares, and therefore the loop product must be positive, but only in one dimension. By comparison, permutation sampling, which does not evaluate the determinant of the antisymmetric propagator exactly, remains plagued by a low-level sign problem, even in one dimension.

Ab initio path integral Monte Carlo simulations of warm dense two-component systems without fixed nodes: Structural properties.

We present extensive new abinitio path integral Monte Carlo (PIMC) results for a variety of structural properties of warm dense hydrogen and beryllium. To deal with the fermion signproblem-an exponential computational bottleneck due to the antisymmetry of the electronic thermal density matrix-we employ the recently proposed [Y. Xiong and H. Xiong, J. Chem. Phys. 157, 094112 (2022); T. Dornheim et al., J. Chem. Phys. 159, 164113 (2023)] ξ-extrapolation method and find excellent agreement with the exact direct PIMC reference data where available. This opens up the intriguing possibility of studying a gamut of properties of light elements and potentially material mixtures over a substantial part of the warm dense matter regime, with direct relevance for astrophysics, material science, and inertial confinement fusion research.

Microscopic self-dynamics in liquid Ne-D_{2} mixtures: Quantum features and itinerant oscillators reexamined.

In this paper, we report the results of a centroid molecular dynamics (CMD) study of the canonical velocity autocorrelation functions (VACFs) in liquid Ne-D_{2} mixtures at a temperature of T=30K and in the full D_{2}-concentration range (0%≤x_{D_{2}}≤100%). This binary system was selected because of its moderate, although sizable, quantum effects which, as far as its equilibrium properties are concerned, are fully described by the path integral Monte Carlo (PIMC) simulations that have been also implemented. A comprehensive test of the VACF spectral moments carried out using three physical quantities (namely, mean kinetic energy, Einstein frequency, and mean-squared force) obtained from PIMC was performed revealing the potentialities, as well as the limitations, of the CMD approach to the single-particle dynamics in these low-T liquid mixtures. Additional physical information was extracted from the canonical VACFs by fitting their spectra via two distinct methods: the Levesque-Verlet model (LV, very flexible but highly heuristic) and the itinerant oscillator model (IO, based on the physical ground of a single particle rattling inside a short-lived diffusing pseudocage). Both provided good fits of the CMD outputs, with LV being always more adequate than IO in the case of the Ne VACFs, while, as for the D_{2} VACFs, the LV superiority is evident only at high x_{D_{2}} values. However, a peculiar and systematic effect was found after analyzing the IO-fitted parameters: the estimated pseudocage masses turned out to be at least one order of magnitude lower than the corresponding values inferred from the PIMC simulations. This outcome concerns both the Ne and the D_{2} rattling molecule and, as we also discovered, had already been observed (but promptly forgotten) in purely classical simulations of liquid Ar. The possible physical origins of this finding have been finally discussed in some detail, also in connection with the result of the more recent exponential expansion theory (EET), which manages to shed more light on the concept of single particles rattling inside short-lived pseudocages, ultimately demonstrating its untenability.

The Harmonic and Gaussian Approximations in the Potential Energy Landscape Formalism for Quantum Liquids.

The potential energy landscape (PEL) formalism has been used in the past to describe the behavior of classical low-temperature liquids and glasses. Here, we extend the PEL formalism to describe the behavior of liquids and glasses that obey quantum mechanics. In particular, we focus on the (i) harmonic and (ii) Gaussian approximations of the PEL, which have been commonly used to describe classical systems, and show how these approximations can be applied to quantum liquids/glasses. Contrary to the case of classical liquids/glasses, the PEL of quantum liquids is temperature-dependent, and hence, the main expressions resulting from approximations (i) and (ii) depend on the nature (classical vs quantum) of the system. The resulting theoretical expressions from the PEL formalism are compared with results from path-integral Monte Carlo (PIMC) simulations of a monatomic model liquid. In the PIMC simulations, every atom of the quantum liquid is represented by a ring-polymer. Our PIMC simulations show that at the local minima of the PEL (inherent structures, or IS), sampled over a wide range of temperatures and volumes, the ring-polymers are collapsed. This considerably facilitates the description of quantum liquids using the PEL formalism. Specifically, the normal modes of the ring-polymer system/quantum liquid at an IS can be calculated analytically if the normal modes of the classical liquid counterpart are known (as obtained, e.g., from classical MC or molecular dynamics simulations of the corresponding atomic liquid).

Virial coefficients of the uniform electron gas from path-integral Monte Carlo simulations.

The properties of plasmas in the low-density limit are described by virial expansions. Analytical expressions are known from Green's function approaches only for the first three virial coefficients. Accurate path-integral Monte Carlo (PIMC) simulations have recently been performed for the uniform electron gas, allowing the virial expansions to be analyzed and interpolation formulas to be derived. The exact expression for the second virial coefficient is used to test the accuracy of the PIMC simulations and the range of validity of the interpolation formula of Groth etal. [Phys. Rev. Lett. 119, 135001 (2017)0031-900710.1103/PhysRevLett.119.135001], and we discuss the fourth virial coefficient, which is not exactly known yet. Combining PIMC simulations with benchmarks from exact virial expansion results would allow us to obtain more accurate representations of the equationof state for parameter ranges of conditions which are of interest, e.g., for helioseismology.

Ab Initio Path Integral Monte Carlo Simulations of the Uniform Electron Gas on Large Length Scales.

The accurate description of non-ideal quantum many-body systems is of prime importance for a host of applications within physics, quantum chemistry, materials science, and related disciplines. At finite temperatures, the gold standard is given by ab initio path integral Monte Carlo (PIMC) simulations, which do not require any empirical input but exhibit an exponential increase in the required computation time for Fermionic systems with an increase in system size N. Very recently, computing Fermionic properties without this bottleneck based on PIMC simulations of fictitious identical particles has been suggested. In our work, we use this technique to perform very large (N ≤ 1000) PIMC simulations of the warm dense electron gas and demonstrate that it is capable of providing a highly accurate description of the investigated properties, i.e., the static structure factor, the static density response function, and the local field correction, over the entire range of length scales.

Bound-State Breaking and the Importance of Thermal Exchange-Correlation Effects in Warm Dense Hydrogen.

Hydrogen at extreme temperatures and pressures is of key relevance for cutting-edge technological applications, with inertial confinement fusion research being a prime example. In addition, it is ubiquitous throughout our universe and naturally occurs in a variety of astrophysical objects. In the present work, we present exact ab initio path integral Monte Carlo (PIMC) results for the electronic density of warm dense hydrogen along a line of constant degeneracy across a broad range of densities. Using the well-known concept of reduced density gradients, we develop a new framework to identify the breaking of bound states due to pressure ionization in bulk hydrogen. Moreover, we use our PIMC results as a reference to rigorously assess the accuracy of a variety of exchange-correlation (XC) functionals in density functional theory calculations for different density regions. Here, a key finding is the importance of thermal XC effects for the accurate description of density gradients in high-energy-density systems. Our exact PIMC test set is freely available online and can be used to guide the development of new methodologies for the simulation of warm dense matter and beyond.

Equation of state of partially ionized hydrogen and deuterium plasma revisited.

We present improved first-principle fermionic path integral Monte Carlo (PIMC) simulation results for a dense partially ionized hydrogen (deuterium) plasma, for temperatures in the range 15000K≤T≤400000K and densities 7×10^{-7}g/cm^{3}≤ρ_{H}≤0.085g/cm^{3} (1.4×10^{-6}g/cm^{3}≤ρ_{D}≤0.17g/cm^{3}), corresponding to 100≥r_{s}≥2, where r_{s}=r[over ¯]/a_{B} is the ratio of the mean interparticle distance to the Bohr radius. These simulations are based on the fermionic propagator PIMC (FP-PIMC) approach in the grand canonical ensemble [Filinov et al., Contrib. Plasma Phys. 61, e202100112 (2021)0863-104210.1002/ctpp.202100112] and fully account for correlation and quantum degeneracy and spin effects. For the application to hydrogen and deuterium, we develop a combination of the fourth-order factorization and the pair product ansatz for the density matrix. Moreover, we avoid the fixed node approximation that may lead to uncontrolled errors in restricted PIMC (RPIMC). Our results allow us to critically reevaluate the accuracy of the RPIMC simulations for hydrogen by Hu etal. [Phys. Rev. B 84, 224109 (2011)1098-012110.1103/PhysRevB.84.224109] and of various chemical models. The deviations are generally found to be small, but for the lowest temperature, T=15640K they reach several percent. We present detailed tables with our first principles results for the pressure and energy isotherms. We expect our updated results will serve as a valuable benchmark for comparison with other methods.

Fermionic physics from abinitio path integral Monte Carlo simulations of fictitious identical particles.

The abinitio path integral Monte Carlo (PIMC) method is one of the most successful methods in statistical physics, quantum chemistry and related fields, but its application to quantum degenerate Fermi systems is severely hampered by an exponential computational bottleneck: the notorious fermion signproblem. Very recently, Xiong and Xiong [J. Chem. Phys. 157, 094112 (2022)] have suggested to partially circumvent the signproblem by carrying out simulations of fictitious systems guided by an interpolating continuous variable ξ ∈ [-1, 1], with the physical Fermi- and Bose-statistics corresponding to ξ = -1 and ξ = 1. It has been proposed that information about the fermionic limit might be obtained by calculations within the bosonic sector ξ > 0 combined with an extrapolation throughout the fermionic sector ξ < 0, essentially bypassing the signproblem. Here, we show how the inclusion of the artificial parameter ξ can be interpreted as an effective penalty on the formation of permutation cycles in the PIMC simulation. We demonstrate that the proposed extrapolation method breaks down for moderate to high quantum degeneracy. Instead, the method constitutes a valuable tool for the description of large Fermi-systems of weak quantum degeneracy. This is demonstrated for electrons in a 2D harmonic trap and for the uniform electron gas (UEG), where we find excellent agreement (∼0.5%) with exact configuration PIMC results in the high-density regime while attaining a speed-up exceeding 11 orders of magnitude. Finally, we extend the idea beyond the energy and analyze the radial density distribution (2D trap), as well as the static structure factor and imaginary-time density-density correlation function (UEG).

Phase separation in binary Bose mixtures at finite temperature

We investigate the magnetic behavior of finite-temperature repulsive two-component Bose mixtures by means of exact path-integral Monte-Carlo simulations. Novel algorithms are implemented for the free energy and the chemical potential of the two components. Results on the magnetic susceptibility suggest that the conditions for phase separation are not modified from the zero temperature case. This contradicts previous predictions based on approximate theories. We also determine the temperature dependence of the chemical potential and the contact parameters for experimentally relevant balanced mixtures.

Dynamical response in strongly coupled uniform electron liquids: Observation of plasmon-roton coexistence using nine sum rules, Shannon information entropy, and path-integral Monte Carlo simulations

Dynamical properties of uniform electron fluids are studied within a nonperturbative approach consisting in the combination of the self-consistent version of the method of moments (SCMM) involving up to nine sum rules and other exact relations, the two-parameter Shannon information entropy maximization procedure, and the $\mathit{ab}\phantom{\rule{0.28em}{0ex}}\mathit{initio}$ path integral Monte Carlo (PIMC) simulations of the imaginary-time intermediate scattering function. The explicit dependence of the dynamic structure factor (DSF) on temperature and density is studied in a broad realm of variation of the dimensionless parameters ($2\ensuremath{\le}{r}_{s}\ensuremath{\le}36$ and $1\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}8$). When the coupling is strong (${r}_{s}\ensuremath{\ge}16$) we clearly observe a bimodal structure of the excitation spectrum with a lower-energy mode possessing a well pronounced rotonlike feature ($\ensuremath{\theta}\ensuremath{\le}2$) and an additional high-energy branch within the roton region which evolves into the strongly overdamped high-frequency shoulder when the coupling decreases (${r}_{s}\ensuremath{\le}10$). We are not aware of any reconstruction of the DSF at these conditions with the effects of dynamical correlations included here via the intermediate scattering and the dynamical Nevanlinna parameter functions. The standard static-local-field approach fails to reproduce this effect. The reliability of our method is confirmed by a detailed comparison with the recent $\mathit{ab}\phantom{\rule{0.28em}{0ex}}\mathit{initio}$ dynamic local field approach by T. Dornheim et al. [Phys. Rev. Lett. 121, 255001 (2018)] available for high/moderate densities (${r}_{s}\ensuremath{\le}10$). Moreover, within the SCMM we are able to construct the modes' dispersion equation in a closed analytical form and find the decrements (lifetimes) of the quasiparticle excitations explicitly. The physical nature of the revealed modes is discussed. Mathematical details of the method are complemented in Appendix. The proposed approach, due to its rigorous mathematical foundation, can find numerous diverse applications in the physics of Fermi and Bose liquids.

Isotopic effect on thermal physical properties of cubic SiC

Using path-integral Monte Carlo (PIMC) and classical Monte Carlo (CMC) simulations in the isothermal–isobaric (NPT) ensemble, we investigate the isotope effect on the thermal properties of the cubic silicon carbide (c-SiC) crystal. At T=50 K, the calculations show that the isotopic substitutions shift up the average vibrational energy and root-mean-square displacements (RMSD) values, with a more significant impact on carbon isotope substitution than the silicon ones. However, the total kinetic to total potential energy ratio suggests that the system behaves harmonically and insensitive to isotopic changes. We also calculated the effect of the quantum contribution on c-SiC crystal, considering the most abundant isotopes. At low temperatures, the c-SiC crystal behaves harmonically, although the C and Si atoms behave anharmonically. The quantum effects increase the lattice constant by almost 0.3%, the average RMSD by 0.0703(4) Å and are responsible for reducing the linear thermal expansion coefficient by about 5.8×10−6 K. The zero-point energy is found about 0.1225(3) eV/atom. A satisfactory agreement between the PIMC values and available theoretical and experimental results are found and discussed.

Energy response and spatial alignment of the perturbed electron gas.

We study the linear energy response of the uniform electron gas to an external harmonic perturbation with a focus on resolving different contributions to the total energy. This has been achieved by carrying out highly accurate abinitio path integral Monte Carlo (PIMC) calculations for a variety of densities and temperatures. We report a number of physical insights into effects such as screening and the relative importance of kinetic and potential energies for different wave numbers. A particularly interesting finding is obtained from the observed non-monotonic behavior of the induced change in the interaction energy, which becomes negative for intermediate wave numbers. This effect is strongly dependent on the coupling strength and constitutes further direct evidence for the spatial alignment of electrons introduced in earlier works [T. Dornheim et al., Commun. Phys. 5, 304 (2022)]. The observed quadratic dependence on the perturbation amplitude in the limit of weak perturbations and the quartic dependence of perturbation amplitude corrections are consistent with linear and nonlinear versions of the density stiffness theorem. All PIMC simulation results are freely available online and can be used to benchmark new methods or as input for other calculations.

Collision-induced three-body polarizability of helium.

We present the first-principles determination of the three-body polarizability and the third dielectric virial coefficient of helium. Coupled-cluster and full configuration interaction methods were used to perform electronic structure calculations. The mean absolute relative uncertainty of the trace of the polarizability tensor, resulting from the incompleteness of the orbital basis set, was found to be 4.7%. Additional uncertainty due to the approximate treatment of triple and the neglect of higher excitations was estimated at 5.7%. An analytic function was developed to describe the short-range behavior of the polarizability and its asymptotics in all fragmentation channels. We calculated the third dielectric virial coefficient and its uncertainty using the classical and semiclassical Feynman-Hibbs approaches. The results of our calculations were compared with experimental data and with recent Path-Integral Monte Carlo (PIMC) calculations [Garberoglio et al., J. Chem. Phys. 155, 234103 (2021)] employing the so-called superposition approximation of the three-body polarizability. For temperatures above 200K, we observed a significant discrepancy between the classical results obtained using superposition approximation and the ab initio computed polarizability. For temperatures from 10K up to 200K, the differences between PIMC and semiclassical calculations are several times smaller than the uncertainties of our results. Except at low temperatures, our results agree very well with the available experimental data but have much smaller uncertainties. The data reported in this work eliminate the main accuracy bottleneck in the optical pressure standard [Gaiser et al., Ann. Phys. 534, 2200336 (2022)] and facilitate further progress in the field of quantum metrology.

A Glimpse into Quantum Triplet Structures in Supercritical 3He.

A methodological study of triplet structures in quantum matter is presented. The focus is on helium-3 under supercritical conditions (4 < T/K < 9; 0.022 < ρN/Å-3 < 0.028), for which strong quantum diffraction effects dominate the behavior. Computational results for the triplet instantaneous structures are reported. Path integral Monte Carlo (PIMC) and several closures are utilized to obtain structure information in the real and the Fourier spaces. PIMC involves the fourth-order propagator and the SAPT2 pair interaction potential. The main triplet closures are: AV3, built as the average of the Kirkwood superposition and the Jackson-Feenberg convolution, and the Barrat-Hansen-Pastore variational approach. The results illustrate the main characteristics of the procedures employed by concentrating on the salient equilateral and isosceles features of the computed structures. Finally, the valuable interpretive role of closures in the triplet context is highlighted.

Ab initio path integral Monte Carlo simulations of hydrogen snapshots at warm dense matter conditions.

We combine ab initio path integral Monte Carlo (PIMC) simulations with fixed ion configurations from density functional theory molecular dynamics (DFT-MD) simulations to solve the electronic problem for hydrogen under warm dense matter conditions [Böhme et al., Phys. Rev. Lett. 129, 066402 (2022)0031-900710.1103/PhysRevLett.129.066402]. The problem of path collapse due to the Coulomb attraction is avoided by utilizing the pair approximation, which is compared against the simpler Kelbg pair potential. We find very favorable convergence behavior towards the former. Since we do not impose any nodal restrictions, our PIMC simulations are afflicted with the notorious fermion sign problem, which we analyze in detail. While computationally demanding, our results constitute an exact benchmark for other methods and approximations within DFT. Our setup gives us the unique capability to study important properties of warm dense hydrogen such as the electronic static density response and exchange-correlation kernel without any model assumptions, which will be very valuable for a variety of applications such as the interpretation of experiments and the development of new XC functionals.

A path integral Monte Carlo (PIMC) method based on Feynman-Kac formula for electrical impedance tomography

A path integral Monte Carlo method (PIMC) based on a Feynman-Kac formula for the Laplace equation with mixed boundary conditions is proposed to solve the forward problem of the electrical impedance tomography (EIT). The forward problem is an important part of iterative algorithms of the inverse EIT problem, and the proposed PIMC provides a local solution to find the potentials and currents on individual electrodes. Improved techniques are proposed to compute with better accuracy both the local time of reflecting Brownian motions (RBMs) and the Feynman-Kac formula for mixed boundary problems of the Laplace equation. Accurate voltage-to-current maps on the electrodes of a model 3-D EIT problem with eight electrodes are obtained by solving a mixed boundary problem with the proposed PIMC method.