Finite-temperature Density Functional Theory Research Articles

Stochastic density functional theory combined with Langevin dynamics for warm dense matter.

This study overviews and extends a recently developed stochastic finite-temperature Kohn-Sham density functional theory to study warm dense matter using Langevin dynamics, specifically under periodic boundary conditions. The method's algorithmic complexity exhibits nearly linear scaling with system size and is inversely proportional to the temperature. Additionally, a linear-scaling stochastic approach is introduced to assess the Kubo-Greenwood conductivity, demonstrating exceptional stability for dc conductivity. Utilizing the developed tools, we investigate the equationof state, radial distribution, and electronic conductivity of hydrogen at a temperature of 30000K. As for the radial distribution functions, we reveal a transition of hydrogen from gaslike to liquidlike behavior as its density exceeds 4g/cm^{3}. As for the electronic conductivity as a function of the density, we identified a remarkable isosbestic point at frequencies around 7eV, which may be an additional signature of a gas-liquid transition in hydrogen at 30000K.

Quantifying ionization in hot dense plasmas.

Ionization is a problematic quantity in that it does not have a well-defined thermodynamic definition and yet it is a key parameter within plasma modeling. One still therefore aims to find a consistent and unambiguous definition for the ionization state. Within this context we present finite-temperature density functional theory calculations of the ionization state of carbon in CH plasmas using two potential definitions: one based on counting the number of continuum electrons, and another based on the optical conductivity. Differences of up to 10% are observed between the two methods. However, including "Pauli forbidden" transitions in the conductivity reproduces the counting definition, suggesting such transitions are important to evaluate the ionization state.

Investigating mechanisms of state localization in highly ionized dense plasmas.

We present experimental observations of K_{β} emission from highly charged Mg ions at solid density, driven by intense x rays from a free electron laser. The presence of K_{β} emission indicates the n=3 atomic shell is relocalized for high charge states, providing an upper constraint on the depression of the ionization potential. We explore the process of state relocalization in dense plasmas from first principles using finite-temperature density functional theory alongside a wave-function localization metric, and find excellent agreement with experimental results.

Shape transition around 222Ra based on finite-temperature covariant density functional theory

The shape evolution and potential energy surfaces of even–even [Formula: see text]Ra are investigated in the [Formula: see text] plane by the covariant density functional theory. For octupole-deformed [Formula: see text]Ra, the free energy surfaces, the deformations, the pairing gaps, the excitation energy as well as the specific heat with increasing temperature are analyzed. Based on the specific heat, [Formula: see text]Ra exhibits three distinct discontinuities as the temperature rises. The first pairing transition occurs at [Formula: see text][Formula: see text]MeV, while the second one happens at [Formula: see text][Formula: see text]MeV when octupole deformation disappears, and the third one takes place at [Formula: see text][Formula: see text]MeV as quadrupole deformation approaches zero. The gaps at [Formula: see text] and [Formula: see text] in the single-particle levels are responsible for the octupole global minima. The shape transitions not only occur between even–even [Formula: see text]Ra, but also occur with increasing temperature for [Formula: see text]Ra.

Microscopic study of neutron-induced fission process of 239Pu via zero- and finite-temperature density functional theory* *Supported by the National Natural Science Foundation of China (11790325, 12275081, 11790320, 11790321, 11961131010, 11605054, 12105369, 12147219, 12047568) and the Continuous Basic Scientific Research Project (WDJC-2019-09)

To study the neutron-induced fission of Pu, potential energy surface (PES) calculations were performed using zero and finite-temperature density functional theory (FT-DFT) with the Skyrme force. The energy of the incident neutron was simulated by the temperature of the FT-DFT. The variations of the least-energy fission path, fission barrier, total kinetic energy, scission line, and mass distribution of fission fragments with the incident neutron energy were analyzed. It was learned that an increase in the temperature lowers the barrier height, the isomeric-state energy, and the ridge between symmetric and asymmetric fission valleys. Additionally, the gaps of the single particle levels become smaller with an increase in the temperature. As the temperature increases, the pre-fission region shrinks, and the scission occurs at smaller deformation around the symmetric fission channel. At low temperatures, the pairing correlations in the collective space are similar to those in zero-temperature DFT, and when the temperature is 0.3 MeV, the pairing gaps decrease rapidly. Two different methods were used to calculate the fission yields of the neutron-induced fission Pu (n, f) with different incident neutron energies, in the framework of time-dependent generator coordinate method (TDGCM). One way to calculate the fission yield of Pu (n, f) is to solve the collective equation of the TDGCM by using the PES from the FT-DFT with the corresponding temperature. The other involves using the PES from the zero-temperature DFT and adjusting the initial collective energy of the wave packet in the TDGCM according to the incident neutron energy. For the cases of the lower incident neutron energies, these two methods gave similar results and reproduced the experimental peak and width of fission fragment distribution. However, for the highest incident neutron energy considered in this study, the results from the TDGCM using the PES from zero-temperature DFT deviated explicitly from the experimental data, whereas those obtained by using the PES from FT-DFT remained close to the experimental data. This indicated that, with the increase in the incident neutron energy, the shell structure of the compound nuclei changed explicitly; thus, it may not be effective to use the PES from zero-temperature to perform the fission dynamic calculation.

Density effect on electronic structure of warm dense matter based on X-ray fluorescence spectroscopy

Warm dense matter (WDM), a kind of transition state of matter between cold condensed matter and high temperature plasma, is one of the main research objects of high energy density physics (HEDP). Compared with the structure of isolated atom, the electron structure of WDM will change significantly because of the influences of density and temperature effect. As WDM is always strongly coupled and partly degenerate, accurate theoretical description is very complicated and the accurate experimental research is also very challenging. In this paper, the density effect on the warm dense matter electron structure based on the X-ray fluorescence spectroscopy is studied. The warm dense titanium with density larger than solid density is produced experimentally based on a specially designed hohlraum. Then, the titanium is pumped to emit fluorescence by using the characteristic line spectrum emitted by the laser irradiating the pump material (Vanadium). The X-ray fluorescence spectra of titanium with different states are diagnosed by changing the delay time between the pump laser and drive laser. The experimental fluorescence spectrum indicates that the difference in energy between <inline-formula><tex-math id="M5">\begin{document}${\mathrm{K}}_{\text{β}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231215_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231215_M5.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ {\mathrm{K}}_{\text{α}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231215_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231215_M6.png"/></alternatives></inline-formula> (<inline-formula><tex-math id="M7">\begin{document}$\Delta E_{{\mathrm{K}}_{\text{β}}\text{-}{\mathrm{K}}_{\text{α}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231215_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231215_M7.png"/></alternatives></inline-formula>) of the compressed titanium (7.2–9.2 g/cm<sup>3</sup>, 1.6–2.4 eV) is about 2 eV smaller than that of cold titanium. Two theoretical methods, i.e. finite-temperature relativistic density functional theory (FTRDFT) and two-step Hartree-Fock-Slater (TSHFS), are used to calculate the fluorescence spectrum of warm dense titanium. The calculated results indicate that the energy difference (<inline-formula><tex-math id="M10">\begin{document}$\Delta E_{{\mathrm{K}}_{\text{β}}\text{-}{\mathrm{K}}_{\text{α}}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231215_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231215_M10.png"/></alternatives></inline-formula>) decreases with the increase of density but changes slowly with the increase of temperature during the calculated state (4.5–13.5 g/cm<sup>3</sup>, 0.03–5 eV). The FTRDFT overestimates the density effect on the line shift, while TSHFS underestimates the density effect. The future work will focus on optimizing the experimental method of X-ray fluorescence spectroscopy, obtaining X-ray fluorescence spectrum of titanium with more states, and then testing the theoretical method for warm dense matter.

A comparison of molecular dynamics potentials used to account for thermal diffuse scattering in multislice simulations

Here we investigate electron scattering simulations with thermal displacements incorporated using molecular dynamics potentials. Specifically, we explore the sensitivity of electron scattering to the phonon band structure, or more explicitly interatomic forces. Silicon serves as the model material where we introduce thermal atomic displacements via empirical and machine-learned molecular dynamics interatomic potentials and compare them to finite-temperature density functional theory interatomic forces. We demonstrate that when molecular dynamics potentials do not sufficiently reproduce the correct phonon band structure, significant errors in the simulated diffraction and image intensities can occur. Moreover, for Si, we find that multislice simulations using machine-learned interatomic potentials are more accurate than empirical ones. In addition to the selected atomic potential, we demonstrate that the sensitivity to the phonon band structure also depends on the crystal zone axis, which can be used to enhance sensitivity to thermal displacements. Finally, we provide a sensitivity analysis with angle-resolved scanning transmission electron microscopy (STEM) to enhance image sensitivity to the details of the phonon band structure.

Plane-wave-based stochastic-deterministic density functional theory for extended systems

Traditional finite-temperature Kohn-Sham density functional theory (KSDFT) has an unfavorable scaling with respect to the electron number or at high temperatures. The evaluation of the ground-state density in KSDFT can be replaced by the Chebyshev trace (CT) method. In addition, the use of stochastic orbitals within the CT method leads to the stochastic density functional theory [Phys. Rev. Lett. 111, 106402 (2013)] (SDFT) and its improved theory, mixed stochastic-deterministic density functional theory [Phys. Rev. Lett. 125, 055002 (2020)] (MDFT). We have implemented the above four methods within the first-principles package ABACUS. All of the four methods are based on the plane-wave basis set with the use of norm-conserving pseudopotentials and the periodic boundary conditions with the use of $k$-point sampling in the Brillouin zone. By using the KSDFT calculation results as benchmarks, we systematically evaluate the accuracy and efficiency of the CT, SDFT, and MDFT methods via examining a series of physical properties, which include the electron density, the free energy, the atomic forces, stress, and density of states for a few condensed phase systems. The results suggest that our implementations of CT, SDFT, and MDFT not only reproduce the KSDFT results with a high accuracy, but also exhibit several advantages over the tradition KSDFT method. We expect these methods can be of great help in studying high-temperature and large-size extended systems such as warm dense matter and dense plasma.

Intrinsic local symmetry breaking in nominally cubic paraelectric BaTiO3

Whereas at low-temperatures ferroelectrics have a well understood ordered spatial dipole arrangement, the fate of these dipole configurations in the higher temperature paraelectric (PE) phase remains poorly understood. Using density functional theory (DFT), we find that, unlike the case in nonpolar $AB{\mathrm{O}}_{3}$, perovskites such as cubic $\mathrm{BaZr}{\mathrm{O}}_{3}$ that do not lower their energy by any form of positional symmetry breaking, the origin of distribution of the $B$-site off-centering in cubic PE such as $\mathrm{BaTi}{\mathrm{O}}_{3}$ is an intrinsic, energy lowering due to symmetry breaking. Minimizing the internal energy ${E}_{0}$ of a constrained cubic phase represented by a large enough supercell to accommodate symmetry breaking already reveals the presence of a distribution of local displacements (i.e., a polymorphous network) that (i) locally mimics the symmetries of the low temperature phases, while (ii) being the precursors of what finite temperature density functional theory (DFT) molecular dynamics (MD) finds as thermal motifs when equilibrating the free energy ${E}_{0}\ensuremath{-}TS$. Analyzing the DFT-derived configurations of the PE cubic supercell by projecting its displacements onto irreducible representations reveals that it is best described as a temperature-dependent superposition of numerous modes, including ferroelectric $({{\mathrm{\ensuremath{\Gamma}}}_{4}}^{--})$ and antiferroelectric (${{M}_{2}}^{--}$ and ${{X}_{5}}^{+}$), rather than a single mode, (e.g., ${{X}_{5}}^{+}$) representing a well-defined long-range ordered configuration. This suggests that the electronic and dielectric properties of such PE phases are best calculated from a polymorphous distribution of interrelated local motifs in large supercells rather than from either purely disordered or long-range ordered models. In this respect, PE phases share a common feature with paramagnetic (PM) and paraelastic (PEL) perovskite phases whose central feature is a polymorphous distribution of local motifs---dipole moments in PE, magnetic moments in PM, and ordinary octahedral distortion modes in PEL---all computable by DFT supercells and useable in calculating electronic and magnetic properties of paraphases.

Formation of Metallic Ag on AgBr by Femtosecond Laser Irradiation

Laser irradiation of materials induces changes in their structure and functional properties. In this work, lattice heating and electronic excitation on silver bromide (AgBr), provoked by femtosecond laser irradiation, have been investigated by finite-temperature density functional theory and ab initio molecular dynamics calculations by using the two-temperature model. According to our results, the electronic temperature of 0.25 eV is enough to excite the electrons from the valence to the conduction band, whereas 1.00 eV changes the structural properties of the irradiated AgBr material. Charge density simulations also show that an Ag clustering process and the formation of Br3− complexes take place when the electronic temperature reaches 2.00 eV and 5.00 eV, respectively. The present results can be used to obtain coherent control of the extreme nonequilibrium conditions due to femtosecond laser irradiation for designing new functional materials.

Air Stable Chalcogen-Doped Rubicenes with Diradical Character

Air Stable Chalcogen-Doped Rubicenes with Diradical Character

Approximate bounds and temperature dependence of adiabatic connection integrands for the uniform electron gas.

Thermal density functional theory is commonly used in simulations of warm dense matter, a highly energetic phase characterized by substantial thermal effects and by correlated electrons demanding quantum mechanical treatment. Methods that account for temperature dependence, such as Mermin-Kohn-Sham finite-temperature density functional theory and free energy density functional theory, are now employed with more regularity and available in many standard code packages. However, approximations from zero-temperature density functional theory are still often used in temperature-dependent simulations using thermally weighted electronic densities as an input to exchange-correlation functional approximations, a practice known to miss temperature-dependent effects in the exchange-correlation free energy of these systems. In this work, the temperature-dependent adiabatic connection is demonstrated and analyzed using a well-known parameterization of the uniform electron gas free energy. Useful tools based on this formalism for analyzing and constraining approximations of the exchange-correlation at zero temperature are leveraged for the finite-temperature case. Inspired by the Lieb-Oxford inequality, which provides a lower bound for the ground-state exchange-correlation energy, bounds for the exchange-correlation at finite temperatures are approximated for various degrees of electronic correlation.

Energy Relaxation and Electron–Phonon Coupling in Laser-Excited Metals

The rate of energy transfer between electrons and phonons is investigated by a first-principles framework for electron temperatures up to = 50,000 K while considering the lattice at ground state. Two typical but differently complex metals are investigated: aluminum and copper. In order to reasonably take the electronic excitation effect into account, we adopt finite temperature density functional theory and linear response to determine the electron temperature-dependent Eliashberg function and electron density of states. Of the three branch-dependent electron–phonon coupling strengths, the longitudinal acoustic mode plays a dominant role in the electron–phonon coupling for aluminum for all temperatures considered here, but for copper it only dominates above an electron temperature of = 40,000 K. The second moment of the Eliashberg function and the electron phonon coupling constant at room temperature K show good agreement with other results. For increasing electron temperatures, we show the limits of the approximation for the Eliashberg function. Our present work provides a rich perspective on the phonon dynamics and this will help to improve insight into the underlying mechanism of energy flow in ultra-fast laser–metal interaction.

Development of the temperature-dependent interatomic potential for molecular dynamics simulation of metal irradiated with an ultrashort pulse laser

Laser ablation is often explained by a two-temperature model (TTM) with different electron and lattice temperatures. To realize a classical molecular dynamics simulation of the TTM, we propose an extension of the embedded atom method to construct an interatomic potential that is dependent on the electron temperature. This method is applied to copper, and its validity is demonstrated by comparison of several physical properties, such as the energy–volume curve, phonon dispersion, electronic heat capacity, ablation threshold, and mean square displacement of atoms, with those of finite-temperature density functional theory.

Effect of Nonequilibrium Transient Electronic Structures on Lattice Stability in Metals: Density Functional Theory Calculations

The electronic structures of metals undergo transient nonequilibrium states during the photoexcitation process caused by isochoric heating of X-ray free-electron laser, and their lattice stability is, thus, significantly affected. By going beyond frozen core approximation, we manually introduced nonequilibrium electron distribution function in finite-temperature density functional theory with the framework of Kohn–Sham–Mermin to investigate such transient states, and their effect on lattice stability in metals is demonstrated by phonon dispersion calculated using the finite displacement method. We found that the perfect lattice of a metal collapses due to the exotic electronic structure of nonequilibrium transient state created by isochoric heating of X-ray free-electron laser. Further increase of the number of holes created in the sample (i.e., an increase of laser fluence) still results in lattice instability for aluminum, while for copper, it results in phonon hardening. The potential energy surface is calculated for the extreme case of both Al and Cu with exactly one hole created in its inner shell for each one of the atoms. A double-well structure is clearly observed for Al, while the potential energy surface becomes steeper for Cu.

Stochastic Vector Techniques in Ground-State Electronic Structure.

We review a suite of stochastic vector computational approaches for studying the electronic structure of extended condensed matter systems. These techniques help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. While their scope is vast, we limit our study to ground-state and finite temperature density functional theory (DFT) and second-order many-body perturbation theory. More advanced topics, such as quasiparticle (charge) and optical (neutral) excitations and higher-order processes, are covered elsewhere. We start by explaining how to use stochastic vectors in computations, characterizing the associated statistical errors. Next, we show how to estimate the electron density in DFT and discuss effective techniques to reduce statistical errors. Finally, we review the use of stochastic vectors for calculating correlation energies within the second-order Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods.

Ionization state and dielectric constant in cold rarefied hydrocarbon plasmas of inertial confinement fusion.

A combined approach to study cold rarefied matter is introduced that includes a semianalytical method based on the free-energy minimization and ab initio calculations based on the finite-temperature density-functional theory. The approach is used to calculate the ionization state of hydrocarbon (CH) under the shock-release conditions in inertial confinement fusion. The dielectric constant of CH is calculated using the Kubo-Greenwood formulation and contribution from atomic polarizabilities is found to be as important as the free-electron contribution. Using the ionization state and dielectric constant, the electron density profile in the rarefaction wave of the shock-release plasma is obtained.

Excited-state potentials for modelling dense plasmas from first principles

The modelling of dense plasmas using finite-temperature density functional theory has proven very successful in determining transport properties and the equation of state of systems where quantum many-body effects and correlations play a key role in their structure. Here we show how excited-state projector augmented-wave potentials can be used to extend these calculations to explicitly model core-hole states, allowing for the calculation of the electronic structure of a range of integer charge configurations embedded in a dense plasma environment. Our excited-state potentials show good agreement with all-electron calculations at finite-temperatures, motivating their use as an efficient approach in modelling from first principles both the structure of strongly-coupled non-equilibrium plasmas and their interaction with intense x-rays.

Accelerating finite-temperature Kohn-Sham density functional theory with deep neural networks

We present a numerical modeling workflow based on machine learning (ML) which reproduces the the total energies produced by Kohn-Sham density functional theory (DFT) at finite electronic temperature to within chemical accuracy at negligible computational cost. Based on deep neural networks, our workflow yields the local density of states (LDOS) for a given atomic configuration. From the LDOS, spatially-resolved, energy-resolved, and integrated quantities can be calculated, including the DFT total free energy, which serves as the Born-Oppenheimer potential energy surface for the atoms. We demonstrate the efficacy of this approach for both solid and liquid metals and compare results between independent and unified machine-learning models for solid and liquid aluminum. Our machine-learning density functional theory framework opens up the path towards multiscale materials modeling for matter under ambient and extreme conditions at a computational scale and cost that is unattainable with current algorithms.

Number of populated electronic configurations in a hot dense plasma.

In hot dense plasmas of intermediate or high-Z elements in the state of local thermodynamic equilibrium, the number of electronic configurations contributing to key macroscopic quantities such as the spectral opacity and equation of state can be enormous. In this work we present systematic methods for the analysis of the number of relativistic electronic configurations in a plasma. While the combinatoric number of configurations can be huge even for mid-Z elements, the number of configurations which have non-negligible population is much lower and depends strongly and nontrivially on temperature and density. We discuss two useful methods for the estimation of the number of populated configurations: (i) using an exact calculation of the total combinatoric number of configurations within superconfigurations in a converged super-transition-array (STA) calculation, and (ii) by using an estimate for the multidimensional width of the probability distribution for electronic population over bound shells, which is binomial if electron exchange and correlation effects are neglected. These methods are analyzed, and the mechanism which leads to the huge number of populated configurations is discussed in detail. Comprehensive average-atom finite-temperature density functional theory (DFT) calculations are performed in a wide range of temperature and density for several low-, mid-, and high-Z plasmas. The effects of temperature and density on the number of populated configurations are discussed and explained.