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arXiv:1403.5358 · Physical Review A

Generalized local-density approximation and one-dimensional finite uniform electron gases

Pierre‐François Loos

Published May 30, 2014• 20 citations
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Abstract

We explicitly build a generalized local-density approximation (GLDA) correlation functional based on one-dimensional uniform electron gases (UEGs). The fundamental parameters of the GLDA---a generalization of the widely known local-density approximation used in density-functional theory---are the electronic density $\ensuremath{\rho}$ and a two-electron local parameter called the hole curvature $\ensuremath{\eta}$. The UEGs considered in this study are finite versions of the conventional infinite homogeneous electron gas and consist of $n$ electrons on an infinitely thin wire with periodic boundary conditions. We perform a comprehensive study of these finite UEGs at high, intermediate, and low densities using perturbation theory and quantum Monte Carlo calculations. We show that the present GLDA functional yields accurate estimates of the correlation energy for both weakly and strongly correlated one-dimensional systems and can be easily generalized to higher-dimensional systems.

DOI
10.1103/physreva.89.052523

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We explicitly build a generalized local-density approximation (GLDA) correlation functional based on one-dimensional uniform electron gases (UEGs). The fundamental parameters of the GLDA---a generalization of the widely known local-density approximation used in density-functional theory---are the electronic density \ensuremathρ\ensuremath{\rho}\ensuremathρ and a two-electron local parameter called the hole curvature \ensuremathη\ensuremath{\eta}\ensuremathη. The UEGs considered in this study are finite versions of the conventional infinite homogeneous electron gas and consist of nnn electrons on an infinitely thin wire with periodic boundary conditions. We perform a comprehensive study of these finite UEGs at high, intermediate, and low densities using perturbation theory and quantum Monte Carlo calculations. We show that the present GLDA functional yields accurate estimates of the correlation energy for both weakly and strongly correlated one-dimensional systems and can be easily generalized to higher-dimensional systems.
A density-functional theory is developed for fermions in one dimension, interacting via a $\ensuremath{\delta}$ function. Such systems provide a natural testing ground for questions of principle, as the local-density approximation should be highly accurate since for this interaction type the exchange contribution to the local-density approximation is intrinsically self-interaction-free. The exact-exchange contribution to the total energy is a local functional of the density. A local-density approximation for correlation is obtained using perturbation theory and Bethe ansatz results for the one-dimensional contact-interacting uniform Fermi gas. The ground-state energies are calculated for two finite systems, the analogs of helium and of Hooke's atom. The local-density approximation is shown to be excellent as expected.
We calculate the ground-state energy, pair correlation function, static structure factor, and momentum density of the one-dimensional electron fluid at high density using variational quantum Monte Carlo simulation. For an infinitely thin cylindrical wire the predicted correlation energy is found to fit nicely with a quadratic function of coupling parameter rs{r}_{s}rs​. The extracted exponent $\ensuremath{\alpha}$ of the momentum density for $k\ensuremath{\sim}{k}_{F}$ is used to determine the Tomonaga-Luttinger parameter ${K}_{\ensuremath{\rho}}$ as a function of rs{r}_{s}rs​ in the high-density regime. We find that the simulated static structure factor and pair correlation function for infinitely thin wires agree with our recent high-density theory [K. Morawetz et al., Phys. Rev. B 97, 155147 (2018)].
Accurate treatment of the electronic correlation in inhomogeneous electronic systems, combined with the ability to capture the correlation energy of the homogeneous electron gas, allows to reach high predictive power in the application of density-functional theory. For two-dimensional systems we can achieve this goal by generalizing our previous approximation [Phys. Rev. B 79, 085316 (2009)] to a parameter-free form, which reproduces the correlation energy of the homogeneous gas while preserving the ability to deal with inhomogeneous systems. The resulting functional is shown to be very accurate for finite systems with an arbitrary number of electrons with respect to numerically exact reference data.
Scaling arguments show that the exchange-correlation energy density of a spin-unpolarized homogeneous electron gas equals $A{\ensuremath{\omega}}^{4∕3}$, where $\ensuremath{\omega}$ is given by a simple functional of the momentum distribution n(k)n(k)n(k) and AAA is a constant. This functional possesses the property of yielding the true n(k)n(k)n(k) (including the discontinuity at the Fermi momentum, which is not reproduced by any of the presently known density matrix functionals) upon minimization of the energy density eee constrained by the electron density $\ensuremath{\rho}$. At the limit of $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\omega}$ coincides with $\ensuremath{\rho}$ and the dependence of $\ensuremath{\omega}$ on n(k)n(k)n(k), which can be readily recovered from the actual values of eee and n(k)n(k)n(k), is given by an explicit analytical formula. Application of this formalism to general Coulombic systems yields a local version of the density matrix theory, in which the exchange-correlation energy is given by a spatial integral of an expression that depends on the amplitudes of natural orbitals at a given point and the respective occupation numbers.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
Semilocal exchange-correlation functionals are the most accurate, realistic and widely used ones to describe the complex many-electron effects of two-dimensional quantum systems. Beyond local density approximation, the generalized gradient approximations(GGAs) are designed using reduced density gradient as main ingredient. An enhancement factor is constructed using the inhomogeneity parameter of GGAs by taking care of the low and high density behaviors of it. Thus, the exchange energy functional proposed by making use of the aforementioned enhancement factor, significantly reduces the error compare to the previously proposed gradient approximations. Another enhancement factor and corresponding energy functional is also constructed using the inhomogeneity parameter originally introduced by Becke [J. Chem. Phys. 109, 2092 (1998)]. Comprehensive testing and performance of both the functionals are demonstrated with respect to the exact exchange formalism by considering two-dimensional parabolically confined quantum dots with varying particle number and confinement strength as a test case.