a question on the Kohn-Sham orbital expressed as Gaussian basis function, is the Kohn-Sham orbital translational invariant (obey periodicity)?

[Fangyong Yan]

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

[hut...@chem.uzh.ch]

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...@chem.uzh.ch
Winterthurerstrasse 190
CH-8057 Zürich, Switzerland
---------------------------------------------------------------

-----cp...@googlegroups.com wrote: -----
To: "cp2k" <cp...@googlegroups.com>
From: "Fangyong Yan"
Sent by: cp...@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)?

--
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 cp2k+uns...@googlegroups.com.
To view this discussion on the web visit https://groups.google.com/d/msgid/cp2k/48567acf-57f7-4751-86f6-2b5b128796d6%40googlegroups.com.

[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?

To view this discussion on the web visit https://groups.google.com/d/msgid/cp2k/OF366936E3.28162C3A-ONC1258573.002CFDC1-C1258573.002CFDC2%40lotus.uzh.ch.

[hut...@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...@chem.uzh.ch
Winterthurerstrasse 190
CH-8057 Zürich, Switzerland
---------------------------------------------------------------

-----cp...@googlegroups.com wrote: -----

To: cp...@googlegroups.com
From: "Lucas Lodeiro"
Sent by: cp...@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)?

To view this discussion on the web visit https://groups.google.com/d/msgid/cp2k/CAOFT4PLKMV8og%3D-fFP1ib9TW8Qi7MVzi4omOds90Xyo2oC5Evw%40mail.gmail.com.

[Fangyong Yan]

Dear Professor Hutter,

Thank you very much for your advice! 

I will read Carla Roeti's paper, and his book on periodic Hartree Fock: Hartree-Fock Ab Initio Treatment of Crystalline Systems, Cesare Pisani, Roberto Dovesi, Carla Roetti. 

Best wishes,

Fangyong

To view this discussion on the web visit https://groups.google.com/d/msgid/cp2k/OFC668D3DA.627DC07D-ONC1258573.003FA34C-C1258573.003FA34D%40lotus.uzh.ch.