[CP2K-user] [CP2K:14356] how does CP2K construct electron density in real-space FFT mesh point? Is the Gaussian product a periodic Gaussian product?

[Thomas Kühne]

Dear Fangyong, 

within the GPW scheme, the orbitals are described by contracted Gaussian functions, 
whereas electron density is expanded in plane-waves. You are right that the extent of 
the Gaussian functions is finite, which is controlled by eps values that are eventually 
related to EPS_DEFAULT. You are also right wrt to the plane-waves, whose truncation 
is controlled by CUTOFF. However, contrary to the cutoff of conventional PW codes, 
the latter differs by a factor of 4, which due to the fact that the density is the sum of the 
orbitals squared combined with the application of the Shannon-Nyquist sampling theorem. 
Lastly, within the GAPW approach, the Gaussian functions are indeed also periodic 
Gaussians. 

Best, 
Thomas

> Am 08.12.2020 um 06:22 schrieb Fangyong Yan <fyya... at gmail.com>:
> 
> Hi, Professor Hutter,
> 
> I read your paper, and I think when you calculate the electron density based on periodic Gaussian, you need to truncate the periodic Gaussian because it is an infinite series, but it decays fast since the periodic Gaussian is a localized basis set. Am I understanding correctly? 
> 
> For electron density represented by planewaves, it also has the same problem, it is an infinite series, and has to be truncated, which is the planewave cutoff. 
> 
> Another question, in the Gaussian augmented planewave method, the atomic hard and soft electron density, they are atom-based electron density, and are represented by Gaussian functions, they are also periodic Gaussian, in order to fulfill the periodicity, am I correct? Thanks!
> 
> Fangyong
> 
> 
> 
> 
> On Fri, Dec 4, 2020 at 9:49 AM Fangyong Yan <fyya... at gmail.com <mailto:fyya... at gmail.com>> wrote:
> Dear Professor Hutter, 
> 
> I will read the paper. Thanks!
> 
> Fangyong
> 
> On Fri, Dec 4, 2020 at 4:37 AM <hut... at chem.uzh.ch <mailto:hut... at chem.uzh.ch>> wrote:
> Hi
> 
> yes, it is periodic Gaussians. For details of the calculation
> read the publications:
> 
>  Lippert, G; Hutter, J; Parrinello, M.
>  MOLECULAR PHYSICS, 92 (3), 477-487 (1997).
>  A hybrid Gaussian and plane wave density functional scheme.
>  https://doi.org/10.1080/002689797170220 <https://doi.org/10.1080/002689797170220>
> 
>  VandeVondele, J; Krack, M; Mohamed, F; Parrinello, M; Chassaing, T;
>  Hutter, J. COMPUTER PHYSICS COMMUNICATIONS, 167 (2), 103-128 (2005).
>  QUICKSTEP: Fast and accurate density functional calculations using a
>  mixed Gaussian and plane waves approach.
>  https://doi.org/10.1016/j.cpc.2004.12.014 <https://doi.org/10.1016/j.cpc.2004.12.014>
> 
> --------------------------------------------------------------
> Juerg Hutter                         Phone : ++41 44 635 4491
> Institut für Chemie C                FAX   : ++41 44 635 6838
> Universität Zürich                   E-mail: hut... at chem.uzh.ch <mailto:hut... at chem.uzh.ch>
> Winterthurerstrasse 190
> CH-8057 Zürich, Switzerland
> ---------------------------------------------------------------
> 
> -----cp... at googlegroups.com <mailto:cp... at googlegroups.com> wrote: -----
> To: "cp2k" <cp... at googlegroups.com <mailto:cp... at googlegroups.com>>
> From: "fy... at gmail.com <mailto:fy... at gmail.com>" 
> Sent by: cp... at googlegroups.com <mailto:cp... at googlegroups.com>
> Date: 12/04/2020 08:42AM
> Subject: [CP2K:14345] how does CP2K construct electron density in real-space FFT mesh point? Is the Gaussian product a periodic Gaussian product?
> 
> Hi, 
> 
> The electron density in real-space FFT mesh point is defined as:
> 
> n(Ri) = summation of m, n {P_mn * Gaussian_mn(Ri), 
> 
> Ri is the grid point, m and n are the index for the Gaussian basis set, P_mn is the density matrix, Gaussian_mn(Ri) is the product of Gaussian basis set m and Gaussian basis set n,
> 
> my question is:
> 1) 
> is Gaussian_mn(Ri) a periodic Gaussian? (Product Gaussian is also a Gaussian). 
> That is,
> 
> Gaussian_mn(Ri) = Gaussian_mn(Ri+L_unitcell), where L_unitcell is the length for the unit cell in either 1,2 or 3 dimension. 
> 2) if it is the periodic Gaussian, and it should be in order to fulfill the periodicity of electron density, how do you do the calculation? 
> Because the periodic Gaussian is actually an infinite series, that is,
> Gaussian(Ri) = summation of L (Gaussian(Ri+L), where L = n*L_unitcell, and n is an integer and span from minus infinite to positive infinite.
> 
> Thanks!
> 
> Fangyong
> 
> 
> 
> 
> 
> 
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==============================
Thomas D. Kühne
Dynamics of Condensed Matter
Chair of Theoretical Chemistry
University of Paderborn
Warburger Str. 100
D-33098 Paderborn
Germany
tdku... at mail.upb.de
+49/(0)5251/60-5726

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