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Physical Review B

Relationship between the random-phase approximation ground-state total energy and GW quasiparticle energy

Lin‐Wang Wang

Published Sep 10, 2010• 12 citations
Publisher

Abstract

A connection between the random-phase-approximation (RPA) total energy ${E}_{\text{RPA}}$ and the $GW$ quasiparticle energy ${ϵ}_{GW}$ has been pointed out. More specifically we show that: $\ensuremath{\delta}{E}_{\text{RPA}}(N)/\ensuremath{\delta}N={ϵ}_{GW}$, where $N$ is the total number of electron. The variational property of the RPA total energy is also discussed and the variational equations for the orbital wave functions are derived. We argue that the RPA formalism is a method which can provide both accurate ground-state energies and single quasiparticle energies.

DOI
10.1103/physrevb.82.115111

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A connection between the random-phase-approximation (RPA) total energy ERPA{E}_{\text{RPA}}ERPA​ and the GWGWGW quasiparticle energy ϵGW{ϵ}_{GW}ϵGW​ has been pointed out. More specifically we show that: \ensuremathδERPA(N)/\ensuremathδN=ϵGW\ensuremath{\delta}{E}_{\text{RPA}}(N)/\ensuremath{\delta}N={ϵ}_{GW}\ensuremathδERPA​(N)/\ensuremathδN=ϵGW​, where NNN is the total number of electron. The variational property of the RPA total energy is also discussed and the variational equations for the orbital wave functions are derived. We argue that the RPA formalism is a method which can provide both accurate ground-state energies and single quasiparticle energies.
A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.
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 GWGWGW-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.
We derive analytical expressions for chemical potentials within the random phase approximation (RPA), equivalently the GWGWGW energy functional evaluated using non interacting Green's functions (GsG_sGs​). The chemical potential is obtained using two formally equivalent approaches: a direct derivative of the total energy with respect to particle number, and a functional derivative via the chain rule through GsG_sGs​, both validated with finite difference benchmarks. We show that the functional derivative of the GWGWGW correlation energy$\unicode{x2013}$i.e., the GWGWGW correlation self energy$\unicode{x2013}$exhibits a discontinuity at integer particle numbers with finite jumps. This resolves the apparent inconsistency between accurate GWGWGW quasiparticle energies and the large delocalization errors observed in RPA total energies, as standard GWGWGW self energies neglect this nonanalytic behavior. Our results suggest that derivative discontinuities are a fundamental feature of correlation energy functionals, analogous to the known discontinuity in the exact exchange correlation energy.
The well-known and well-used Density Functional Theory (DFT) is a variational theory for the ground-state. There are no guarantees that it delivers good results for excited state properties such as quasiparticle energies or wave function. All that is guaranteed is that the total energy is exact, the chemical potential is correct, and that the eective electronic states (Kohn-Sham states) generate the right electron density at the variational minimum. Luttinger-Ward approaches to electronic structure are attractive as beyond-DFT schemes since, in principle, they oer a variational functional of the one-electron Green’s function that yields both the correct total energy and all one-particle properties at the variational extremum. However, compared to DFT, these methods face a number of additional challenges beyond the familiar problem of choosing an exchange-correlation functional: (a) they have a much greater computational cost compared to DFT, (b) one must choose some set of trial Green’s functions, and (c) one must locate an extremum (not necessarily a minimum). This talk aims both to be pedagogical and to provide a summary of our most recent work in this area. We focus mainly on the random-phase approximation (RPA) for the correlation functional which yields the GW self-energy for the Green’s function. We will describe ecient approaches for computing the RPA correlation energy; how various types of approximations to the correlation energy create a ladder of approximate self-energy operators and what type of physics is included in these approximations (exchange, screening, Hubbard U, etc.); and how the extremization is in fact highly problematic. We will end with some ideas for making practical progress with the extremization and their relation to the successful Quasiparticle Self-consistent GW (QSGW) scheme [1].
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 ERPA(N){E}^{\mathrm{RPA}}(N)ERPA(N) is the total DFT-RPA energy of the NNN-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 ``G0W0{G}_{0}{W}_{0}G0​W0​'' self-energy correction (i.e., a GWGWGW self-energy correction computed using Kohn-Sham orbitals and energies as input). This provides a well-defined and meaningful interpretation to G0W0{G}_{0}{W}_{0}G0​W0​ 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 random phase approximation (RPA) builds in correlations left out by mean-field theory. In full $0\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model spaces we calculate the Hartree-Fock (HF)+RPA(\mathrm{HF})+\mathrm{RPA}(HF)+RPA binding energy, and compare it to exact diagonalization. We find that often, but not always, the HF+RPA\mathrm{HF}+\mathrm{RPA}HF+RPA gives a good approximation to the ``exact'' ground state energy. In those cases where the RPA is less satisfactory, however, there is no obvious correlation with properties of the HF state, such as deformation or overlap with the exact ground state wave function. This weakens the reliability of the RPA for computing binding energies.