Simple analytical particle and kinetic energy densities for a dilute fermionic gas in a d-dimensional harmonic trap

We derive simple analytical expressions for the particle density rho(r) and the kinetic energy density tau(r) for a system of noninteracting fermions in a d-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation tau[rho] using the exact rho(r) and show that it locally reproduces the exact kinetic energy density tau(r), including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of tau(TF)[rho(r)] yields the exact total kinetic energy.     

Push it to the limit: comparing periodic and local approaches to density functional theory for intermolecular interactions

ABSTRACT

With the aim of systematically comparing two popular approaches to density functional theory – all-electron calculations with local basis sets, and periodic calculations employing plane wave basis sets and norm-conserving pseudopotentials – we have computed complete-basis binding energies across the S22 set of intermolecular interactions, a dataset consisting of noncovalent interactions of small- and medium-sized molecules containing first- and second-row atoms, using the Troullier-Martins norm-conserving pseudopotentials with SPW92, a local spin-density approximation; and PBE, a generalised gradient approximation. We have found that it is challenging to reach the basis set limit with these periodic calculations; for the methods and systems examined, a minimum vacuum distance of 30?Å between a system and its nearest images is necessary – unless some form of dipole correction is employed – as is a kinetic energy cutoff of at least 80 Ry. The trends in convergence with respect to vacuum size and kinetic energy cutoff are largely independent of the level of density functional approximation employed. A sense of the impact of each hyperparameter on basis set error provides a foundation for ensuring quality calculations in future studies and allows us to quantify the basis set errors incurred in existing studies on similar systems.     

Kinetic energy functionals for molecular calculations

The orbitals and eigenvalues of a Kohn-Sham density functional theory calculation can be used to determine the kinetic potential, the functional derivative of the non-interacting kinetic energy. Thus, approproximate kinetic energy functionals can be systematically parameterized to improve their functional derivatives. Fitting procedures have been applied to various functional forms and the quality of the resulting functionals investigated using variationally optimized densities. The best functionals include the full correction of Weizsäcker and a modulation of the Thomas-Fermi p^5/3 term by a function of the distance from the nucleus. These atom-specific functionals predict virtually exact shell structure, and may be combined readily into a functional which supports molecular binding.     

The kinetic energy partition method applied to a confined quantum harmonic oscillator in a one-dimensional box

A new, physically motivated, basis set expansion method for solving quantum eigenvalue problems with competing interaction potentials is presented. In contrast to the usual dissection of the potential energy into unperturbed and perturbing terms, we divide the kinetic energy into partial terms by modifying the mass factor. The partition scheme results in partial kinetic energies with their effective mass factors. By distributing each partial kinetic energy to a respective potential energy to form a subsystem, the total Hamiltonian is written as the sum of subsystem Hamiltonians. Using a linear combination of the subsystem wave-functions to represent the system wave-function we obtain a set of coupled equations for the expansion coefficients, by solving these energies and wave-functions can be obtained. We demonstrate the solution scheme with a standard model system: a confined harmonic oscillator in a one-dimensional box. With only a few (less than ten) basis functions from each subsystem, we can reproduce the exact solutions very accurately, thus showing the applicability of this method.     

Climbing the density functional ladder: nonempirical meta-generalized gradient approximation designed for molecules and solids

The electron density, its gradient, and the Kohn-Sham orbital kinetic energy density are the local ingredients of a meta-generalized gradient approximation (meta-GGA). We construct a meta-GGA density functional for the exchange-correlation energy that satisfies exact constraints without empirical parameters. The exchange and correlation terms respect two paradigms: one- or two-electron densities and slowly varying densities, and so describe both molecules and solids with high accuracy, as shown by extensive numerical tests. This functional completes the third rung of "Jacob's ladder" of approximations, above the local spin density and GGA rungs.     

Particle density and non-local kinetic energy density functional for two-dimensional harmonically confined Fermi vapors

We evaluate analytically some ground state properties of two-dimensional harmonically confined Fermi vapors with isotropy and for an arbitrary number of closed shells. We first derive a differential form of the virial theorem and an expression for the kinetic energy density in terms of the fermion particle density and its low-order derivatives. These results allow an explicit differential equation to be obtained for the particle density. The equation is third-order, linear and homogeneous. We also obtain a relation between the turning points of kinetic energy and particle densities, and an expression of the non-local kinetic energy density functional. Received 27 March 2001 and Received in final form 12 June 2001     

Meta-generalized gradient approximation: non-empirical construction and performance of a density functional

The local ingredients of a meta-generalized gradient approximation (meta-GGA) include the electron density, its gradient, and the Kohn–Sham orbital kinetic energy density. We discuss the strategy of constructing a successful meta-GGA density functional for the exchange-correlation energy that satisfies exact constraints without empirical parameters. The new feature of this functional is that it simultaneously respects the two paradigms of electronic structure theory: one- or two-electron densities and slowly-varying densities, and so is uniformly accurate for atoms, molecules and solids. Results of extensive numerical tests of the new functional are summarized and evaluated.     

Degenerate ground states and a fractional number of electrons in density and reduced density matrix functional theory

For a linear combination of electron densities of degenerate ground states, it is shown that the value of any energy functional is the ground state energy, if the energy functional is exact for ground state densities, size consistent, and translational invariant. The corresponding functional of kinetic and interaction energy is the linear combination of the functionals of the degenerate densities. Without invoking ensembles, it is shown that the energy functional of fractional number electrons is a series of straight lines interpolating its values at integers. These results underscore the importance of grand canonical ensemble formulation in density functional theory.     

Energy basis via decoherence

The question of the emergence of a preferred basis which is generally understood as that basis in which the reduced density matrix is driven to a diagonal (classically interpretable) form via environment induced decoherence is addressed. The exact solutions of the Caldeira-Leggett Master Equation are analyzed for a free particle and a harmonic oscillator system. In both cases, we see that the reduced density matrix is driven diagonal in the energy basis, which is momentum for the free particle and the number states for the harmonic oscillator. This seems to single out the energy basis as the preferred basis which is contrary to the general notion that it is the position basis which is selected since the coupling to the environment is via the position coordinates     

Investigation of the correlation potential from Kohn-Sham perturbation theory

Perturbation theory on the basis of the Kohn-Sham Hamiltonian leads to an implicit density functional for the correlation energy E(c). In this contribution we investigate the corresponding correlation potential v(c). It is shown that for finite systems the v(c) obtained by direct application of the optimized potential method diverges in the asymptotic region. The presence of unoccupied states, inherent in any perturbative form of E(c), is identified as the origin of this unphysical behavior. An approximate variational procedure is developed in order to avoid this difficulty. The potential resulting from this method qualitatively reproduces the shell structure of the exact atomic v(c).     

Kinetic energy of an inhomogeneous electron gas

It is well known that the kinetic energy of a system of N noninteracting particles in an external field V(r) in the nondegenerate ground state is a universal functional of the particle density (r) [1]. However, the explicit form of this functional is defined only for a certain class of functions (r). In particular, in the case of an almost constant or a slowly varying density we obtain the wellknown Thomas-Fermi-Weizsäcker-Kirzhnits functional [1, 2]. In [3] the kinetic energy functional of an electron in an atom is obtained in the WKB approximation. The kinetic energy of electrons with quantum numbers n andl is represented as the sum of two terms. The first term is written in the form of the density of the electrons with the given quantum numbers times some orthogonalized pseudopotential, which takes into account the orthogonality of their wave functions with respect to the core (its form is discussed below). The second term is the intrinsic kinetic energy of the electrons. The introduction of an orthogonalized pseudopotential is very convenient for the calculation of atomic properties [4, 5]. It is therefore of great interest to extend the results of [3] to the case of an electron gas in a metal and to obtain an expression for its orthogonalized crystalline pseudopotential. The solution of these problems is the aim of the present paper.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 117–120, July, 1977.     

The Pauli energy grows exponentially with the electronic localisation

We investigate the Pauli energy in atoms and molecules as a measure of electron localisation. Our results indicate that the Pauli energy has an exponential dependence on the number of localised electrons. This relationship yields to a kinetic energy density expression that depends on the electron density ρ(r) and the pair density ρ₂(r, r′). The proposed equation shows certain advantages over a similar orbital-free kinetic energy functional recently proposed by Delle Site and co-workers. The methodology introduced here is a novel approach for exploring electronic quantities with a partition scheme that might be useful for research in density functional theory.     

五边形石墨烯负载Pt单原子稳定性能的应变调控

基于密度泛函理论的第一性原理计算方法,研究了Pt原子在五边形石墨烯(PG)上的吸附与动力学行为.研究结果表明,单个Pt原子在PG上虽然具有较大的吸附能及较高的扩散势垒,却不能够在衬底形成均匀分散的单原子.这是因为,随Pt原子数增加,Pt_n(n=1, 2, 3)在PG上的平均结合能也逐渐增加,更倾向于形成团簇,该发现有效否定了之前的报道称Pt能在PG上形成稳定的单原子催化剂这一结论(Phys. Chem. Chem. Phys. 21, 12201 (2019)).基于此,我们考虑对PG施加双轴应变,随着拉伸应力增加,Pt金属原子间的平均结合能逐渐降低,当拉伸应变施加至12%左右时,单个Pt在衬底上的结合能与Pt₂在衬底上的平均结合能相等,从而实现PG上均匀分散的Pt单原子催化剂.该结果对五边形石墨烯基材料应变调控实现单原子催化剂提供理论借鉴.     

Treatment of Confinement in the Faddeev Approach to Three-Quark Problems

We present a method to solve the Faddeev integral equations of the semirelativistic constituent quark model. In such a model the quark–quark interaction is modeled by an infinitely rising confining potential and the kinetic energy is taken in a relativistic form. We solve the integral equations in Coulomb–Sturmian basis. This basis facilitates an exact treatment of the confining potentials.     

Derivation of the density functional theory from the cluster expansion

The density functional theory is derived from a cluster expansion by truncating the higher-order correlations in one and only one term in the kinetic energy. The formulation allows self-consistent calculation of the exchange correlation effect without imposing additional assumptions to generalize the local density approximation. The pair correlation is described as a two-body collision of bound-state electrons, and modifies the electron- electron interaction energy as well as the kinetic energy. The theory admits excited states, and has no self-interaction energy.     

On adhesive energies at bimetallic interfaces

The adhesive energies between densely packed planes of simple metal slabs are calculated as a function of separation. The assumed simple form of the electron density profile makes the calculation semianalytical and simple. Due to the improved treatment of the kinetic energy functional through the inclusion of the fourth-order density gradients, the results are in reasonable agreement with the results of self-consistent calculations and experiment.     

The exchange-correlation energy of a metallic surface

The exchange-correlation energy of a metal surface is analyzed in terms of the wavelength of the fluctuations which contribute to it. A scheme is proposed to interpolate between the short-wavelength region properly described by the local density functional approximation and the long-wavelength region for which an exact limiting form is established. This interpolation scheme is tested and verified for the soluble infinite barrier model and then applied to realistic density profiles.     

Deviations from hydrodynamics on the nanoscale

This study has examined the decay of an instantaneously imposed heat pulse on an equilibrium model of a dense fluid. The spatial extent of the initial pulse is quite small, of the order of 100 cubic nanometres; the amount of energy added to the system is only 5% of the total system kinetic energy. This small pulse decays quite rapidly, within several picoseconds, but the decay proceeds more slowly than predicted by the hydrodynamic equations. During the first picosecond of the decay, the kinetic energy is not equipartitioned, and a rapid process of energy transfer from kinetic energy to potential energy via interparticle interactions takes place. A new transport theory is developed that includes the ‘pre-hydrodynamic’ stage of evolution of non-equilibrium systems. Formally exact expressions for the local density, velocity, and kinetic energy (temperature) fields are developed in terms of Green's functions that depend on dynamic quantities, such as the mean-square displacement, averaged over the ensemble of initial states. No partial differential equations are involved.     

Finite Temperature Semiclassical Approach for Local Potentials

The nucleon distribution, kinetic energy density, free energy density and entropy density up to the second order in h are derived in a finite temperature semiclassical approach for two local potentials, harmonic oscillator and Woods-Saxon potentiale. The preaent results are compared with the exact quantum reaults for both potentials. The comparison justifies the extension of semiclassical approximation from zero temperature to finite temperature.