Physical Review A

Second-order response Bethe-Salpeter equation

Hannes Hübener

Published Jun 28, 2011• 5 citations

Abstract

A Bethe-Salpeter equation for the three-particle correlation function is derived, representing a second-order response generalization of the usual Bethe-Salpeter equation. The equation can be solved formally, giving the three-particle correlation function in terms of the two-particle correlation function and many-body interaction kernels. The similarity to second-order response time-dependent density-functional theory can be used to improve the understanding of the higher-order exchange and correlation kernels of this theory. An exact expression for such a kernel is derived.

DOI: 10.1103/physreva.83.062122

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Physical Review A

Second-order response Bethe-Salpeter equation

Hannes Hübener

Published Jun 28, 2011• 5 citations

Publisher

Abstract

DOI: 10.1103/physreva.83.062122

Related papers

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Norman Henry March, Thomas A. Niehaus, Sáandor Suhai

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Physical Review AOct 6, 20061 citation

The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function

L

. 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

W

in the BSE for

L

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.

h(k)

and direct correlation function

c(k)

. The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ gives the symmetrized scattering function $S(k\ensuremath{\omega})$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.

(x)

-correlation

(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

{f}_{xc}

, 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

{f}_{xc}

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 ${\ensuremath{\chi}}_{0}$.