Correlation energy functional within the $GW$-RPA: Exact forms, approximate forms, and challenges

Correlation energy functional within the $G W$ -RPA: Exact forms, approximate forms, and challenges

Sohrab Ismail-Beigi

Phys. Rev. B 81, 195126 – Published 27 May 2010

Abstract

In principle, the Luttinger-Ward Green’s-function formalism allows one to compute simultaneously the total energy and the quasiparticle band structure of a many-body electronic system from first principles. We present approximate and exact expressions for the correlation energy within the $G W$ -random-phase approximation that are more amenable to computation and allow for developing efficient approximations to the self-energy operator and correlation energy. The exact form is a sum over differences between plasmon and interband energies. The approximate forms are based on summing over screened interband transitions. We also demonstrate that blind extremization of such functionals leads to unphysical results: imposing physical constraints on the allowed solutions (Green’s functions) is necessary. Finally, we present some relevant numerical results for atomic systems.

Received 13 February 2010

DOI:https://doi.org/10.1103/PhysRevB.81.195126

Authors & Affiliations

Sohrab Ismail-Beigi

Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA

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Vol. 81, Iss. 19 — 15 May 2010

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Images

Figure 1
Schematic figure showing the simplest likely scenario for the Klein total-energy functional. What is shown are level curves of the total-energy functional $F [G]$ as a function of the self-energy $Σ_{x c} (ω)$ that determines the Green’s function $G$ via the Dyson Eq. (4) (not necessarily self-consistently). The horizontal axis represents self-energies that are static and Hermitian operators $U_{0}$ that will generate noninteracting Green’s functions $G_{0}$ via Eq. (3). The vertical axis represents the deviation of the self-energy from a static and Hermitian operator. The black circle represents the extremum of the total-energy functional corresponding to the true self-consistent self-energy which must occur for a dynamic and non-Hermitian $Σ_{x c} (ω)$ since $F$ has no extremum along the horizontal axis.Reuse & Permissions
Figure 2
(Color online) Correlation energy $Φ_{c}$ as a function of the scaling $λ$ of the transition energies for the boron atom. Single-particle energies are from a ground-state LSDA calculation. The blue squares with solid line are the exact RPA correlation energies from Eq. (38) when using scaled transition energies in the RPA/Casida Eq. (36). The green circles with dashed line are correlation energies from the static approximation of Eq. (42) using the scaled transition energies for computing the screening. The dashed or solid lines are guides for the eye. Clearly, the energy decreases monotonically with decreasing $λ$ to large and unphysical negative values with no extrema along this path.Reuse & Permissions
Figure 3
(Color online) Convergence of correlation energies for the boron atom. Both plots show the cumulative sum of the contributions to the correlation energies (vertical) for transitions and/or plasma modes up to some given energy $Δ_{t}$ (horizontal). LSDA wave functions and eigenenergies are used with a radial grid of size $r_{max} = 40 bohr$ radii and $N_{max} = 400$ eigenstates. The lowest solid black curve is for the static approximation of Eq. (42), the middle red dashed curve is for the dynamic approximation of Eq. (41), and the uppermost green dotted curve is for the exact RPA formula of Eq. (38). The plots show the same data with the only difference being a linear (top plot) or a logarithmic scale (bottom plot) of the horizontal axis. The dominant transitions in LSDA boron are the $2 s − 2 p$ at 0.21 hartree and $1 s − 2 p$ at 6.4 hartree, both visible in the lower plot as energies where the correlation contributions have a sudden jump.Reuse & Permissions

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