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Physical Review B

Semilocal dynamical correlation with increased localization

Lucian A. Constantin, Eduardo Santos Fabiano, Fabio Della Sala

Published Jul 20, 2012• 50 citations
Publisher

Abstract

Using a localization technique for the correlation energy density and a linear response constraint, we construct a localized semilocal model for dynamical correlation. The model is incorporated into a meta-generalized gradient approximation functional which is very accurate for jellium surfaces, Hooke's atom at all regimes, and works well both in conjunction with a semilocal exchange and together with local and nonlocal exact exchange.

DOI
10.1103/physrevb.86.035130

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Using a localization technique for the correlation energy density and a linear response constraint, we construct a localized semilocal model for dynamical correlation. The model is incorporated into a meta-generalized gradient approximation functional which is very accurate for jellium surfaces, Hooke's atom at all regimes, and works well both in conjunction with a semilocal exchange and together with local and nonlocal exact exchange.
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.
The frequency-dependent exchange-correlation potential, which appears in the usual Kohn-Sham formulation of a time-dependent linear response problem, is a strongly nonlocal functional of the density, so that a consistent local density approximation generally does not exist. This problem can be avoided by choosing the current density as the basic variable in a generalized Kohn-Sham theory. This theory admits a local approximation which, for fixed frequency, is exact in the limit of slowly varying densities and perturbing potentials.
The exact time-dependent correlation energy in time-dependent density-functional theory is shown to sometimes become positive, which is impossible with most present TDDFT approximations. Both the correlation potential and energy can be strongly nonlocal in time. A new inequality is derived for the time-dependent exchange-correlation energy. The correlation energy appears to scale to a constant function of scaled time in the high-density limit. In the linear response regime, the correlation energy is shown to become purely adiabatic, but the correlation potential is generally nonadiabatic. The usefulness of the virial theorem as a test of numerical accuracy is demonstrated. All results are found or inspired by exact numerical solution of a simple model system (Hooke’s atom), and inversion of the corresponding Kohn–Sham equations.
We have recently proposed an approximation for the dynamical exchange-correlation (XC) potentials of time-dependent current-density functional theory beyond the local density approximation [Phys. Rev. Lett. 97, 036403 (2006)]. The novel feature of the approximation is that the dependence of the dynamical XC potentials upon the density gradient and other inhomogeneity parameters (e.g., the Laplacian of the density and the kinetic energy density) is introduced by applying the generalized gradient approximation (GGA) and meta-GGA to the calculation of the XC stress tensor. The scheme may allow a more accurate treatment of the dynamical properties of an inhomogeneous system, while reducing to the exact form for slowly varying densities and slowly varying external potentials. In this work, we present in detail the derivation of this XC potential, spell out the underlying assumptions, and explain their physical basis.
Approximations for the ground-state exchange-correlation potential of density-functional theory have reached a high level of sophistication. By contrast, time- or frequency-dependent exchange-correlation potentials are still being treated in a local approximation. Here we propose a novel approximation scheme, which effectively brings the power of the generalized gradient approximation (GGA) and meta-GGA to time-dependent density-functional theory. The theory should allow a more accurate treatment of strongly inhomogeneous electronic systems (e.g. molecular junctions) while remaining essentially exact for slowly varying densities and slowly varying external potentials.
Rigorous extension of semilocal collinear functionals to noncollinear DFT using SU(2)SU(2)SU(2) rotations
In the presence of spin-orbit coupling and in geometrically frustrated materials, a noncollinear treatment the magnetization density is essential. However, in density functional theory most exchange--correlation functional approximations were originally developed for locally collinear magnetization. Many practical approaches to noncollinear DFT have emerged over the past decade. However, a first-principles connection between widely used semilocal collinear functionals and their noncollinear generalizations remains lacking. In this work, a locally exact relation between collinear and noncollinear exchange--correlation functionals is derived at the level of gradient expansions within a u(2)u(2)u(2) matrix representation of the energy functional. Within this framework, collinear semilocal variables naturally acquire distinct dependencies on transverse and longitudinal magnetization gradient components. The widely used Scalmani--Frisch scheme emerges as a first-order approximation. The transformation of collinear functional derivatives to noncollinear space is implemented through numerically robust SU(2)SU(2)SU(2) rotations. A consistent description of local magnetic torques is demonstrated for the prototypical spin-frustrated Cr3_33​ cluster. The approach further extends to fully nonlocal functionals and provides a direct route towards numerically stable relativistic response calculations. The influence on magnetic properties in presence of spin-orbit coupling is illustrated through calculations of hyperfine couplings in the high-spin ground states of uranium and the uranium ion.