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Bulletin of the American Physical Society

New exact and approximate forms for the Luttinger-Ward correlation energy functional within the GW-RPA approximation

Sohrab Ismail‐Beigi

Published Mar 16, 2009• 0 citations
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New expressions for the correlation energy functional within the GW-RPA approximation

Sohrab Ismail‐Beigi

Bulletin of the American Physical Society•Mar 17, 2010•0 citations

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

Sohrab Ismail‐Beigi

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 $GW$-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.

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Progress and Challenges with Luttinger-Ward Approaches for Going Beyond DFT

Sohrab Ismail‐Beigi

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].

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The RPA and the GW approximation for the self-energy

Richard M. Martin, Lucia Reining, David M. Ceperley

Besides the proof of a modified Luttinger–Ward–Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. L. Hedin, Phys. Rev . 139, A796–823 (1965) Summary In this chapter a set of equations is formulated that determine the self-energy and the one-body Green's function in terms of the screened Coulomb interaction between classical charges. The equations contain a correction to the classical picture in terms of a vertex function. The physical meaning of the various contributions is discussed. The simplest approximation for the vertex yields the random phase approximation for the polarizability and the GW approximation for the self-energy. Various aspects of the GWA are analyzed, with a focus on the physics that is added beyond Hartree– Fock. A brief summary of model cases illustrates the domain of validity and the limits of the GWA. In this chapter we elaborate in more detail on the question of how to calculate the onebody Green's function from a Dyson equation with a self-energy kernel. In the previous chapter a scheme was introduced to design approximations to the self-energy. However, the question of where to stop, which pieces of physics to include and which to neglect, is not yet settled. Of course, there is no unique answer, besides the exact solution, but different strategies can be more or less advantageous in practice. In a system with a few electrons, for example, different aspects will be important than in a system with many electrons. Here we are mostly interested in solids, or more generally in extended systems. In such systems, screening plays an essential role: the interaction between two charges is strongly modified, in general reduced, by the rearrangement of all the other charges. It is therefore most convenient to reformulate the equations such that screening appears explicitly. Some steps in this direction can be found in earlier chapters, in particular in Sec. 8.3, the formulation of the Ψ [ G , W ]-functional of the screened interaction W instead of the Ф[ G , v c ]-functional of the bare v c . In Sec. 10.5 the screened interaction approximation for the self-energy is derived, with the self-energy as a product of the one-body Green's function and the screened interaction W .

Cambridge University Press eBooks•May 31, 2016•0 citations

Exact Correlation Exponents of Luttinger Liquids in One Dimension

N. Kawakamai, Sung-Kil Yang

Springer proceedings in physics•Jan 1, 1992•0 citations
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.
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].
Besides the proof of a modified Luttinger–Ward–Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. L. Hedin, Phys. Rev . 139, A796–823 (1965) Summary In this chapter a set of equations is formulated that determine the self-energy and the one-body Green's function in terms of the screened Coulomb interaction between classical charges. The equations contain a correction to the classical picture in terms of a vertex function. The physical meaning of the various contributions is discussed. The simplest approximation for the vertex yields the random phase approximation for the polarizability and the GW approximation for the self-energy. Various aspects of the GWA are analyzed, with a focus on the physics that is added beyond Hartree– Fock. A brief summary of model cases illustrates the domain of validity and the limits of the GWA. In this chapter we elaborate in more detail on the question of how to calculate the onebody Green's function from a Dyson equation with a self-energy kernel. In the previous chapter a scheme was introduced to design approximations to the self-energy. However, the question of where to stop, which pieces of physics to include and which to neglect, is not yet settled. Of course, there is no unique answer, besides the exact solution, but different strategies can be more or less advantageous in practice. In a system with a few electrons, for example, different aspects will be important than in a system with many electrons. Here we are mostly interested in solids, or more generally in extended systems. In such systems, screening plays an essential role: the interaction between two charges is strongly modified, in general reduced, by the rearrangement of all the other charges. It is therefore most convenient to reformulate the equations such that screening appears explicitly. Some steps in this direction can be found in earlier chapters, in particular in Sec. 8.3, the formulation of the Ψ [ G , W ]-functional of the screened interaction W instead of the Ф[ G , v c ]-functional of the bare v c . In Sec. 10.5 the screened interaction approximation for the self-energy is derived, with the self-energy as a product of the one-body Green's function and the screened interaction W .