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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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