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International Journal of Modern Physics B

EXCHANGE–CORRELATION KERNEL IN TIME-DEPENDENT DENSITY FUNCTIONAL THEORY DERIVED FROM MANY-BODY THEORY

K. KARLSSON, F. Aryasetiawan

Published Mar 20, 2004• 2 citations
Publisher

Abstract

We derive a simplified Bethe–Salpeter equation for calculating optical absorption based on the assumption of a local electron–hole interaction. The original four-point equation for the kernel is reduced to a two-point one. A connection to the exchange–correlation kernel in time-dependent density functional theory can be established. The resulting fxc is found to be -W/2 where W contains only the short-range (local) part of the Coulomb screened interaction. This simple approximation was successfully applied to optical absorption spectra of some excitonic crystals, reproducing not only the continuum excitons but also the bound ones.

DOI
10.1142/s0217979204024537

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We derive a simplified Bethe–Salpeter equation for calculating optical absorption based on the assumption of a local electron–hole interaction. The original four-point equation for the kernel is reduced to a two-point one. A connection to the exchange–correlation kernel in time-dependent density functional theory can be established. The resulting fxc is found to be -W/2 where W contains only the short-range (local) part of the Coulomb screened interaction. This simple approximation was successfully applied to optical absorption spectra of some excitonic crystals, reproducing not only the continuum excitons but also the bound ones.
We investigate the description of excitonic effects within time-dependent density-functional theory (TDDFT). The exchange-correlation kernel fxc{f}_{xc}fxc​ introduced in TDDFT allows a clear separation of quasiparticle and excitonic effects. Using a diagrammatic representation for fxc{f}_{xc}fxc​, we express its excitonic part fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ in terms of the effective vertex function $\ensuremath{\Lambda}$. The latter fulfills an integral equation that thereby establishes the exact correspondence between TDDFT and the standard many-body approach based on the Bethe-Salpeter equation (BSE). The diagrammatic structure of the kernel in the equation for $\ensuremath{\Lambda}$ suggests the possibility of strong cancellation effects. Should the cancellation take place, already the first-order approximation to fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ is sufficient. A potential advantage of TDDFT over the many-body BSE method is thus dependent on the efficiency of the above-quoted cancellation. We explicitly verify this for an analytically solvable two-dimensional two-band model. The calculations confirm that the low-order fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ perfectly describes the bound exciton as well as the excitonic effects in the continuous spectrum in a wide range of the electron-hole coupling strength.
This chapter presents various techniques from many-body theory to construct time-dependent exchange-correlation (xc) functionals. The first section introduces a perturbation theory along the so-called adiabatic connection. This leads to a perturbative expansion of the xc potential, with exact exchange representing the lowest order. The following section introduces nonequilibrium Green's functions along the Keldysh contour. A Keldysh action principle is formulated which yields a formally exact representation of the time-dependent xc potential. As an alternative, the xc potential is expressed through the so-called Sham-Schlüter equation. The third section establishes a connection between the xc kernels of time-dependent density-functional theory in linear response and many-body perturbation theory. A diagrammatic expansion of the xc kernel is given. Finally, an expression for the xc kernel is derived starting from the Bethe-Salpeter equation. This xc kernel gives excellent agreement with experimental optical absorption spectra of insulators, including excitonic features.
We develop a method to calculate the polarizability of a many-electron system within Green's function theory in a similar way as within time-dependent density functional theory and apply it to Si and GaAs. The method joins the computational simplicity of the latter with the accuracy of the former approach. The resulting exchange-correlation many-body kernel fxcFQP{f}_{\mathrm{xc}}^{\mathrm{FQP}}fxcFQP​ and optical spectra are in good agreement with those obtained by the solution of the Bethe-Salpeter equation.
For the calculation of neutral excitations, time-dependent density functional theory (TDDFT) is an exact reformulation of the many-body time-dependent Schrödinger equation, based on knowledge of the density instead of the many-body wavefunction. The density can be determined in an efficient scheme by solving one-particle non-interacting Schrödinger equations—the Kohn–Sham equations. The complication of the problem is hidden in the—unknown—time-dependent exchange and correlation potential that appears in the Kohn–Sham equations and for which it is essential to find good approximations. Many approximations have been suggested and tested for finite systems, where even the very simple adiabatic local-density approximation (ALDA) has often proved to be successful. In the case of solids, ALDA fails to reproduce optical absorption spectra, which are instead well described by solving the Bethe–Salpeter equation of many-body perturbation theory (MBPT). On the other hand, ALDA can lead to excellent results for loss functions (at vanishing and finite momentum transfer). In view of this and thanks to recent successful developments of improved linear-response kernels derived from MBPT, TDDFT is today considered a promising alternative to MBPT for the calculation of electronic spectra, even for solids. After reviewing the fundamentals of TDDFT within linear response, we discuss different approaches and a variety of applications to extended systems.
In time-dependent density-functional theory, an important quantity is the exchange-correlation kernel whose knowledge allows for the calculations of the excitation spectrum of electronic systems. In most applications, the exchange-correlation kernel is approximated by the adiabatic local-density approximation, ignoring the frequency dependence and nonlocality. To gain insight into the nature of the exact exchange-correlation kernel, we have considered a two-dimensional one-band Hubbard model. The calculated exact exchange-correlation kernel reveals a number of striking features. It has a weak energy dependence up to the main excitation energy. On the other hand, the exact kernel shows a strong energy dependence in energy regions, where there are many-body excitations that are not contained in the local-density approximation, even though these excitations have small weight. It is found that a static approximation can well reproduce the main excitation peaks, but the satellite structures arising from many-body interaction are unlikely to be accounted for with a static approximation.
For two prototype systems, we calculate the exact exchange-correlation kernels ${f}_{\mathrm{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of time-dependent density functional theory. fxc{f}_{\mathrm{xc}}fxc​, the key quantity for optical absorption spectra of electronic systems, is normally subject to uncontrolled approximation. We find that, up to the first excitation energy, the exact fxc{f}_{\mathrm{xc}}fxc​ has weak frequency dependence and a simple, though nonlocal, spatial form. For higher excitations, the spatial behavior and frequency dependence become more complex. The accuracy of the underlying exchange-correlation potential is of crucial importance.