Convergence behavior of the random phase approximation renormalized correlation energy

Based on the random phase approximation (RPA), RPA renormalization [J. E. Bates and F. Furche, J. Chem. Phys. 139, 171103 (2013)] is a robust many-body perturbation theory that works for molecules and materials because it does not diverge as the Kohn-Sham gap approaches zero. Additionally, RPA renormalization enables the simultaneous calculation of RPA and beyond-RPA correlation energies since the total correlation energy is the sum of a series of independent contributions. The first-order approximation (RPAr1) yields the dominant beyond-RPA contribution to the correlation energy for a given exchange-correlation kernel, but systematically underestimates the total beyond-RPA correction. For both the homogeneous electron gas model and real systems, we demonstrate numerically that RPA renormalization beyond first order converges monotonically to the infinite-order beyond-RPA correlation energy for several model exchange-correlation kernels and that the rate of convergence is principally determined by the choice of the kernel and spin polarization of the ground state. The monotonic convergence is rationalized from an analysis of the RPA renormalized correlation energy corrections, assuming the exchange-correlation kernel and response functions satisfy some reasonable conditions. For spin-unpolarized atoms, molecules, and bulk solids, we find that RPA renormalization is typically converged to 1 meV error or less by fourth order regardless of the band gap or dimensionality. Most spin-polarized systems converge at a slightly slower rate, with errors on the order of 10 meV at fourth order and typically requiring up to sixth order to reach 1 meV error or less. Slowest to converge, however, open-shell atoms present the most challenging case and require many higher orders to converge.

Figure 1

Convergence of RPA renormalization to the infinite-order correlation energy for selected densities of the electron gas of spin polarization $\xi$ with the NEO, rALDA, and CP07 kernels. $\mathrm{\Delta }{E}_c^n={\sum }_i^n\mathrm{\Delta }{E}_c^{\text{RPAr-}i}-\mathrm{\Delta }{E}_c^{\text{bRPAr}}$ is the error for a given order $n$ of RPA renormalization in comparison to the infinite-order result. The dashed line corresponds to 1 meV error. For typical densities between $r_s=1$ and 10, RPAr converges to meV accuracy between second and fourth order for each kernel. As the density decreases, the convergence slows, and at very low densities (large $r_s$) the form the xc kernel makes a noticeable impact on the number of orders needed to reach convergence.

Figure 2

Convergence of RPA renormalization for real systems near their equilibrium geometries using the rAPBE and CP07 kernels. For spin-unpolarized systems, convergence to the infinite-order result is fast and largely independent of the details of the system such as band gap or dimensionality. For an atom, a molecule, several semiconductors, an insulator, three bulk metals, and one metallic surface, RPAr is converged to less than 1 meV error by fourth order.

Figure 3

Convergence of RPA renormalization using the spin-dependent (left) and spin-independent (right) rAPBE kernel for several first-row atoms with unpaired electrons, oxygen molecule (${\mu }_B=2$), and ferromagnetic iron in the bcc phase and the Co(0001) surface. For B, C, and O, the convergence of RPA renormalization is extremely slow when combined with the spin-dependent kernel. For the spin-independent rAPBEns kernel, all systems converge much more rapidly, although B, C, and O still converge more slowly than the other examples. Other than the open-shell atoms, the convergence of rAPBE and rAPBEns are similar, highlighting a unique challenge of RPAr combined with these kernels for atomic systems.

Figure 4

Convergence (left) and bRPA correlation corrections (right) from RPAr for ${\mathrm{N}}_2$ at four different stretched bond lengths with the rAPBE kernel. Near equilibrium ($R_{eq}=110\mathrm{pm}$) RPAr is converged to less than 1 meV error by fourth order, however, convergence slows as $R$ increases and terms up to eighth order or higher are required as the atoms dissociate. Still, even at double the equilibrium bond length, by fifth order RPAr is converged to approximately 10 meV error or less. Since RPA is the underlying approximation, the kernel corrections from RPAr remain finite even as the molecule dissociates and the Kohn-Sham gap goes to zero.