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Physical review. B, Condensed matter

Exact exchange and correlation corrections for large wave vectors

Xiaodong Zhu, A. W. Overhauser

Published Sep 15, 1984• 30 citations
Publisher

Abstract

We extend Niklasson's method in order to study the spin response of an electron liquid in a weak sinusoidal magnetic field. The many-body correction $G\ensuremath{-}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ caused by exchange and correlation is introduced to describe the correct wave-vector- and frequency-dependent spin susceptibility. The exact behavior of $G\ensuremath{-}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})$ in the large-$q$ limit is related to the pair distribution function $g(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ at $r=0$. ${G}_{\ensuremath{-}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})\ensuremath{\rightarrow}\frac{[4g(0)\ensuremath{-}1]}{3}$, as $q\ensuremath{\rightarrow}\ensuremath{\infty}$. At metallic densities this value is negative, opposite in sign to the limit at small wave vector. Thus the spin susceptibility for large wave vectors is suppressed, rather than enhanced, by many-body effects.

DOI
10.1103/physrevb.30.3158

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We extend Niklasson's method in order to study the spin response of an electron liquid in a weak sinusoidal magnetic field. The many-body correction G\ensuremath−(q\ensuremath→,\ensuremathω)G\ensuremath{-}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})G\ensuremath−(q\ensuremath→​,\ensuremathω) caused by exchange and correlation is introduced to describe the correct wave-vector- and frequency-dependent spin susceptibility. The exact behavior of G\ensuremath−(q\ensuremath→,\ensuremathω)G\ensuremath{-}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})G\ensuremath−(q\ensuremath→​,\ensuremathω) in the large-qqq limit is related to the pair distribution function g(r\ensuremath→)g(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})g(r\ensuremath→) at r=0r=0r=0. G\ensuremath−(q\ensuremath→,\ensuremathω)\ensuremath→[4g(0)\ensuremath−1]3{G}_{\ensuremath{-}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}},\ensuremath{\omega})\ensuremath{\rightarrow}\frac{[4g(0)\ensuremath{-}1]}{3}G\ensuremath−​(q\ensuremath→​,\ensuremathω)\ensuremath→3[4g(0)\ensuremath−1]​, as q\ensuremath→\ensuremath∞q\ensuremath{\rightarrow}\ensuremath{\infty}q\ensuremath→\ensuremath∞. At metallic densities this value is negative, opposite in sign to the limit at small wave vector. Thus the spin susceptibility for large wave vectors is suppressed, rather than enhanced, by many-body effects.
We analyze the spin and charge susceptibilities of a spin-polarized electron gas subject to a weak space- and time-dependent field coupled to the electronic spins, with the main attention given to the case of space dimensionality D=2.D=2.D=2. Exchange and correlations enter the dynamic susceptibilities through spin-dependent local-field factors ${G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(\mathbf{q},\ensuremath{\omega}).$ For an arbitrary degree of polarization, we determine the exact analytic expressions of ${G}_{\ensuremath{\sigma}}^{\ifmmode\pm\else\textpm\fi{}}(\mathbf{q},\ensuremath{\omega})$ in two limiting cases: (i) the limit of large wave number q at finite frequency $\ensuremath{\omega},$ already considered in D=3D=3D=3 by D. C. Marinescu and J. J. Quinn [Phys. Rev. B 56,56,56, 1114 (1997)]; and (ii) the static limit at small wave number. In the latter case, we obtain thermodynamic sum rules of general validity in both dimensionalities. Our work gives insight into many-body vertex corrections and basic information for calculations of the effects of the electron-electron interactions on physical properties.
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We have derived the exact limiting behavior of the many-body local fields ${G}_{\ifmmode\pm\else\textpm\fi{}}$(q,\ensuremath{\omega}) in a two-dimensional electron gas. At large wave vectors and finite frequencies these functions can be expressed in terms of g(0), the value at the origin of the pair correlation function of the electron gas. We find in particular that in this limit G+{G}_{+}G+​(\ensuremath{\infty},\ensuremath{\omega})=1-g(0), and ${G}_{\mathrm{\ensuremath{-}}}$(\ensuremath{\infty},\ensuremath{\omega})=g(0). Our results provide new insight into the problem of the many-body vertex corrections in this interacting system.
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.
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.
The effects of exchange and correlations on the diamagnetic susceptibility of an interacting electron gas are investigated using the double-time Green's-function method. The equation of motion of the appropriate Green's function is linearized using a moment-conserving decoupling approximation. The many-body corrections to the diamagnetic susceptibility at metallic densities are found to be small. The present expression reduces to the exact result in the high-density limit. Many of the results of earlier calculations also follow from our expression for the diamagnetic susceptibility.