[CP2K-user] [CP2K:13365] a question on the Kohn-Sham orbital expressed as Gaussian basis function, is the Kohn-Sham orbital translational invariant (obey periodicity)?

[Lucas Lodeiro]

Hi Juerg,

Reading your answer, I had a question. I understand the case at gamma
point, the orbitals are traslational invariant as the local functions, due
to the modulation function of Bloch function is equal to unity. But, what
happened with non-gamma points? In this case the orbitals are Bloch
functions, where the modulation function is a complex non-invariant one...
Do you construct the invariant part of orbital with local functions and use
a PW for modulation function?

El lun., 25 may. 2020 a las 4:11, <hut... at chem.uzh.ch> escribió:

> Hi
>
> there is nothing special about local basis functions and PBC
> in the GPW method. You can find the same reasonings in any
> approach using local basis functions (Gaussians, STO, numerical).
>
> At the gamma point the basis functions can be considered infinite
> sums of local functions replicated in all cells.
> Energy is defined per unit cell and this means integrals are over
> the unit cell only. As this is inconvenient, one tries to get them
> converted into all space integrals over single functions only and then
> summed over all combinations.
>
> Check the many papers for details.
>
> regards
>
> Juerg Hutter
> --------------------------------------------------------------
> 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: "Fangyong Yan"
> Sent by: cp... at googlegroups.com
> Date: 05/23/2020 11:25AM
> Subject: [CP2K:13360] a question on the Kohn-Sham orbital expressed as
> Gaussian basis function, is the Kohn-Sham orbital translational invariant
> (obey periodicity)?
>
> Dear CP2K developers,
>
> I have a question regarding the translational invariant of Kohn-Sham
> orbital in the Gaussian plane-wave method.
>
> The Kohn-Sham orbital should be translational invariant, and if we express
> the orbital using Gaussian basis function, these Gaussian basis function
> needs also be translational invariant, in the paper "A hybrid Gaussian and
> plane wave density functional scheme", by LIPPERT, HUTTER and PARRINELLO,
> MOLECULAR PHYSICS, 1997, VOL. 92, NO. 3, 477-487. The Kohn-Sham orbital has
> been expanded by Gaussian basis function which includes the periodicity.
> (Eq. 4 and 5 in the paper).
>
> However, in the recent paper of CP2K, "QUICKSTEP: Fast and accurate
> density functional calculations using a mixed Gaussian and plane waves
> approach", Joost VandeVondele, Matthias Krack , Fawzi Mohamed , Michele
> Parrinello , Thomas Chassaing , Jürg Hutter, Computer Physics
> Communications 167 (2005) 103–128, from Eq. 1, the definition of electron
> density, has been expanded based on Gaussian basis function, but these
> Gaussian basis function has not mentioned to take account of periodicity.
> But if the Gaussian basis function has not taken account the periodicity,
> the electron density is not translational invariant.
>
> Could you please help me with Eq. 1 regarding the periodicity of electron
> density in this recent paper of C2PK?
>
> Thank you very much!
>
> Fangyong
>
>
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