(gw-frequency-dependence-tutorial)=
# Frequency dependence in the GW approximation

In this tutorial we present and compare the methods available in Yambo to treat the frequency dependence of key quantities for a GW calculation. 
We will use bulk hBN as example system.

````{admonition} Requirements
- `SAVE` folder of bulk hBN: download and extract `hBN.tar.gz` from the [Tutorial files](#tutorial-files-target) page.
- `yambo` executable (version {math}`\geq`5.2)
- `gnuplot` for plotting some of the results

It is assumed that you are familiar with the concepts related to the [SAVE folder](#database-initialization-tutorial-target) and [handling files with Yambo](#input-file-generation-tutorial-target).

```{seealso}
:class: dropdown
- [Quasiparticles corrections in the GW approximation](#gw-on-hbn-tutorial-target)
```
````

In Yambo, the frequency integration appearing in the expression of the GW self-energy can be handled in different ways:

- **Plasmon Pole Approximation (PPA)**: the dynamic screening matrix {math}`\epsilon^{-1}_{\boldsymbol{GG}'}(\boldsymbol{q},\omega)` is expressed as a single-pole function. The parameters for this representation are obtained through a fit enabled by the explicit evaluation of the inverse dielectric matrix at two frequencies, the static limit {math}`\omega = 0` and another imaginary frequency {math}`\omega = i\omega_p` - given in the input file - which is ideally comparable with the plasma frequency of the material. With this representation the solution of the frequency integral is analytical.

- **Full Frequency Real-Axis (FF-RA)**: here the frequency integral is calculated numerically by explicitely evaluating {math}`\epsilon^{-1}_{\boldsymbol{GG}'}(\boldsymbol{q},\omega)` for a number of real frequencies - given in the input, typically in the order of 100 - uniformly distrbuted in the energy range given by the range of bands included in the calculation of the response function.

- **Multi-Pole Approximation (MPA)**: this can be thought as a generalization of the PPA able to provide FF-RA accuracy at a much lower computational cost. This approach builds from the representation of the polarizability {math}`\chi` as the sum of a finite and small number of poles. The parameters for this representation are obtained solving a nonlinear system of equations defined by the explicit evaluation of the polarizability on a number of frequencies that is twice the number of poles. This provides an analytical representation of {math}`\chi` over the whole complex plane which in turn allows to compute the self-energy analytically.

For a more detailed description of these methods see the Yambo code papers[^YamboPapers] (PPA, FF-RA) and the papers presenting the MPA method[^MPAPapers].

[^YamboPapers]: A. Marini, et al., ["yambo: An ab initio tool for excited state calculations"](https://doi.org/10.1016/j.cpc.2009.02.003), Computer Physics Communications 180.8 (2009): 1392-1403; 
D. Sangalli, et al., ["Many-body perturbation theory calculations using the yambo code"](https://doi.org/10.1088/1361-648X/ab15d0), J.Phys.:Condens. Matter **31** 325902 (2019).

[^MPAPapers]: D. A. Leon, et al., ["Frequency dependence in GW made simple using a multipole approximation"](https://doi.org/10.1103/PhysRevB.104.115157), Phys. Rev. B **104** 115157 (2021); 
D. A. Leon, et al., ["Efficient full frequency GW for metals using a multipole approach for the dielectric screening"](https://doi.org/10.1103/PhysRevB.107.155130), Phys. Rev. B **107** 155130 (2023).