Thomas-Fermi Kinetic Energy Research Articles

Phenomenological potentials for the refractory metals Cr, Mo and W

Cohesion in the refractory metals Cr, Mo, and W is phenomenologically described in this work via a n-body energy functional with a set of physically motivated parameters that were optimized to reproduce selected experimental properties characteristic of perfect and defective crystals. The functional contains four terms accounting for the hard-core repulsion, the Thomas–Fermi kinetic energy repulsion and for contributions to the binding energy of s and d valence electrons. Lattice dynamics, molecular statics, and molecular dynamics calculations show that this model describes satisfactorily thermodynamic properties of the studied metals whereas, unlike other empirical approaches from the literature, predictions of phonon dispersion relations and of surface and point defect energetics reveal in fair good agreement with experiments. These results suggest that the present model is well adapted to large-scale simulations and whenever total energy calculations of thermodynamic properties are unfeasible.

Phase transitions of repulsive two-component Fermi gases in two dimensions

We predict the phase separations of two-dimensional Fermi gases with repulsive contact-type interactions between two spin components. Using density-potential functional theory with systematic semiclassical approximations, we address the long-standing problem of itinerant ferromagnetism in realistic settings. We reveal a universal transition from the paramagnetic state at small repulsive interactions toward ferromagnetic density profiles at large interaction strengths, with intricate particle-number dependent phases in between. Building on quantum Monte Carlo results for uniform systems, we benchmark our simulations against Hartree–Fock calculations for a small number of trapped fermions. We thereby demonstrate that our employed corrections to the bare contact interaction energy and especially to the Thomas–Fermi kinetic energy functional are necessary for reliably predicting properties of trapped mesoscopic Fermi gases. The density patterns of the ground state survive at low finite temperatures and confirm the Stoner-type polarization behavior across a universal interaction parameter, albeit with substantial quantitative differences that originate in the trapping potential and the quantum-corrected kinetic energy. We also uncover a plethora of metastable configurations that are energetically comparable to the ground-state density profiles and are thus likely to be observed in experiments. We argue that our density-functional approach can be easily applied to interacting multi-component Fermi gases in general.

Variational Principle for Eigenmodes of Reactivity in Conceptual Density Functional Theory.

In conceptual density functional theory, reactivity indexes as the Fukui function, the global hardness/softness, and hardness/softness kernels are fundamental linear responses extensively studied to predict the nucleophilic and electrophilic propensities of atoms in molecules. We demonstrate that the hardness/softness kernels of an isolated system can be expanded in eigenmodes, solutions of a variational principle. These modes are divided into two groups: the polarization modes and the charging modes. The eigenvectors of the polarization modes are orthogonal to the Fukui function and can be interpreted as densities induced at a constant chemical potential. The charging modes of an isolated system are associated with virtual charge transfers weighted by the Fukui function and obey an exact nontrivial sum rule. The exact relation between these charging eigenmodes and those of the polarizability kernel is established. The physical interpretation of the modes is discussed. Applications of the present findings to the Thomas–Fermi and von Weizacker kinetic energy functionals are presented. For a confined free quantum gas, described by the von Weizacker kinetic energy functional, we succeed to derive an approximate analytical solution for the Fukui function and for hardness/softness and polarizability kernels. Finally, we indicate how numerical calculations of the hardness kernel of a molecule could be performed from the Kohn–Sham orbitals.

Exact orbital-free kinetic energy functional for general many-electron systems

The exact form of the kinetic energy functional has remained elusive in orbital-free models of density functional theory (DFT). This has been the main stumbling block for the development of a general-purpose framework on this basis. Here, we show that on the basis of a two-density model, which represents many-electron systems by mass density and spin density components, we can derive the exact form of such a functional. The exact functional is shown to contain previously suggested functionals to some extent, with the notable exception of the Thomas-Fermi kinetic energy functional.

Structure and dynamics of warm dense aluminum: a molecular dynamics study with density functional theory and deep potential

We perform a systematic study on the structure and dynamics of warm dense aluminum (Al) at temperatures ranging from 0.5 to 5.0 eV with molecular dynamics utilizing both density functional theory (DFT) and the deep potential (DP) method. On one hand, unlike the Thomas–Fermi kinetic energy density functional (KEDF), we find that the orbital-free DFT method with the Wang–Teter non-local KEDF yields properties of warm dense Al that agree well with the Kohn–Sham DFT method, enabling accurate orbital-free DFT simulations of warm dense Al at relatively low temperatures. On the other hand, the DP method constructs a deep neural network that has a high accuracy in reproducing short- and long-ranged properties of warm dense Al when compared to the DFT methods. The DP method is orders of magnitudes faster than DFT and is well-suited for simulating large systems and long trajectories to yield accurate properties of warm dense Al. Our results suggest that the combination of DFT methods and the DP model is a powerful tool for accurately and efficiently simulating warm dense matter.

Analysis of over-magnetization of elemental transition metal solids from the SCAN density functional

Recent investigations have found that the strongly constrained and appropriately normed (SCAN) meta-GGA exchange-correlation functional significantly over-magnetizes elemental Fe, Co, and Ni solids. For the paradigmatic case, bcc Fe, the error relative to experiment is $\gtrsim 20 \%$. Comparative analysis of magnetization results from SCAN and its \textit{deorbitalized} counterpart, SCAN-L, leads to identification of the source of the discrepancy. It is not from the difference between Kohn-Sham (SCAN-L) and generalized Kohn-Sham (SCAN) procedures. The key is the iso-orbital indicator $\alpha$ (the ratio of the local Pauli and Thomas-Fermi kinetic energy densities). Its \textit{deorbitalized} counterpart, $\alpha_L$, has more dispersion in both spin channels with respect to magnetization in an approximate region between 0.6 Bohr and 1.2 Bohr around an Fe nucleus. The overall effect is that the SCAN switching function evaluated with $\alpha_L$ reduces the energetic disadvantage of the down channel with respect to up compared to the original $\alpha$, which in turn reduces the magnetization. This identifies the cause of the SCAN magnetization error as insensitivity of the SCAN switching function to $\alpha$ values in the approximate range $0.5 \lesssim \alpha \lesssim 0.8$ and oversensitivity for $\alpha \gtrsim 0.8$.

Leading gradient correction to the kinetic energy for two-dimensional fermion gases

Density-functional theory (DFT) is notorious for the absence of gradient corrections to the two-dimensional (2D) Thomas-Fermi kinetic-energy functional; it is widely accepted that the 2D analog of the 3D von Weizs\"acker correction vanishes, together with all higher-order corrections. Contrary to this long-held belief, we show that the leading correction to the kinetic energy does not vanish, is unambiguous, and contributes perturbatively to the total energy. This insight emerges naturally in a simple extension of standard DFT, which has the effective potential energy as a functional variable on equal footing with the single-particle density.

Hydrodynamic theory for quantum plasmonics: Linear-response dynamics of the inhomogeneous electron gas

We investigate the hydrodynamic theory of metals, offering systematic studies of the linear-response dynamics for an inhomogeneous electron gas. We include the quantum functional terms of the Thomas-Fermi kinetic energy, the von Weizs\acker kinetic energy, and the exchange-correlation Coulomb energies under the local density approximation. The advantages, limitations, and possible improvements of the hydrodynamic theory are transparently demonstrated. The roles of various parameters in the theory are identified. We anticipate that the hydrodynamic theory can be applied to investigate the linear response of complex metallic nanostructures, including quantum effects, by adjusting theory parameters appropriately.

Derivative discontinuity and exchange-correlation potential of meta-GGAs in density-functional theory.

We investigate fundamental properties of meta-generalized-gradient approximations (meta-GGAs) to the exchange-correlation energy functional, which have an implicit density dependence via the Kohn-Sham kinetic-energy density. To this purpose, we construct the most simple meta-GGA by expressing the local exchange-correlation energy per particle as a function of a fictitious density, which is obtained by inverting the Thomas-Fermi kinetic-energy functional. This simple functional considerably improves the total energy of atoms as compared to the standard local density approximation. The corresponding exchange-correlation potentials are then determined exactly through a solution of the optimized effective potential equation. These potentials support an additional bound state and exhibit a derivative discontinuity at integer particle numbers. We further demonstrate that through the kinetic-energy density any meta-GGA incorporates a derivative discontinuity. However, we also find that for commonly used meta-GGAs the discontinuity is largely underestimated and in some cases even negative.

On the relationship among Ghosh-Berkowitz-Parr entropy, Shannon entropy and Fisher information

In their seminal work thirty years ago, Ghosh, Berkowitz, and Parr reformulated the ground-state density-functional theory into a local version of thermodynamics, where a new concept, called Ghosh-Berkowitz-Parr entropy, was proposed. Employing the non-interacting kinetic energy density and Thomas-Fermi kinetic energy density, this entropy was formulated like the Sackur-Tetrode equation in classical thermodynamics. Not much has been known about its properties. In this contribution, we investigate its relationship with Shannon entropy and Fisher information from the numerical perspective. To that end, we have examined 36 neutral atoms and 42 molecular systems. We have considered both molecular and atomic contributions with Bader’s zero-flux criterion to partition atoms in molecules. Our results show that these quantities are closely correlated, and yet their correlations might be complicated since no universal relationship among them has been observed.

How does the ambiguity of the electronic stress tensor influence its ability to serve as bonding indicator

The electronic stress tensor is not uniquely defined. Possible bonding indicators originating from the quantum stress tensor may inherit this ambiguity. Based on a general formula of the stress tensor this ambiguity can be described by an external parameterλfor indicators derived from the scaled trace of the stress tensor (whereby the scaling function is proportional to the Thomas–Fermi kinetic energy density). The influence ofλis analyzed and the consequences for the representation of chemical bonding are discussed in detail. It is found that the scaled trace of the stress tensor may serve as suitable bonding indicator over a wide range ofλvalues, excluding the value range between −0.15 and −0.48. Focusing on the eigenvalues of the stress tensor, it is found that the sign of the eigenvalues heavily depends on the chosen representation of the stress tensor. Therefore, chemical bonding analyses which are based on the interpretation of the eigenvalue sign (e.g., the spindle structure) are strongly dependent on the chosen form of the stress tensor. © 2014 Wiley Periodicals, Inc.

How does the ambiguity of the electronic stress tensor influence its ability to reveal the atomic shell structure

The electronic stress tensor is not uniquely defined. Therefore, shell indicators stemming from the quantum stress tensor may inherit this ambiguity. Based on a general formula of the stress tensor this ambiguity can be described by an external parameter λ. Two functions derived from the quantum stress tensor have been evaluated according to their ability to serve as shell indicators. The influence of λ is analyzed and the consequences for the representation of the atomic shell structure are discussed in detail. It is found that the trace of the stress tensor does not fully reveal the atomic shell structure. In contrast, the scaled trace (whereby the scaling function is proportional to the Thomas-Fermi kinetic energy density) produces fairly good representation of the atomic shell structure over a wide range of λ values.

Is it possible to construct excited-state energy functionals by splitting k-space?

We show that our procedure of constructing excited-state energy functionals by splitting k-space, employed so far to obtain exchange energies of excited-states, is quite general. We do so by applying the same method to construct modified Thomas–Fermi kinetic energy functional and its gradient-expansion up to the second order for the excited-states. We show that the resulting kinetic energy functional has the same accuracy for the excited-states as the ground-state functionals do for the ground-states.

Local behavior of the first-order gradient correction to the Thomas–Fermi kinetic energy functional

The first-order gradient correction to the Thomas-Fermi functional proposed by Haq et al. [Chem. Phys. Lett. 111, 79 (1984)] has been tested by evaluating both the total kinetic energy and the local kinetic energy density. For the kinetic energy density, we have evaluated its deviation from the exact orbital-based result through a quality factor that reflects the quality of the functionals in a better way than their relative errors. The study is performed on two different systems: Light atoms (up to Z=18) and a noninteracting model of fermions confined in a Coulombic-type potential, a system that provides useful insights about the performance of the functionals when the ground state is degenerate. It is found that this approximation gives very low relative errors and a better local behavior than any other kinetic energy density functional.

Interesting properties of Thomas–Fermi kinetic and Parr electron–electron‐repulsion DFT energy functional generated compact one‐electron density approximation for ground‐state electronic energy of molecular systems

The reduction of the electronic Schrodinger equation or its calculating algorithm from 4N-dimensions to a (nonlinear, approximate) density functional of three spatial dimension one-electron density for an N-electron system, which is tractable in the practice, is a long desired goal in electronic structure calculation. If the Thomas-Fermi kinetic energy ( approximately integralrho(5/3)dr(1)) and Parr electron-electron repulsion energy ( approximately integralrho(4/3)dr(1)) main-term functionals are accepted, and they should, the later described, compact one-electron density approximation for calculating ground state electronic energy from the 2nd Hohenberg-Kohn theorem is also noticeable, because it is a certain consequence of the aforementioned two basic functionals. Its two parameters have been fitted to neutral and ionic atoms, which are transferable to molecules when one uses it for estimating ground-state electronic energy. The convergence is proportional to the number of nuclei (M) needing low disc space usage and numerical integration. Its properties are discussed and compared with known ab initio methods, and for energy differences (here atomic ionization potentials) it is comparable or sometimes gives better result than those. It does not reach the chemical accuracy for total electronic energy, but beside its amusing simplicity, it is interesting in theoretical point of view, and can serve as generator function for more accurate one-electron density models.

Yukawa Monte Carlo and Orbital Free Molecular Dynamics approaches for the equation of state and structural properties of hot dense matter

Molecular Dynamics (MD) and Monte Carlo (MC) simulations techniques have been extensively employed to obtain accurate results on the structural and dynamical properties of hot dense matter. Among them, Yukawa MC (YMC) and MD simulations are powerful techniques to compute plasma properties such as equation of state (EOS) or transport coefficients, but are, in principle, limited to applications where the linear electronic screening assumption is valid. On the contrary, it has been recently shown that a modified scheme using density functional theory with a Thomas–Fermi kinetic energy functional for the electrons may be well suited to perform MD simulations on high Z elements at high densities and temperature, without any assumption on the electronic screening. Since Yukawa theory is based on several assumptions and parameters, our purpose in this paper is to check some of these approximations by the Orbital Free Molecular Dynamics (OFMD) scheme, that goes beyond the classical models. For several selected iron plasma conditions, we have compared YMC and OFMD pair correlation functions and equation of state. This provides the opportunity to discuss the influence of the primary parameter for Yukawa potentials, i.e. the ionization of the plasma in the hot and dense regime.

Very-high-temperature molecular dynamics

It is shown that a modified scheme of density functional theory, using the Thomas-Fermi kinetic energy functional for the electrons, is well suited to perform very-high-temperature molecular dynamics simulations on high-Z elements. As an example, iron on the principal Hugoniot is simulated up to 5 keV and 5 times the normal density, giving an equation of state in agreement with current models. Ionic structure is obtained and is given to an excellent level of precision by the structure of the one-component plasma computed for a coupling parameter corresponding to Thomas-Fermi ionization.

The extended Thomas–Fermi kinetic energy density functional with position-dependent effective mass in one dimension

The point canonical transformations map the Schrodinger equation with constant mass to a wave equation with a position-dependent effective mass. Using such a technique we derive, for a one-dimensional inhomogeneous system of noninteracting fermions with density ρ(x) and spatially dependent effective mass distribution m(x), the semiclassical kinetic energy density functional τ(ρ) in the so-called extended Thomas–Fermi model up to order 2. For a given position-dependent mass, we compare numerically the total semiclassical kinetic energy with its exact quantum mechanical counterpart. The qualitative agreement is excellent.

Fully local orbital-free calculation of electronic structure using pseudopotentials

An exactly solvable orbital-free technique is applied to the calculation of the electronic structure of polyatomic systems. The Thomas–Fermi kinetic energy, local exchange, local electrostatic energy functionals, and pseudopotentials are used. Given the potential, the cost of the calculation of the electronic density and the total energy is linear in the number of atoms. A first order estimate of Kohn–Sham orbitals using the effective potential from the orbital-free calculation and wavelets as basis set is also given. Numerical results are shown for several homonuclear and heteronuclear metallic clusters. The atomic structure of medium size sodium clusters was obtained minimizing the orbital-free energy by varying the nuclear coordinates using a simulated annealing procedure.

Molecular dynamics study of electron gas models for liquid water

The frozen density electron gas model proposed by Gordon and Kim for rare gas systems has been implemented in a molecular dynamics code. This code has been applied to investigate various options for extending this scheme to inter-molecular interactions in liquid water. We have compared a number of gradient corrections to the Thomas-Fermi kinetic energy. We also explored a more empirical approach based on adaptation of the frozen molecular electron density to the condensed phase environment. Consistent with experience from force field methods, enhancement of the molecular dipole moment proved to be necessary to reproduce the properties of the liquid. The best models we investigated are a gradient corrected expansion of the simple local density Hamiltonian applied in the original Gordon and Kim model. In addition, these models observed a modified molecular electron density carrying the same dipole moment of 2.95 D as has been observed by recent ab initio molecular dynamics studies based on fully self-consistent Kohn-Sham methods. Possible implications of this finding for force field models are discussed.