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arXiv:0902.1416 · Physical Review Letters

Lower Bounds on the Exchange-Correlation Energy in Reduced Dimensions

E. Räsänen, Stefano Pittalis, Klaus Capelle, César R. Proetto

Published May 20, 2009• 43 citations
arXivHTMLPDF

Abstract

Bounds on the exchange-correlation energy of many-electron systems are derived and tested. By using universal scaling properties of the electron-electron interaction, we obtain the exponent of the bounds in three, two, one, and quasione dimensions. From the properties of the electron gas in the dilute regime, the tightest estimate to date is given for the numerical prefactor of the bound, which is crucial in practical applications. Numerical tests on various low-dimensional systems are in line with the bounds obtained and give evidence of an interesting dimensional crossover between two and one dimensions.

DOI
10.1103/physrevlett.102.206406

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Bounds on the exchange-correlation energy of many-electron systems are derived and tested. By using universal scaling properties of the electron-electron interaction, we obtain the exponent of the bounds in three, two, one, and quasione dimensions. From the properties of the electron gas in the dilute regime, the tightest estimate to date is given for the numerical prefactor of the bound, which is crucial in practical applications. Numerical tests on various low-dimensional systems are in line with the bounds obtained and give evidence of an interesting dimensional crossover between two and one dimensions.
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Reduced density-matrix functional theory (RDMFT) has become an appealing alternative to density-functional theory to describe electronic properties of strongly correlated systems. Here we derive exact conditions for the suitability of RDMFT to describe the two-dimensional homogeneous electron gas, which is the base system for semiconductor quantum dots and quantum Hall devices, for example. Following the method of Cioslowski and Pernal [J. Chem. Phys. 111, 3396 (1999)] we focus on the properties of power functionals of the form $f(n,{n}^{\ensuremath{'}})={(n{n}^{\ensuremath{'}})}^{\ensuremath{\alpha}}$ for the scaling function in the exchange-correlation energy. We show that in order to have stable and analytic solutions, and for fff to satisfy the homogeneous scaling constraint, the power is restricted to $1/4\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}3/4$. Applying a reasonable ansatz for the momentum distribution and the lower bound for the exchange-correlation energy tightens the physical regime further to $0.64\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\leqslant}0.75$.
We show that the effective electron-electron interaction ${v}_{\text{eff}}[n](\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})=\mathrm{erf}\left[|\mathbf{r}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}}|b(n)\right]/|\mathbf{r}\ensuremath{-}{\mathbf{r}}^{\ensuremath{'}}|$, with nnn being the electronic density and b(n)b(n)b(n) fixed from the compressibility sum rule, is remarkably accurate in both three and two dimensions, competing with the actual state-of-the-art exchange-correlation static kernels. Using this result, we propose a simple isotropic approximation for the quasi-two-dimensional regime, which gives a realistic wave vector analysis of the correlation energy, in the context of the linear response time-dependent density functional theory.
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.