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Bessel expansion of PAO/VPS
Date: 2017/04/17 06:03
Name: Artem Pulkin

I am trying to reproduce calculations described at

http://openmx-square.org/exx/CPC2009/

There, the complexity of expansion N log N originates from fast Fourier transform on a regular grid. However, OpenMX PAO and VPS are expressed on a logarithmic grid due to features at the origin. Thus, I assumed that the values on a regular grid are interpolated. I have a few conceptual questions here:

(i) Do I have to choose a very fine grid to include wavefunction / PAO features at the origin?
(ii) If yes, why is it N log N complexity? I would rather call it N exp N provided N is the number of points in the logarithmic grid.
(iii) Isn't it more efficient to do the Fourier integration of Eq. 8 on the logarithmic grid with complecity N^2 rather than having a very fine grid and effective complexity N exp N?
(iv) Did you consider Logarithmic Fourier Transform with even smallest complexity "N" for this problem?
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Re: Bessel expansion of PAO/VPS ( No.1 )
Date: 2017/04/20 13:05
Name: T. Ozaki

Hi Artem,

> (i) Do I have to choose a very fine grid to include wavefunction / PAO features
> at the origin?

Since PAOs are basis functions for pseudopotentials, the variation near the origin
is not so large compared to all electron cases. So, we do not need to have a very
fine grid near the origin.


> (ii) If yes, why is it N log N complexity? I would rather call it N exp N provided N
> is the number of points in the logarithmic grid.

The complexity comes from the FFT in Eqs. (8), (19), and (20), resulting in N log N.


> (iii) Isn't it more efficient to do the Fourier integration of Eq. 8 on the logarithmic
> grid with complecity N^2 rather than having a very fine grid and effective complexity N
> exp N?
> (iv) Did you consider Logarithmic Fourier Transform with even smallest complexity
> "N" for this problem?

If the Siegman-Talman method is used, we can keep the logarithmic grid.
The case is shown in Fig. 7. From the comparison, we found that regular mesh is
more efficient than the logarithmic grid for PAOs.

Regards,

TO
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Re: Bessel expansion of PAO/VPS ( No.2 )
Date: 2017/04/29 01:49
Name: Artem Pulkin

Dear Taisuke,

May I use this thread and your attention to enlighten myself in DFT basics.

As I understood, the calculations of overlap matrix and nonlocal potential matrix can be done with the same set of tools. However, the data found in VPS files has to be transformed into separable Kleinman-Bylander form. The latter relies on some initial spherical basis set. I found that Siesta uses eigenstates of an atomic Hamiltonian. Is it also the case in OpenMX? Can I use different spherical functions, say, those from PAO files?
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Re: Bessel expansion of PAO/VPS ( No.3 )
Date: 2017/05/18 10:57
Name: T. Ozaki

Hi Artem,

> Can I use different spherical functions, say, those from PAO files?

Data stored in the VPS files, *.vps are already in the separable form.
The data structure of the VPS file can be found at
http://t-ozaki.issp.u-tokyo.ac.jp/winter-school16/2-PP-Ozaki.pdf
If multiple valance states for a L-channel are included in PP,
the separable form is generated by a modified procedure of Vanderbilt,
while the separable form is generated by Blochl's technique if a valence
state for a L-channel is included in PP. For the latter case, the radial
functions of PAO can be utilized.

Anyway, you need to deeply touch the pseudopotential generator: ADPACK.

Regards,

TO
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Re: Bessel expansion of PAO/VPS ( No.4 )
Date: 2017/05/24 08:46
Name: Artem Pulkin

Dear Taisuke,

Do I understand it right: the non-local part of the pseudopotential is presented in the form Eq.(7) of Bolchl PRB (1990) where the radial part of the product 1/sqrt(c_i) |v*phi_i> are columns in VPS file (units are square roots of energy)?

Otherwise where do I find the angular momentum of the corresponding spherical harmonic? What about the spin part?

For example, there are 12 non-local components in

http://www.jaist.ac.jp/~t-ozaki/vps_pao2013/H/H_CA13.vps

As far as I understood this number is factorized into 3 Blochl projectors ( Blochl.projector.num) times two shells, L=0 (1s) and L=1 (2p), times 2 spins. Am I right? What is the column order in "Pseudo.Potentials" with respect to these numbers?

Thank you in advance.

Artem
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Re: Bessel expansion of PAO/VPS ( No.5 )
Date: 2017/05/24 09:45
Name: T. Ozaki

Dear Artem,

> the non-local part of the pseudopotential is presented in the form Eq.(7)
> of Bolchl PRB (1990)

Yes, for the both cases: Blochl and MBK we have the same form.
One can see in case of hydrogen that there are 15 columns in between
<Pseudo.Potentials and Pseudo.Potentials>, where the sequence is as follows:

x, log(x), Vlocal, Vs1(l+1/2), Vs1(l-1/2), Vs2(l+1/2), Vs2(l-1/2), Vs3(l+1/2), Vs3(l-1/2),
Vp1(l+1/2), Vp1(l-1/2), Vp2(l+1/2), Vp2(l-1/2), Vp3(l+1/2), Vp3(l-1/2).

Vlocal: local potential
Vs1(l+1/2): 1st non-local potential of s-channel with l+1/2, and the other cases can be deduced from the notation.

Note that though Vs does not depend on j=l+-1/2, we have two columns for l+-1/2
for the format to be consistent for other L-channels.

Regards,

Taisuke
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