Correlation beyond the random phase approximation: A consistent many-body perturbation theory approach

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.

Figure 1

Correlation energy (per particle) of a spin-polarized homogeneous electron gas versus the density parameter $r_s$ in various approximations: QMC [23, 24] (black lines), RPA (red lines), RPA + DSOSEX (blue lines), RPA + 2DSOSEX (dotted blue lines), and RPA + (static) SOSEX (dotted red lines). (a) Paramagnetic case; (b) ferromagnetic case.

Figure 2

Universal function $f(x,y)$ characterizing the screened exchange of energy and momentum between particles in one spin channel of the electron gas. A perspective view (upper panel) and its projection onto the $xy$ plane (lower panel) are displayed. The line distance in the contour plot is 0.005.