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Physical review. B./Physical review. B

Simple effective interaction for dimensional crossover

Lucian A. Constantin

Published Mar 9, 2016• 31 citations
Publisher

Abstract

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 $n$ being the electronic density and $b(n)$ 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.

DOI
10.1103/physrevb.93.121104

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Physical Review A•Oct 6, 2006•1 citation
We show that the effective electron-electron interaction veff[n](r,r\ensuremath′)=erf[∣r\ensuremath−r\ensuremath′∣b(n)]/∣r\ensuremath−r\ensuremath′∣{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{'}}|veff​[n](r,r\ensuremath′)=erf[∣r\ensuremath−r\ensuremath′∣b(n)]/∣r\ensuremath−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.
The interacting and the Kohn-Sham static density-density response functions for different one-dimensional two-electron singlet systems are reconstructed numerically. From their inverse we obtain the exact static exchange-correlation kernel. This quantity represents the adiabatically exact approximation of the frequency-dependent exchange-correlation kernel that is crucial for time-dependent linear density-response theory. We investigate its performance for nonlocal perturbations and analyze its sum rule properties. We also compute the adiabatically exact transition energies that follow from the static kernel within linear-response theory.
We have evaluated the density-density response of the two-dimensional electron gas at zero temperature by solving the Dyson equation for the particle-hole Green's function, including exchange Coulomb matrix elements and short-range contributions in the ladder approximation. We study the effect of these correlations on the total energy, compressibility per particle, local field factor G ( q ), static structure factor and pair-correlation function. Results are compared with the normal random-phase approximation, local field theories and quantum Monte Carlo calculations.
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.
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.
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.
A formally exact expression for the interaction density response function $\ensuremath{\chi}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$ exists in terms of (i) its noninteracting counterpart ${\ensuremath{\chi}}_{0}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$ and (ii) an exchange (x)(x)(x)-correlation (c)(c)(c) kernel ${f}_{xc}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$. In the absence of a first-principles theory for the $\ensuremath{\omega}$ dependence of fxc{f}_{xc}fxc​, the adiabatic approximation is most frequently made in this term, to construct a workable time-dependent density-functional theory. In the present study, a proposal is put forward to avoid the adiabatic approximation by working in the exchange-only limit in which fxc{f}_{xc}fxc​ is set equal to ${f}_{x}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$. We then refer to a result for the exchange energy given by Pines and Nozieres to motivate the assumption that ${f}_{x}={f}_{x}[\mathrm{Im}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\chi}}_{0}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})]$. The essential proposal here is therefore that the integral equation to be solved for the interacting density response function $\ensuremath{\chi}$ is in the exchange-only case characterized entirely by the noninteracting response function ${\ensuremath{\chi}}_{0}$.