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Physical review. B, Condensed matter

Electron correlation energies from scaled exchange-correlation kernels: Importance of spatial versus temporal nonlocality

Manfred Lein, Eberhard K. U. Gross, John P. Perdew

Published May 15, 2000• 119 citations
Publisher

Abstract

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 ${f}_{\mathrm{xc}}.$ 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 ${(f}_{\mathrm{xc}}$ $=$ 0) makes the correlation energy per electron too negative by about 0.5 eV, the adiabatic local-density approximation $[{f}_{\mathrm{xc}}$ $=$ ${f}_{\mathrm{xc}}(q$ $=$ $0,\ensuremath{\omega}$ $=$ 0)] makes a comparable error in the opposite direction. The adiabatic nonlocal approximation $[{f}_{\mathrm{xc}}$ $=$ ${f}_{\mathrm{xc}}(q,\ensuremath{\omega}$ $=$ 0)] reduces this error to about 0.1 eV, and inclusion of the full frequency dependence $[{f}_{\mathrm{xc}}$ $=$ ${f}_{\mathrm{xc}}(q,\ensuremath{\omega})]$ 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.

DOI
10.1103/physrevb.61.13431

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Within density functional theory, a coordinate-scaling relation for the coupling-constant dependence of the exchange-correlation kernel fxc(r,r\ensuremath′;\ensuremathω){f}_{\mathrm{xc}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})fxc​(r,r\ensuremath′;\ensuremathω) 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,\ensuremathω0,\ensuremath{\omega}0,\ensuremathω === 0)] makes a comparable error in the opposite direction. The adiabatic nonlocal approximation [fxc[{f}_{\mathrm{xc}}[fxc​ === fxc(q,\ensuremathω{f}_{\mathrm{xc}}(q,\ensuremath{\omega}fxc​(q,\ensuremathω === 0)] reduces this error to about 0.1 eV, and inclusion of the full frequency dependence [fxc[{f}_{\mathrm{xc}}[fxc​ === fxc(q,\ensuremathω)]{f}_{\mathrm{xc}}(q,\ensuremath{\omega})]fxc​(q,\ensuremathω)] 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.
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.
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.
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.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function LLL. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential WWW in the BSE for LLL leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.