I'm running task 330 to calculate the response function for a set of q vectors, with no xc kernel (fxctype=0). My expectation in this case was that the non-interacting response, CHI0_ij.OUT, would be the same as the interacting response, CHI_ij.OUT. But this isn't the case, all of the non-zero elements are different.
I'm far from an expect on these types of calculations so I feel like my understanding is wrong. Can anyone help?
Harry
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fxctype=0 is RPA. In case of task 330 this would mean adding all RPA diagrams BUT including local field effects. These local fields then lead to chi0 being different from chi
chi=chi0{1+v.chi0}^-1
Sangeeta
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Thanks for the clarification Sangeeta. If I'm not interested in the interacting response function, is there a way of calculating only chi0, to save some computation time?
Harry
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Just reviving this thread to ask another related question about the interacting response function, chi, defined above by Sangeeta.
Is chi calculated element-wise, or as a full matrix? So are the elements given by:
chi_ij = chi0_ij/( 1+v.chi0_ij )
Or is that denominator a full matrix inversion? So the interaction mixes all of the elements of chi0.
In case it wasn't already obvious, I'm still learning my way around response functions so any help is appreciated!
Harry
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That is a very good question. In older version of Elk we did it component wise (i am guessing i and j reffer to x,y and z, i.e. direction of perturbation and response). But in the latest version we have implemented the full form i.e. the whole matrix is inverted.
Best
Sangeeta
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Thanks for the reply. Yes, my indices were referring to response in components of the density (charge, mx,my,mz) to components of the field (v,Bx,By,Bz). Which I'm pretty sure is the convention Elk uses.
Presumably the choice of whether or not to do it component-wise has a big impact on the result. Is there a good physical reason why plugging the full matrix into the equation is the 'correct' thing to do?
Regards,
Harry
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Looks like I'm missing something then; it's not obvious to me why the off-diagonal can't be done element-wise as in (with element (0,1) for example):
chi_01=chi0_01/(1+v.chi0_01)
Because chi_01 and chi0_01 are just scalar functions of q and omega aren't they?
Harry
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Hi Harry and Sangeeta,
I'd like to add my voice to this discussion too. Let us for simplicity first consider the nonzero q case. Then chi_0, chi and eps are all scalars ( no x,y,z indices) because they are the longitudinal response functions (e.g., rho = chi * v_ext). However, chi0, chi, and epsilon are infinite matrices in the reciprocal space, indexed with the reciprocal lattice vectors G,G'. Then, the equality
chi=chi0(1+vchi0)^{-1} holds as a matrix equality, not, of course, element-wise with respect to the G,G' vectors.
Let us first agree (or disagree) on this, and after that pass to a little more complicated q=0 case.
Vladimir.
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I believe elk returns only the G=G'=0 case, does it not? So is it correct to apply the operation to go from chi0 to chi, but only on the GG'=00 component? Or should we compute the interacting response function in real space, then transform and take the GG'=00 part?
I have to admit I wasn't thinking about the G and G' aspect of chi when I phrased my question. I considered the matrix nature to be in the different components of the density and the perturbing field. So does this make chi an even larger object? And the problem of computing chi from chi0 even less trivial?
Harry
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Please see Fig. 1 of the following link for the structure of the response function. Note that each grey-lines marked square is a matrix in and G' i.e. the response fuction is a matrix of a matrix of a matrix. This also means 1 is a identity matrix. https://arxiv.org/abs/1808.00215
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About what elk returns: it prints the macrioscopic response function. However all microscopic parts like LFE are included in inversion of the matrix and in the end you just look at the macroscopic part that ca be compared to the experiments.
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Hi Harry,
yes, microscopic chi0 and chi are matrices with rows and columns numbered with G and G'. The relation
chi=chi0 (1+v*chi0)^(-1) (1)
(which holds in RPA only) is a matrix relation. We first compute the chi0 matrix, then by (1) obtain the chi matrix. For certain applications, as, e.g., optics, EELS, we are interested in the upper left element (G=0,G'=0) of this matrix only, but it doesn't mean that we can use (1) as a relation between chi0_00 and chi_00: all elements chi0_GG' are involved in finding of chi_00. This is known as local-field effects (LFE), as Sangeeta mentions. In the case of the homogeneous electron gas only, the matrices are diagonal, and (1) can be used as a scalar relation.
It is easy to understand LFE: since
rho_G= sum_G' chi0_GG' V^{KS}_G' (2)
and
rho_G= sum_G' chi_GG' V^{ext}_G' (3)
where rho_G, V^{KS}_G, and V^{ext}_G are the induced density, Kohn-Sham potential, and the external potential, respectively, (1) immediately follows from (2) and (3) if, in RPA, we put V^{KS}_G=V^{ext}_G+4 pi/(G+q)^2 * rho_G. Usually, only V^{ext}_0 is nonzero, but all V^{KS}_G are nonzero in a crystal.
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Hi All,
I'm running task 330 to calculate the response function for a set of q vectors, with no xc kernel (fxctype=0). My expectation in this case was that the non-interacting response, CHI0_ij.OUT, would be the same as the interacting response, CHI_ij.OUT. But this isn't the case, all of the non-zero elements are different.
I'm far from an expect on these types of calculations so I feel like my understanding is wrong. Can anyone help?
Harry
Hi Harry,
fxctype=0 is RPA. In case of task 330 this would mean adding all RPA diagrams BUT including local field effects. These local fields then lead to chi0 being different from chi
chi=chi0{1+v.chi0}^-1
Sangeeta
Thanks for the clarification Sangeeta. If I'm not interested in the interacting response function, is there a way of calculating only chi0, to save some computation time?
Harry
Hi All,
Just reviving this thread to ask another related question about the interacting response function, chi, defined above by Sangeeta.
Is chi calculated element-wise, or as a full matrix? So are the elements given by:
chi_ij = chi0_ij/( 1+v.chi0_ij )
Or is that denominator a full matrix inversion? So the interaction mixes all of the elements of chi0.
In case it wasn't already obvious, I'm still learning my way around response functions so any help is appreciated!
Harry
Hi Harry,
That is a very good question. In older version of Elk we did it component wise (i am guessing i and j reffer to x,y and z, i.e. direction of perturbation and response). But in the latest version we have implemented the full form i.e. the whole matrix is inverted.
Best
Sangeeta
Hi Sangeeta,
Thanks for the reply. Yes, my indices were referring to response in components of the density (charge, mx,my,mz) to components of the field (v,Bx,By,Bz). Which I'm pretty sure is the convention Elk uses.
Presumably the choice of whether or not to do it component-wise has a big impact on the result. Is there a good physical reason why plugging the full matrix into the equation is the 'correct' thing to do?
Regards,
Harry
For diagonal part it has almost no impact. For off-diagonal, there is no other way to do it.
Looks like I'm missing something then; it's not obvious to me why the off-diagonal can't be done element-wise as in (with element (0,1) for example):
chi_01=chi0_01/(1+v.chi0_01)
Because chi_01 and chi0_01 are just scalar functions of q and omega aren't they?
Harry
Hi Harry and Sangeeta,
I'd like to add my voice to this discussion too. Let us for simplicity first consider the nonzero q case. Then chi_0, chi and eps are all scalars ( no x,y,z indices) because they are the longitudinal response functions (e.g., rho = chi * v_ext). However, chi0, chi, and epsilon are infinite matrices in the reciprocal space, indexed with the reciprocal lattice vectors G,G'. Then, the equality
chi=chi0(1+vchi0)^{-1} holds as a matrix equality, not, of course, element-wise with respect to the G,G' vectors.
Let us first agree (or disagree) on this, and after that pass to a little more complicated q=0 case.
Vladimir.
Thanks Vladimir,
I believe elk returns only the G=G'=0 case, does it not? So is it correct to apply the operation to go from chi0 to chi, but only on the GG'=00 component? Or should we compute the interacting response function in real space, then transform and take the GG'=00 part?
I have to admit I wasn't thinking about the G and G' aspect of chi when I phrased my question. I considered the matrix nature to be in the different components of the density and the perturbing field. So does this make chi an even larger object? And the problem of computing chi from chi0 even less trivial?
Harry
Please see Fig. 1 of the following link for the structure of the response function. Note that each grey-lines marked square is a matrix in and G' i.e. the response fuction is a matrix of a matrix of a matrix. This also means 1 is a identity matrix.
https://arxiv.org/abs/1808.00215
About what elk returns: it prints the macrioscopic response function. However all microscopic parts like LFE are included in inversion of the matrix and in the end you just look at the macroscopic part that ca be compared to the experiments.
Hi Harry,
yes, microscopic chi0 and chi are matrices with rows and columns numbered with G and G'. The relation
chi=chi0 (1+v*chi0)^(-1) (1)
(which holds in RPA only) is a matrix relation. We first compute the chi0 matrix, then by (1) obtain the chi matrix. For certain applications, as, e.g., optics, EELS, we are interested in the upper left element (G=0,G'=0) of this matrix only, but it doesn't mean that we can use (1) as a relation between chi0_00 and chi_00: all elements chi0_GG' are involved in finding of chi_00. This is known as local-field effects (LFE), as Sangeeta mentions. In the case of the homogeneous electron gas only, the matrices are diagonal, and (1) can be used as a scalar relation.
It is easy to understand LFE: since
rho_G= sum_G' chi0_GG' V^{KS}_G' (2)
and
rho_G= sum_G' chi_GG' V^{ext}_G' (3)
where rho_G, V^{KS}_G, and V^{ext}_G are the induced density, Kohn-Sham potential, and the external potential, respectively, (1) immediately follows from (2) and (3) if, in RPA, we put V^{KS}_G=V^{ext}_G+4 pi/(G+q)^2 * rho_G. Usually, only V^{ext}_0 is nonzero, but all V^{KS}_G are nonzero in a crystal.