derivation of BSE oscillator strengths and dielectric function

[awang95]

Hi,

I've been trying to understand how the implemented BSE oscillator strengths are related to the BSE dielectric function from the posted slides (https://www.vasp.at/wiki/images/2/29/VA ... re_BSE.pdf; image also attached) and the relevant references, specifically the implementation paper (https://journals.aps.org/prb/pdf/10.110 ... .92.045209) and this paper (https://journals.aps.org/prl/pdf/10.110 ... tt.80.4510). However, there are some things that I hope to ask and clarify:

1. Do the f_ck and f_vk correspond to the occupancies of the associated conduction and valence bands, as mentioned here (https://journals.aps.org/prl/pdf/10.110 ... tt.80.4510), or something else about the conduction and valence bands? The slides didn't explicitly mention what they are so just like to make sure.

2. Is equation 50 in the implementation paper equivalent to equation 8 from here (https://journals.aps.org/prl/pdf/10.110 ... tt.80.4510), and therefore a generalization of the equation for BSE imaginary dielectric function shown on the slide "BSE dielectric function"? If so, if you could also point me in the right direction to seeing how they're equivalent that would be awesome. The oscillator strength is given by equation 51 - is that correct?

3. The TDHF routine (used for hybrid TD-DFT calculations) also follow the same procedure outlined in the slides and the implementation paper, right?

thanks!

[alexey.tal]

Dear awang95,

1. Do the f_ck and f_vk correspond to the occupancies of the associated conduction and valence bands, as mentioned here (https://journals.aps.org/prl/pdf/10.110 ... tt.80.4510), or something else about the conduction and valence bands? The slides didn't explicitly mention what they are so just like to make sure.

Yes, these are the occupancies.

2. Is equation 50 in the implementation paper equivalent to equation 8 from here (https://journals.aps.org/prl/pdf/10.110 ... tt.80.4510), and therefore a generalization of the equation for BSE imaginary dielectric function shown on the slide "BSE dielectric function"? If so, if you could also point me in the right direction to seeing how they're equivalent that would be awesome. The oscillator strength is given by equation 51 - is that correct?

In principle, these expressions are equivalent. One should keep in mind that in Sander's paper the eigenvectors are split into X and Y, so a vector (X,Y) corresponds to an eigenvector A in Albrecht's paper. Also, in Albrecht's paper the band indices run over all included bands, whereas indices a and i in Sander's paper correspond to empty and occupied states. In the Tamm-Dancoff approximation this expression simplifies to Eq. (9) of Albrecht or the expression used in the slides. You can also have a look at this thread for the details on the oscillator strength.

3. The TDHF routine (used for hybrid TD-DFT calculations) also follow the same procedure outlined in the slides and the implementation paper, right?

The BSE and TDHF use the same algorithm and only differ in the way the screened interaction is treated.

[awang95]

ok thanks so much for your response! it was super helpful!