[QE-developers] [SPAM] Re: The mathematical formulation of g(:, igk_k(ig, ik)) in epsilon.f90

[Paolo Giannozzi]

On 9/25/24 02:17, Yuvam Bhateja wrote:

> caux = caux + conjg(evc(ig+npwx,iband1))*evc(ig+npwx,iband1)
> dipole_aux(:,iband1,iband1) = dipole_aux(:,iband1,iband1) + ( 
> g(:,igk_k(ig,ik))+ xk(:,ik) ) * caux
> 
> I couldn't understand this relation between the dipole matrix element 
> dipole_aux and the reciprocal lattice vector g(:,igk_k(ig,ik))+ xk(:,ik).

the off-diagonal dipole matrix elements are related to the  matrix 
elements of the momentum p via a well-known identity: see e.g. 
http://astro1.panet.utoledo.edu/~ljc/fsum16.jpg . What is computed in 
epsilon.f90, subroutine dipole_calc, is exactly the matrix elements of 
p=-i\hbar\nabla on a plane-wave basis set.

"dipole_calc" also computes the diagonal matrix elements of p. Not sure 
what they are used for: the diagonal matrix elements of the dipole are 
ill-defined so the relation between matrix elements of the position r 
and of the momentum p is no longer valid in general.

Paolo

> I even read the epsilon.x user manual trying to understand the 
> underlying maths behind this code. I ultimately want to know the 
> mathematical relation between the epsilon and the reciprocal lattice 
> vector g(:,igk_k(ig,ik))+ xk(:,ik) that is used in this module.
> 
> If anyone can please direct me to any source material, developer guide 
> or papers, it would be great!
> 
> Regards,
> Yuvam
> 
> 
> 
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-- 
Paolo Giannozzi, Dip. Scienze Matematiche Informatiche e Fisiche,
Univ. Udine, via delle Scienze 206, 33100 Udine Italy, +39-0432-558216
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