Correlation energy within exact-exchange adiabatic connection fluctuation-dissipation theory: Systematic development and simple approximations

We have calculated the correlation energy of the homogeneous electron gas (HEG) and the dissociation energy curves of molecules with covalent bonds from an efficient implementation of the adiabatic connection fluctuation dissipation expression including the exact-exchange (EXX) kernel. The EXX kernel is defined from first-order perturbation theory and used in the Dyson equation of time-dependent density-functional theory. Within this approximation (RPAx), the correlation energies of the HEG are significantly improved with respect to the random phase approximation (RPA) up to densities of the order of $r_s\approx 10$. However, beyond this value, the RPAx response function exhibits an unphysical divergence and the approximation breaks down. Total energies of molecules at equilibrium are also highly accurate, but we find a similar instability at stretched geometries. Staying within an exact first-order approximation to the response function, we use an alternative resummation of the higher-order terms. This slight redefinition of RPAx fixes the instability in total energy calculations without compromising the overall accuracy of the approach.

Figure 1

Correlation energy per particle in the homogeneous electron gas as a function of the Wigner-Seitz radius evaluated with different kernels: RPA (red squares), RPAx (green diamonds), and QMC calculation (black circles).

Figure 2

Critical behavior of the static density-density RPAx response function when the density decreases. For $r_s>10.6$, the system becomes unstable with respect to charge modulation with wave vector $\approx 2k_F$.

Figure 3

Correlation energies per particle as a function of $r_s$ evaluated from the RPA, original and modified RPAx response functions, and compared to accurate QMC calculations.

Figure 4

Approximate static response functions for the HEG at $r_s=5$ as compared to the exact QMC calculations [46]. The alternative RPAx approximations show a better agreement with QMC results.

Figure 5

Spin-polarization function $\gamma$ for $r_s=2$ from RPA (red line), RPAX (green squares), Perdew-Zunger parametrization [48] (blue solid line), Perdew-Wang parametrization [49] (brown dashed line), and Quantum Mote Carlo calculation [50] (open circles).

Figure 6

Dissociation curve of the ${\mathrm{H}}_2$ molecule. PBE, RPA, and RPAx (original and alternative) results are compared with accurate calculations [53].

Figure 7

Dissociation curve of the ${\mathrm{N}}_2$ molecule. The plot compares results from PBE, RPA, and RPAx (original and alternative) calculations.