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Physical Review B

Dynamical exchange-correlation potentials in the spin channel for the two-dimensional electron liquid

Zhixin Qian

Published Aug 10, 2005• 3 citations
Publisher

Abstract

The exchange-correlation kernel in the spin channel in an electron liquid has the structure ${f}_{xc,\ensuremath{-}}^{L,T}(q,\ensuremath{\omega})\stackrel{q\ensuremath{\rightarrow}0}{\ensuremath{\rightarrow}}A(\ensuremath{\omega})∕{q}^{2}+{B}^{L,T}(\ensuremath{\omega})$ in the limit of the long wavelength. Here $L$ denotes the longitudinal component and $T$ the transverse component relative to the direction of the wave vector $\mathbf{q}$. A collection of exact results for $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ is obtained at limiting low and high frequency, respectively, in two dimensions. Based on these results, we further give approximate interpolations for $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ at all frequencies in the paramagnetic case.

DOI
10.1103/physrevb.72.075115

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The exchange-correlation kernel in the spin channel in an electron liquid has the structure fxc,\ensuremath−L,T(q,\ensuremathω)\ensuremath→q\ensuremath→0A(\ensuremathω)∕q2+BL,T(\ensuremathω){f}_{xc,\ensuremath{-}}^{L,T}(q,\ensuremath{\omega})\stackrel{q\ensuremath{\rightarrow}0}{\ensuremath{\rightarrow}}A(\ensuremath{\omega})∕{q}^{2}+{B}^{L,T}(\ensuremath{\omega})fxc,\ensuremath−L,T​(q,\ensuremathω)\ensuremath→q\ensuremath→0​A(\ensuremathω)∕q2+BL,T(\ensuremathω) in the limit of the long wavelength. Here LLL denotes the longitudinal component and TTT the transverse component relative to the direction of the wave vector q\mathbf{q}q. A collection of exact results for A(\ensuremathω)A(\ensuremath{\omega})A(\ensuremathω) and BL,T(\ensuremathω){B}^{L,T}(\ensuremath{\omega})BL,T(\ensuremathω) is obtained at limiting low and high frequency, respectively, in two dimensions. Based on these results, we further give approximate interpolations for A(\ensuremathω)A(\ensuremath{\omega})A(\ensuremathω) and BL,T(\ensuremathω){B}^{L,T}(\ensuremath{\omega})BL,T(\ensuremathω) at all frequencies in the paramagnetic case.
The components of the exchange-correlation kernel tensor of an isotropic electron liquid in the spin channel have the structure ${f}_{\mathrm{xc},\ensuremath{-}}^{L,T}(q,\ensuremath{\omega})\stackrel{\stackrel{\ensuremath{\rightarrow}}{q}0}{\ensuremath{\rightarrow}}A(\ensuremath{\omega}{)/q}^{2}{+B}^{L,T}(\ensuremath{\omega}),$ where L denotes the longitudinal component and T the transverse component relative to the direction of the wave vector q.\mathbf{q}.q. In this paper we calculate analytically the high- and low-frequency limits of $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ and combine these limiting forms with the Kramers-Kr\"onig dispersion relations to obtain approximations for $A(\ensuremath{\omega})$ and ${B}^{L,T}(\ensuremath{\omega})$ at all frequencies.
The high-frequency limits of the singular component $A(\ensuremath{\omega})$ of the small wave-vector expansion of the longitudinal (L)(L)(L) and transverse (T)(T)(T) components of the spin-resolved exchange-correlation kernel tensor ${f}_{xc,\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})=\ensuremath{-}v(q){G}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})$ in a two-dimensional isotropic electron liquid with arbitrary spin polarization are studied. Here ${G}_{\ensuremath{\sigma}{\ensuremath{\sigma}}^{\ensuremath{'}}}^{L,T}(q,\ensuremath{\omega})$ is the spin-resolved local-field factor, v(q)v(q)v(q) is the Coulomb interaction in momentum space, and $\ensuremath{\sigma}$ denotes spin. Particularly, the real part of $A(\ensuremath{\omega})$ is found to be logarithmically divergent at large $\ensuremath{\omega}$. The large wave-vector structure of the corresponding spin-resolved static structure factor is also established.
Spin correlations in an interacting electron liquid are studied in the high-frequency limit and in both two and three dimensions. The third-moment sum rule is evaluated and is used to derive exact limiting forms (at both long and short wavelengths) for the spin-antisymmetric local-field factor, ${limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega}).$ In two dimensions ${limop}_{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\omega}}\ensuremath{\infty}}{G}_{\ensuremath{-}}(\mathbf{q},\ensuremath{\omega})$ is found to diverge as 1/q1/q1/q at long wavelengths, and the spin-antisymmetric exchange-correlation kernel of time-dependent spin density functional theory diverges as 1/q2{1/q}^{2}1/q2 in both two and three dimensions. These signal a failure of the local-density approximation, one that can be redressed by alternative approaches.
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.
The structure of the space-time-dependent spin-correlation functions for an anisotropic Heisenberg paramagnet with cylindrical symmetry at elevated temperature is analyzed in terms of a two-parameter Gaussian representation representation of the generalized diffusivity. The parameters are chosen so as to exactly preserve the frequency moments, i.e., ${{〈{\ensuremath{\omega}}^{2n}〉}_{\mathrm{K}}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ ($\ensuremath{\alpha}=z, x, or y$; n=0,1,2n=0, 1, 2n=0,1,2) of the longitudinal and the transverse spectral functions. These moments are calculated for arbitrary spin SSS, arbitrary range of the exchange interactions, and arbitrary dimensionality. The results for the correlation functions are worked out for one, two, and three dimensions. In the extreme transverse limit, the results are compared with the exactly soluble XY\mathrm{XY}XY model for a one-dimensional spin-\textonehalf{} system with only nearest-neighbor (transverse) exchange. The agreement of the results is found to be satisfactory.
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.