# GW calculations, including self-consistency

### This page gives hints on how to perform a GW calculation, including self-consistency with the ABINIT package.

##### Copyright (C) 2016-2017 ABINIT group (MG)

Mentioned in topic_Susceptibility, help_features#7.

### Table of content:

- 1. Introduction.
- 2. Related lesson(s) of the tutorial.
- 3. Related input variables.
- 4. Selected input files.
- 5. References.

### 1. Introduction.

DFT performs reasonably well for the determination of structural properties, but fails to predict accurate band gaps. A more rigorous framework for the description of excited states is provided by many-body perturbation theory (MBPT) [Fetter1971],[Abrikosov1975], based on the Green's functions formalism and the concept of quasi-particles [Onida2002].

Within MBPT, one can calculate the quasi-particle (QP) energies, E, and amplitudes, Ψ, by solving a nonlinear equation involving the non-Hermitian, nonlocal and frequency dependent self-energy operator Σ.

This equation goes beyond the mean-field approximation of independent KS particles as it accounts for the dynamic many-body effects in the electron-electron interaction.

Details about the GW implementation in ABINIT can be found here.

A typical GW calculation consists of two different steps (following a DFT calculation): first the screened interaction ε-1 is calculated and stored on disk (optdriver=3), then the KS band structure and W are used to evaluate the matrix elements of Σ, finally obtaining the QP corrections (optdriver=4).

The computation of the screened interaction is described in topic_Susceptibility, while the computation of the self-energy is described in topic_SelfEnergy. The frequency meshes, used e.g. for integration along the real and imaginary axes are described in topic_FrequencyMeshMBPT.
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### 2. Related lesson(s) of the tutorial.

- The first lesson on GW (GW1) deals with the computation of the quasi-particle band gap of Silicon (semiconductor), in the GW approximation (much better than the Kohn-Sham LDA band structure), with a plasmon-pole model.
- The second lesson on GW (GW2) deals with the computation of the quasi-particle band structure of Aluminum, in the GW approximation (so, much better than the Kohn-Sham LDA band structure) without using the plasmon-pole model.
- Parallelism of Many-Body Perturbation calculations (GW) allows to speed up the calculation of accurate electronic structures (quasi-particle band structure, including many-body effects).
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### 3. Related input variables.

Compulsory input variables:

... optdriver [OPTions for the DRIVER]

Basic input variables:

... bdgw [BanDs for GW calculation]
... gw_nstep [GW Number of self-consistent STEPs]
... gw_sctype [GW, Self-Consistency TYPE]
... gw_toldfeig [GW TOLerance on the DiFference of the EIGenvalues]
... gwcalctyp [GW CALCulation TYPe]
... gwpara [GW PARAllelization level]

Useful input variables:

... getqps [GET QuasiParticle Structure]
... getscr [GET SCReening (the inverse dielectric matrix) from ...]
... getsuscep [GET SUSCEPtibility (the irreducible polarizability) from ...]
... gwfockmix [GW FOCK exchange MIXing parameter]
... irdqps [Integer that governs the ReaDing of QuasiParticle Structure]
... irdscr [Integer that governs the ReaDing of the SCReening]
... irdsuscep [Integer that governs the ReaDing of the SUSCEPtibility]
... mbpt_sciss [Many Body Perturbation Theory SCISSor operator]
... nbandkss [Number of BANDs in the KSS file]
... nsym [Number of SYMmetry operations]
... rhoqpmix [RHO QuasiParticle MIXing]
... symmorphi [SYMMORPHIc symmetry operation selection]
... usepawu [USE PAW+U (spherical part)]

Input variables for experts:

... fftgw [FFT for GW calculation]
... gw_nqlwl [GW, Number of Q-points for the Long Wave-Length Limit]
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### 4. Selected input files.

WARNING : as of ABINITv8.6.x, the list of input files provided in the specific section of the topics Web pages is still to be reviewed/tuned. In some cases, it will be adequate, and in other cases, it might be incomplete, or perhaps even useless.
The user can find some related example input files in the ABINIT package in the directory /tests, or on the Web:

tests/v3/Input: t30.in t31.in

tests/v4/Input: t84.in t85.in

tests/v5/Input: t65.in t66.in t69.in
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### 5. References.

[Abrikosov1975] A.A. Abrikosov, L.P. Gorkov and E. Dzyaloshinskii, "Methods of quantum field theory in statistical physics", Dover, New-York , (1975).
[Fetter1971] A.L. Fetter and J.D. Walecka, "Quantum Theory of Many-Particle Systems", McGraw-Hill, New York , (1971).
[Onida2002] G. Onida, L. Reining and A. Rubio, "Electronic excitations: density-functional versus many-body Green's-function approaches", Rev. Mod. Phys. 74, 601–659 (2002).
DOI: 10.1103/RevModPhys.74.601.

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