[CP2K-user] [CP2K:13370] a question on the Kohn-Sham orbital expressed as Gaussian basis function, is the Kohn-Sham orbital translational invariant (obey periodicity)?

[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

On Mon, May 25, 2020 at 7:35 AM <hut... at chem.uzh.ch> wrote:

> 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
>
>
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