[CP2K-user] [CP2K:14389] how does CP2K construct electron density in real-space FFT mesh point? Is the Gaussian product a periodic Gaussian product?

[Fangyong Yan]

Dear Thomas,

I forget to mention, the periodic Gaussian is not truncated in cp2k,
because when we truncate it, it is no longer periodic.

Best,

Fangyong

On Mon, Dec 14, 2020 at 3:30 PM Fangyong Yan <fyya... at gmail.com> wrote:

> Dear Thomas,
>
> Thanks for your explanation! Yes, periodicity has to be kept for condensed
> phase calculations.
>
> Best,
>
> Fangyong
>
> On Mon, Dec 14, 2020 at 5:12 AM Thomas Kühne <tku... at gmail.com> wrote:
>
>> 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> 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> 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
>>>>
>>>>  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
>>>>
>>>> --------------------------------------------------------------
>>>> 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
>>>> Winterthurerstrasse 190
>>>> CH-8057 Zürich, Switzerland
>>>> ---------------------------------------------------------------
>>>>
>>>> -----cp... at googlegroups.com wrote: -----
>>>> To: "cp2k" <cp... at googlegroups.com>
>>>> From: "fy... at gmail.com"
>>>> Sent by: 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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