# Implementation of the GW Approximation

### Table of Contents

- Introduction
- How the 1-particle and many-particle codes synchronise
- Analysis of many-body spectral functions
- Other Resources

### Introduction

The *GW* approximation is implemented in two forms: as a 1-shot perturbation to the LDA and as quasiparticle self-consistent *GW* (QS*GW*).

In 1-shot mode, the diagonal part of Σ is evaluated at the one-particle (usually LDA) energies, yielding a correction to the LDA levels in 1st order perturbation theory.

In QSGW mode, the full Σ matrix is calculated and “quasiparticlized:” coupling one-electron states *i* and *j*, is evaluated at the one-particle energies and , and the average value is taken. This results in a static but orbital-dependent potential, a “quasiparticlized” derived from , and defines a one-particle hamiltonian which is the LDA one-particle hamiltonian with substituting for .

The new is used to mke a new , and the process is iterated until stops changing. This is quasiparticle self-consistency.

QS*GW* theory is an elegant way to find the optimum noninteracting hamiltonian for *GW* calculations. It is vastly better than basing on the LDA, as is customary, albeit at some computational cost. A particularly valuable property of this optimum starting point is that the peaks of the interacting Green’s function *G* coincide with the poles of . The eigenfunctions of are as close as possible to those of *G*, by construction. What are true poles in get broadened by the interactions, so quasiparticles lose weight. Thus the density-of-states, or spectral function, is composed of a superposition of δ-functions for , but are broadened for *G*. Also the eigenvalues acquire an imaginary part, making the QP lifetime finite.

In QS*GW* theory and *G* are closely linked. Associated with the two kinds of *G* are two kinds of density-of-states (DOS). There is “noninteracting” or “coherent” DOS, namely the spectral function of which is associated with DOS in one-particle description, and is what is typically calculated by a band program such as *lmf*. There is also the true DOS (spectral function of *G*) which is approximately what is approximately measured by e.g. a photoemission experiment. The *GW* package has a facility to generate both kinds of DOS through the normal *lmf* process for the noninteracting DOS or by analyzing the spectral functions for the interacting case.

A detailed description of QS*GW* theory and the way in which Questaal implements it can be found in the references in “Other Resources” below.

### How the 1-particle and many-particle codes synchronise

The *GW* package comprise a separate set of codes from the density-functional code, *lmf*. It uses the single-particle basis set of *lmf* to calculate the screened coulomb interaction *W* and the self-energy, Σ=*iGW*.

Thus, the *GW* package handles the many-body part, *lmf* the 1-body part. The two connect through special purpose interfaces: *lmfgwd* sets up the inputs needed by GW, supplying information about the wave functions in the augmentation spheres and in the interstitial. The *GW* part estimates shifts in QP levels and makes the quasiparticlized . Apart from these linkages, the *GW* package is completely separate from *lmf*. The *GW* package has a separate input file, *GWinput* and does not depend on the ctrl file *lmf* and *lmfgwd* use; only on the eigenfunctions they generate.

In reality the *GW* package makes . *lmf* can read this potential and add it to the LDA hamiltonian. This construction enables *lmf* to do the same kinds of calculations it performs with the LDA potential.

The QS*GW* flowchart is shown in the Figure. Initially *lmf* reads a trial charge density ρ (`start`) and makes a self-consistent ρ, which in DFT is sufficient to make a one-body *H*^{0}. This is sufficient to start the QS*GW* cycle, which is managed by shell script *lmgw*,

*lmgw* performs the following:

*lmfgwd*constructs this*H*^{0}(initially*H*^{0}=*H*^{DFT}) to make eigenfunctions and eigenvalues (), which is sufficient to make*ψ, ε**G*^{0}(in practice*G*^{0}is never calculated).- Matrix elements of the Hartree potential are made (
`V`). - The bare polarizability (
`Π`) is made. - Π is combined with
`V`to make the susceptibility*χ*and the RPA screened coulomb interaction (`W`). - At this stage there is an option to improve on
*W*by adding a vertex (`χ`).^{BSE} *W*is (formally) convolved with*G*^{0}to make the self-energy Σ(*ω*) (`Σ`).- Σ(
*ω*) is quasiparticlized to make a static, but spatially nonlocal self-energy (Σ^{0}). *lmf*can be run again, now with a potential added to .*lmf*can iterate until the density is self-consistent (an inner self-consistent loop) and the entire cycle repeated until stops changing (outer loop).

*lmgw* manages one outer loop; *lmgwsc* calls *lmgw* and cycles through outer loops until self-consistency is reached.

Note that when making the density self-consistent, in is kept frozen, while is updated. So there is an inexact cancellation of these two terms until self-consistency in is reached.

#### Operation of *lmfgwd*

*lmfgwd* generates the same *H*^{0} as *lmf*, but its purpose is to create files needed for the *GW* cycle. After setting up the potential it prompts for a job, which tells *lmfgwd* what to do.

- Job −1 generates a
*GWinput*and exits. The structure of this file is documented on this page.*lmfgwd*will make a reasonable*GWinput*, but it should be kept in mind that this file is only a template and it should be checked. Many of tags written to*GWinput*can be read from tags in the GW category of the*ctrl*file. (While tags in*GWinput*and the*ctrl*files are kept separate, this switch generates a*GWinput*that synchronises with the*ctrl*file.) Particularly important are**QpGcut_cou**and the**k**mesh (**n1n2n3**). Execution time is sensitive to both, so they are not given conservative values. - Job 0 is an ‘initializer’ mode. It creates several input files the
*GW*package requires:*SYMOPS*,*LATTC*,*CLASS*,*NLAindx*,*ldima* - Job 1 generates files required by the
*GW*package (e.g. eigenfunctions, eigenvalues, and wave function information) - Job −2 performs sanity checks on
*GWinput*

Even while the *GW* package is independent of *lmf*, the *GWinput* file is complicated and in practice almost always autogenerated by *lmfgwd*. What *lmfgwd* writes to *GWinput* *is* modified by contents in the ctrl file. *lmfgwd* reads a special **GW** category from the ctrl file; through tags in this category you can set some of the parameters it writes into *GWinput* .

### Analysis of many-body spectral functions

A pair of executables *spectral* and *lmfgws* are included in the Questaal suite. They are postprocessing tools used to analyze spectral functions made by the GW package, derived by either individual QP levels or integrated over the Brillouin zone to make the interacting density-of-states. See this page for a tutorial on spectral functions.

### Other Resources

See this tutorial for a basic introduction to doing a QS*GW* calculation.

This is Questaal’s main methods paper, which includes a detailed discussion of *GW* and QS*GW*:

Dimitar Pashov, Swagata Acharya, Walter R. L. Lambrecht, Jerome Jackson, Kirill D. Belashchenko, Athanasios Chantis, Francois Jamet, Mark van Schilfgaarde, *Questaal: a package of electronic structure methods based on the linear muffin-tin orbital technique*, Comp. Phys. Comm. **249**, 107065 (2020).

This paper presents the first description of an all-electron *GW* implementation in a mixed basis set:

T. Kotani and M. van Schilfgaarde, *All-electron GW approximation with the mixed basis expansion based on the full-potential LMTO method*, Sol. State Comm. 121, 461 (2002).

These papers established the framework for QuasiParticle Self-Consistent *GW* theory:

Sergey V. Faleev, Mark van Schilfgaarde, Takao Kotani, *All-electron self-consistent GW approximation: Application to Si, MnO, and NiO*, Phys. Rev. Lett. 93, 126406 (2004);

M. van Schilfgaarde, Takao Kotani, S. V. Faleev, *Quasiparticle self-consistent* GW *theory*, Phys. Rev. Lett. 96, 226402 (2006)

Questaal’s GW implementation is based on this paper:

Takao Kotani, M. van Schilfgaarde, S. V. Faleev, *Quasiparticle self-consistent GW method: a basis for the independent-particle approximation*, Phys. Rev. B76, 165106 (2007)

This paper shows all-electron results from LDA-based GW, and its limitations:

M. van Schilfgaarde, Takao Kotani, S. V. Faleev, *Adequacy of Approximations in GW Theory*, Phys. Rev. B74, 245125 (2006)