## Calculation of W

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### Calculation of W

Dear Daniele,

The dynamically screened interaction W is given as the integral over the inverse of the dielectric matrix eps_G,G'(q, omega) multiplied by the bare interaction v.

my question is whether W is calculated by YAMBO over a frequency integral for the FULL eps_G,G'(q, omega) matrix OR
is it calculated over a frequency integral for the G=0, G'=0 components of the inversed eps_G,G'(q, omega) matrix?

Best,
Martin
Martin Spenke, PhD Student
Theoretisch-Physikalisches Institut
Universität Hamburg, Germany
martinspenke

Posts: 149
Joined: Tue Apr 08, 2014 6:05 am

### Re: Calculation of W

Dear Martin it is computed over a frequency integral for the FULL eps_G,G'(q, omega) matrix.
In src/pol_function/X_irredux.F
yambo computes the full Xo_G,G'(q,omega).
The latter is then used to get X_G,G'(q,omega) solving the dyson equation for X and finally eps^-1_G,G'(q,w)
eps_M(q,omega) (the macroscopic term) is defined as the inverse of the head when doing "yambo -o c" calculations.

Otherwise the full eps^-1_G,G'(q,w) is used to construct the GW self-energy for example.
Notice that if the PPA is used, eps^-1_G,G' is computed for two frequencies only, 0 and omega_p, and then interpolated for the other frequencies.

Best,
D.
Davide Sangalli, PhD
CNR-ISM, Division of Ultrafast Processes in Materials (FLASHit) and MaX Centre
http://www.ism.cnr.it/en/davide-sangalli-cv/
http://www.max-centre.eu/

Davide Sangalli

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Location: Via Salaria Km 29.3, CP 10, 00016, Monterotondo Stazione, Italy

### Re: Calculation of W

Dear Davide,

That means, that the GW self-energy is constructed in YAMBO as a summation of head and wings of W_G,G'(q,omega), right?

eps_M(q,omega) (the macroscopic term) is defined as the inverse of the head when doing "yambo -o c" calculations.

This is a point I always asked myself. Are the other diagonal and OFF-diagonal elements of
the inverse of the microscopic dielectric matrix really so unimportant for the optical absorption spectra?

I have another question regarding calculation of optical spectra with respect to finite q.
Is it still not possible to run finite q BSE calculations? If I remember correctly, this was implemented by Claudio Attaccalite in his own version of YAMBO, however I can not find it any more on GitHub.

Best wishes,
Martin
Martin Spenke, PhD Student
Theoretisch-Physikalisches Institut
Universität Hamburg, Germany
martinspenke

Posts: 149
Joined: Tue Apr 08, 2014 6:05 am

### Re: Calculation of W

Dear Martin,
in GW the self energy is built using the whole eps^-1_GG' matrix:
Wgg'(q,w)=eps^-1gg'(q,w)v(q+g')

In absorption, as said by Davide, the macroscopic dielectric function is defined as the head of the inverse of eps_gg'
eps_M(w)=1/(eps^-1)_00 which is different from eps_00 (non-interacting case).
In order to invert the eps matrix, you need all the non-diagonal terms: these are what are called the local field effects.

Finite q BSE it is actually in the devel version of the code, we are still testing it, therefore it is not available in the gpl version,
and at this moment I cannot tell you when it will be released.

Best,

Daniele
Dr. Daniele Varsano
S3-CNR Institute of Nanoscience and MaX Center, Italy
MaX - Materials design at the Exascale
http://www.nano.cnr.it
http://www.max-centre.eu/

Daniele Varsano

Posts: 2027
Joined: Tue Mar 17, 2009 2:23 pm

### Re: Calculation of W

Dear Daniele,

in GW the self energy is built using the whole eps^-1_GG' matrix:
Wgg'(q,w)=eps^-1gg'(q,w)v(q+g')

The point is that the integral over frequency is over components of Wgg'(q,w) (spitted into head and wings) because there is a summation over G and G' before the frequency integral (as you can see in the picture).
Is this the way how YAMBO calculates the GW self-energy?

I struggle to accurately understand how the off-diagonal elements of W are treated for the self-energy.
Again by splitting W into head and wings components OR by directly performing a frequency integral over the full Wgg'(q,w) matrix for each component of W?

According to my understanding : the GW self-energy is given as a sum of frequency integralS over G,G' components of W.

Thanks and best wishes
Martin
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Martin Spenke, PhD Student
Theoretisch-Physikalisches Institut
Universität Hamburg, Germany
martinspenke

Posts: 149
Joined: Tue Apr 08, 2014 6:05 am

### Re: Calculation of W

Dear Martin,
when plasmon pole is used the integral is performed analytically.
Otherwise it in real axis integration yambo calculate the full matrix eps^-1gg'(q,w) for each frequency (the number of frequencies is given in input), and then the expression you reported is evaluated by performing the sum_gg' and bands for each frequency and then the integral is evaluated exploiting Kramers Kronig relations. If you are interested you may have a look at the following subroutines:
/src/qp/QP_real_axis.F
/src/qp/QP_W2Sc.F

Best,
Daniele
Dr. Daniele Varsano
S3-CNR Institute of Nanoscience and MaX Center, Italy
MaX - Materials design at the Exascale
http://www.nano.cnr.it
http://www.max-centre.eu/

Daniele Varsano

Posts: 2027
Joined: Tue Mar 17, 2009 2:23 pm