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arXiv:cond-mat/0505142 · Physical Review B

Simple model of the static exchange-correlation kernel of a uniform electron gas with long-range electron-electron interaction

Julien Toulouse

Published Jul 14, 2005• 21 citations
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Abstract

A simple approximate expression in real and reciprocal spaces is given for the static exchange-correlation kernel of a uniform electron gas interacting with the long-range part only of the Coulomb interaction. This expression interpolates between the exact asymptotic behaviors of this kernel at small and large wave vectors which in turn requires, among other things, information from the momentum distribution of the uniform electron gas with the same interaction that have been calculated in the ${G}_{0}{W}_{0}$ approximation. This exchange-correlation kernel as well as its complement analog associated to the short-range part of the Coulomb interaction are more local than the Coulombic exchange-correlation kernel and constitute potential ingredients in approximations for recent adiabatic connection fluctuation-dissipation and/or density functional theory approaches of the electronic correlation problem based on a separate treatment of long- and short-range interaction effects.

DOI
10.1103/physrevb.72.035117

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A simple approximate expression in real and reciprocal spaces is given for the static exchange-correlation kernel of a uniform electron gas interacting with the long-range part only of the Coulomb interaction. This expression interpolates between the exact asymptotic behaviors of this kernel at small and large wave vectors which in turn requires, among other things, information from the momentum distribution of the uniform electron gas with the same interaction that have been calculated in the G0W0{G}_{0}{W}_{0}G0​W0​ approximation. This exchange-correlation kernel as well as its complement analog associated to the short-range part of the Coulomb interaction are more local than the Coulombic exchange-correlation kernel and constitute potential ingredients in approximations for recent adiabatic connection fluctuation-dissipation and/or density functional theory approaches of the electronic correlation problem based on a separate treatment of long- and short-range interaction effects.
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.
Within density functional theory, a coordinate-scaling relation for the coupling-constant dependence of the exchange-correlation kernel ${f}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})$ is utilized to express the correlation energy of a many-electron system in terms of fxc.{f}_{\mathrm{xc}}.fxc​. As a test of several of the available approximations for the exchange-correlation kernel, or equivalently the local-field factor, we calculate the uniform-gas correlation energy. While the random phase approximation (fxc{(f}_{\mathrm{xc}}(fxc​ === 0) makes the correlation energy per electron too negative by about 0.5 eV, the adiabatic local-density approximation [fxc[{f}_{\mathrm{xc}}[fxc​ === fxc(q{f}_{\mathrm{xc}}(qfxc​(q === $0,\ensuremath{\omega}$ === 0)] makes a comparable error in the opposite direction. The adiabatic nonlocal approximation [fxc[{f}_{\mathrm{xc}}[fxc​ === ${f}_{\mathrm{xc}}(q,\ensuremath{\omega}$ === 0)] reduces this error to about 0.1 eV, and inclusion of the full frequency dependence [fxc[{f}_{\mathrm{xc}}[fxc​ === ${f}_{\mathrm{xc}}(q,\ensuremath{\omega})]$ in an approximate parametrization reduces it further to less than 0.02 eV. We also report the wave-vector analysis and the imaginary-frequency analysis of the correlation energy for each choice of kernel.
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.
We provide the expression of the exchange-correlation potential in the adiabatic connection fluctuation-dissipation (ACFD) framework, for arbitrary time-dependent (TD) kernels. We investigate the asymptotic behavior of the ACFD potential in three relevant approximations: the random-phase approximation, the exact-exchange kernel in two-electron systems, and the adiabatic local-density approximation. We show that these potentials have the expected $\ensuremath{-}{1/r+Q/r}^{3}\ensuremath{-}\ensuremath{\alpha}{/(2r}^{4})$ tail (in closed-shell systems with spherical symmetry), where Q and $\ensuremath{\alpha}$ depend on the TD kernel and reflect the physics included in each approximation. We also discuss approximate ACFD potentials that are much simpler to compute than the exact ones while being likely of reasonable accuracy.
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.
We propose a method that employs functionals of the one-electron reduced density matrix (density matrix) to capture long-range effects of electron correlation. The complementary short-range regime is treated with density functionals. In an effort to find approximations for the long-range density-matrix functional, a modified power functional is applied to the homogeneous electron gas with Coulomb interactions replaced by their corresponding long-range counterparts. For the power $\ensuremath{\beta}=1/2$ and the range-separation parameter $\ensuremath{\omega}=1/{r}_{s}$, the functional reproduces the correlation and the kinetic correlation energies with a remarkable accuracy for intermediate and large values of rs{r}_{s}rs​. Analysis of the Euler equation corresponding to this functional reveals correct rs{r}_{s}rs​ expansion of the correlation energy in the limit of large rs{r}_{s}rs​. The first expansion coefficient is in very good agreement with that obtained from the modified Wigner-Seitz model.