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arXiv:1003.0990 · Physical Review B

Antiadiabatic limit of the exchange-correlation kernels of an inhomogeneous electron gas

V. U. Nazarov, I. V. Tokatly, Stefano Pittalis, Giovanni Vignale

Published Jun 2, 2010• 9 citations
arXivHTMLPDF

Abstract

We express the high-frequency (antiadiabatic) limit of the exchange-correlation (xc) kernels of an inhomogeneous electron gas in terms of the following equilibrium properties: the ground-state density, the xc kinetic stress tensor, the pair-correlation function and the ground-state xc potential. Of these quantities, the first three are amenable to exact evaluation by quantum Monte Carlo methods while the last can be obtained from the inversion of the Kohn-Sham equation for the ground-state orbitals. The exact scalar kernel, in this limit, is found to be of very long range in space, at variance with the kernel that is used in the standard local density approximation. The antiadiabatic xc kernels will be useful in calculations of excitation energies by time-dependent density-functional theory in atoms, molecules, and solids, and provide a solid basis for interpolation between the low- and high-frequency limits of the xc kernels.

DOI
10.1103/physrevb.81.245101

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We express the high-frequency (antiadiabatic) limit of the exchange-correlation (xc) kernels of an inhomogeneous electron gas in terms of the following equilibrium properties: the ground-state density, the xc kinetic stress tensor, the pair-correlation function and the ground-state xc potential. Of these quantities, the first three are amenable to exact evaluation by quantum Monte Carlo methods while the last can be obtained from the inversion of the Kohn-Sham equation for the ground-state orbitals. The exact scalar kernel, in this limit, is found to be of very long range in space, at variance with the kernel that is used in the standard local density approximation. The antiadiabatic xc kernels will be useful in calculations of excitation energies by time-dependent density-functional theory in atoms, molecules, and solids, and provide a solid basis for interpolation between the low- and high-frequency limits of the xc kernels.
The dynamical exchange-correlation kernel f{xc} of a nonuniform electron gas is an essential input for the time-dependent density-functional theory of electronic systems. The long-wavelength behavior of this kernel is known to be of the form f{xc}=alpha/q{2} where q is the wave vector and alpha is a frequency-dependent coefficient. We show that in the limit of weak nonuniformity the coefficient alpha has a simple and exact expression in terms of the ground-state density and the frequency-dependent kernel of a uniform electron gas at the average density. We present an approximate evaluation of this expression for Si and discuss its implications for the theory of excitonic effects.
The key quantity in the formulation of linear-response theory within time-dependent density-functional theory is the so-called exchange-correlation (xc) kernel. This chapter discusses the properties of the xc kernel. In general, it is a complex and frequency-dependent object, and a functional of the ground-state density. Several exact properties such as the zero-force theorem, static and high-frequency limit, and the long-range behaviour are discussed. The xc kernel can be related to a time-dependent variational principle provided one pays attention to certain time-dependent boundary conditions to ensure causality. The adiabatic approximation of the xc kernel is discussed. Finally, the third section of the chapter reviews the xc kernel of the homogeneous electron liquid. Many of its exact properties are covered, such as the high- and low-frequency limit and the long-wavevector limit. Three widely used parametrizations are presented and the analytical behaviour in the complex frequency plane is investigated.
The exact exchange-correlation (xc) kernel ${f}_{\text{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of linear response time-dependent density functional theory is computed over a wide range of frequencies for three canonical one-dimensional finite systems. Methods used to ensure the numerical robustness of fxc{f}_{\text{xc}}fxc​ are set out. The frequency dependence of fxc{f}_{\text{xc}}fxc​ is found to be largely due to its analytic structure, i.e., its singularities at certain frequencies, which are required in order to capture particular transitions, including those of double excitation character. However, within the frequency range of the first few interacting excitations, fxc{f}_{\text{xc}}fxc​ is approximately $\ensuremath{\omega}$ independent, meaning the exact adiabatic approximation ${f}_{\text{xc}}(\ensuremath{\omega}=0)$ remedies the failings of the local density approximation and random phase approximation for these lowest transitions. The key differences between the exact fxc{f}_{\text{xc}}fxc​ and its common approximations are analyzed, and cannot be eliminated by exploiting the limited gauge freedom in fxc{f}_{\text{xc}}fxc​. The optical spectrum benefits from using as accurate as possible an fxc{f}_{\text{xc}}fxc​ and ground-state xc potential, while maintaining exact compatibility between the two is of less importance.
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.
The scalar f(xc) and tensor f(xc) exchange-correlation (xc) kernels are key ingredients of the time-dependent density functional theory and the time-dependent current density functional theory, respectively. We derive a comparatively simple relation between these two kernels under the assumption that the dynamic xc can be considered "weak." A calculation of the frequency-dependent dielectric function of silicon using this relation in conjunction with Vignale-Kohn f(xc) demonstrates a potential of our method to account for the dynamic many-body effects within the rigorous scheme of time-dependent density functional theory. Our formula provides a bridge between the scalar f(xc), which directly enters many applications, and the tensor f(xc) which, due to its locality in space, is much easier to approximate.
The interacting and the Kohn-Sham static density-density response functions for different one-dimensional two-electron singlet systems are reconstructed numerically. From their inverse we obtain the exact static exchange-correlation kernel. This quantity represents the adiabatically exact approximation of the frequency-dependent exchange-correlation kernel that is crucial for time-dependent linear density-response theory. We investigate its performance for nonlocal perturbations and analyze its sum rule properties. We also compute the adiabatically exact transition energies that follow from the static kernel within linear-response theory.