[CP2K-user] [CP2K:13366] a question on the Kohn-Sham orbital expressed as Gaussian basis function, is the Kohn-Sham orbital translational invariant (obey periodicity)?

[hut... at chem.uzh.ch]

Hi

integrals are calculated in real space and then we use Fourier 
transformation to get the matrix elements of the KS matrix in
reciprocal space. See the many papers on PBC implementations 
from GAUSSIAN, Turbomole, FHI-AIMS, and of course CRYSTAL groups.

For example

Carla Roetti "The Crystal Code", in C.Pisani, Lecture Notes in Chemistry 67 (1996)

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: cp... at googlegroups.com
From: "Lucas Lodeiro" 
Sent by: cp... at googlegroups.com
Date: 05/25/2020 10:57AM
Subject: Re: [CP2K:13366] a question on the Kohn-Sham orbital expressed as Gaussian basis function, is the Kohn-Sham orbital translational invariant (obey periodicity)?

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
 
 
   -- 
  You received this message because you are subscribed to the Google Groups "cp2k" group.
  To unsubscribe from this group and stop receiving emails from it, send an email to cp... at googlegroups.com.
  To view this discussion on the web visit https://groups.google.com/d/msgid/cp2k/48567acf-57f7-4751-86f6-2b5b128796d6%40googlegroups.com.
 
 
 -- 
 You received this message because you are subscribed to the Google Groups "cp2k" group.
 To unsubscribe from this group and stop receiving emails from it, send an email to cp... at googlegroups.com.
 To view this discussion on the web visit https://groups.google.com/d/msgid/cp2k/OF366936E3.28162C3A-ONC1258573.002CFDC1-C1258573.002CFDC2%40lotus.uzh.ch.
   
  -- 
 You received this message because you are subscribed to the Google Groups "cp2k" group.
 To unsubscribe from this group and stop receiving emails from it, send an email to cp... at googlegroups.com.
 To view this discussion on the web visit https://groups.google.com/d/msgid/cp2k/CAOFT4PLKMV8og%3D-fFP1ib9TW8Qi7MVzi4omOds90Xyo2oC5Evw%40mail.gmail.com.