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Physical review. B./Physical review. B

Convergence behavior of the random phase approximation renormalized correlation energy

Jefferson E. Bates, J. Sensenig, Adrienn Ruzsinszky

Published May 26, 2017• 17 citations

Abstract

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.

DOI: 10.1103/physrevb.95.195158

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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.

We derive a renormalized many-body perturbation theory (MBPT) starting from the random phase approximation (RPA). This RPA-renormalized perturbation theory extends the scope of single-reference MBPT methods to small-gap systems without significantly increasing the computational cost. The leading correction to RPA, termed the approximate exchange kernel (AXK), substantially improves upon RPA atomization energies and ionization potentials without affecting other properties such as barrier heights where RPA is already accurate. Thus, AXK is more balanced than second-order screened exchange [A. Grüneis et al., J. Chem. Phys. 131, 154115 (2009)], which tends to overcorrect RPA for systems with stronger static correlation. Similarly, AXK avoids the divergence of second-order Møller-Plesset (MP2) theory for small gap systems and delivers a much more consistent performance than MP2 across the periodic table at comparable cost. RPA+AXK thus is an accurate, non-empirical, and robust tool to assess and improve semi-local density functional theory for a wide range of systems previously inaccessible to first-principles electronic structure calculations.

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.

Using a second-order approximation to Random Phase Approximation renormalized (RPAr) many-body perturbation theory for the interacting density-density response function, we have developed a so-called higher-order terms (HOT) approximation for the correlation energy. In combination with the first-order RPAr correction, our new method faithfully captures the infinite-order correlation for a given exchange-correlation kernel, yielding errors of the total correlation energy on the order of 1% or less for most systems. For exchange-like kernels, our new method has the further benefit that the coupling-strength integration can be completely eliminated resulting in a modest reduction in computational cost compared to the traditional approach. When the correlation energy is accurately reproduced by the HOT approximation, structural properties and energy differences are also accurately reproduced, as we demonstrate for several periodic solids and some molecular systems. Energy differences involving fragmentation are challenging for the HOT method, however, due to errors that may not cancel between a composite system and its constituent pieces.

We calculate the interacting bandgap energy of a solid within the random-phase approximation (RPA) to density functional theory (DFT). The interacting bandgap energy is defined as ${E}_{g}={E}^{\mathrm{RPA}}(N+1)+{E}^{\mathrm{RPA}}(N\ensuremath{-}1)\ensuremath{-}2{E}^{\mathrm{RPA}}(N)$, where

{E}^{\mathrm{RPA}}(N)

is the total DFT-RPA energy of the

N

-electron system. We compare the interacting bandgap energy with the Kohn-Sham bandgap energy, which is the difference between the conduction and valence band edges in the Kohn-Sham band structure. We show that they differ by an unrenormalized ``

{G}_{0}{W}_{0}

'' self-energy correction (i.e., a

GW

self-energy correction computed using Kohn-Sham orbitals and energies as input). This provides a well-defined and meaningful interpretation to

{G}_{0}{W}_{0}

quasiparticle bandgap calculations, but questions the physics behind the renormalization factors in the expression of the bandgap energy. We also separate the kinetic from the Coulomb contributions to the DFT-RPA bandgap energy, and discuss the related problem of the derivative discontinuity in the DFT-RPA functional. Last we discuss the applicability of our results to other functionals based on many-body perturbation theory.

The correlated random-phase-approximation (CRPA) method is applied to study the ground-state and excited-state properties of an anyon gas. We perform a systematic 1/N expansion to the calculation of elementary excitations, where N is the statistical parameter. We find that the leading-(1/N)-order contributions are the same as those obtained by calculations of the RPA response functions. We then consider corrections to the RPA by a perturbation of the exchange interactions, and show that these higher-order corrections are important at large wave vectors. Our calculation gives an excitation spectrum in good agreement with that obtained by the exact solution of a small system.

The exactly solvable Luttinger model for a system of one-dimensional, massless, two-component fermions interacting via general two-body forces is used as a test case for a systematically generated sequence of decoupling approximations commonly referred to as random-phase approximations (RPA) of which the simple RPA is the lowest. It is found, irrespective of both the nature and strength of the interaction, that the exact (collective) spectrum, as well as the exact correlation energy, can already be obtained in the simple RPA, and furthermore, that the next higher RPA gives a vanishing correction to these quantities. It is also shown, quite in contradistinction to the exact result, that the free-gas momentum distribution

{\overline{n}}_{k}

persists in the second RPA as well as in the first, thereby eliminating the higher RPA as a possible approach to the calculation of

{\overline{n}}_{k}

. Thus, in the context of this particular model, although the first few orders of RPA are highly effective for calculating the excitation spectrum and correlation energy, they are sensibly powerless in their ability to predict any variation in structure beyond that of the free-gas distribution, a result which most probably holds as well in the realistic three-dimensional case.