[CP2K-user] [CP2K:21935] Re: Real space representation of S and KS matrices with k-points

[Fabian Ducry]

Hi Dmitry,

You can easily do it with https://manual.cp2k.org/trunk/CP2K_INPUT/FORCE_EVAL/SUBSYS/TOPOLOGY/CENTER_COORDINATES.html
bo need to manually shift the atoms. But I think it should be better documented. I'll try to find time to do it.

Cheers,
Fabian

On 29 October 2025 23:05:43 GMT+03:00, Dmitry Ryndyk <dmitry.ryndyk at tu-dresden.de> wrote:
>Ah, indeed!
>It works and technically solves my problem.
>Thank you, Fabian!
>Why not do it by default when creating RS image matrices?
>
>Best wishes,
>Dmitry
>
>Fabian Ducry schrieb am Mittwoch, 29. Oktober 2025 um 17:14:47 UTC+1:
>
>> Hi Dmitry,
>>
>> the coordinates need to be in (-L/2, L/2) in each dimension, where L is 
>> the length of the cell.
>>
>> Cheers,
>> Fabian
>>
>>
>> On 29 October 2025 18:21:43 GMT+03:00, Dmitry Ryndyk <
>> dmitry... at tu-dresden.de> wrote:
>>
>>> Meanwhile, I found a very strange problem with the real space matrix 
>>> images themselves.
>>> As an example, a very simple system of a 1D H array, with 4 atoms in the 
>>> unit cell (input file in attachment).
>>> The coordinates are
>>>     &CELL
>>>       ABC    16.0 16.0 6.0
>>>       PERIODIC xyz
>>>     &END CELL
>>>     &COORD
>>>       H        0.00000000       0.00000000       0.00000000 
>>>       H        0.00000000       0.00000000       1.50000000 
>>>       H        0.00000000       0.00000000       3.00000000 
>>>       H        0.00000000       0.00000000       4.50000000 
>>>     &END COORD
>>>
>>> The 5 RS image cells are
>>> S CSR write|   5 periodic images
>>>       Number    X      Y      Z
>>>          1      0      0      0
>>>          2      0      0     -1
>>>          3      0      0      1
>>>          4      0      0     -2
>>>          5      0      0      2
>>>
>>> and the 0,0,0 image for S overlap is (all other in attachment)
>>>
>>>        1       1  0.10000000000000E+001
>>>        1       2  0.39126534955779E+000
>>>        1       3  0.39734112631705E-001
>>>        1       4  0.39126534955779E+000
>>>        2       1  0.39126534955779E+000
>>>        2       2  0.10000000000000E+001
>>>        2       3  0.39126534955779E+000
>>>        2       4  0.39734112631705E-001
>>>        3       1  0.39734112631705E-001
>>>        3       2  0.39126534955779E+000
>>>        3       3  0.10000000000000E+001
>>>        3       4  0.12433452435606E-002
>>>        4       1  0.39126534955779E+000
>>>        4       2  0.39734112631705E-001
>>>        4       3  0.12433452435606E-002
>>>        4       4  0.10000000000000E+001
>>>
>>> As one can see, the values of S correspond to the atom positions in order
>>> 4 1 2 3
>>> not
>>> 1 2 3 4
>>> as I assume to be correct.
>>>
>>> If one makes only one RS image, e.g., taking 
>>> ABC    16.0 16.0 16.0
>>> in the input file, the file looks correct.
>>>
>>>        1       1  0.10000000000000E+001
>>>        1       2  0.39126534955779E+000
>>>        1       3  0.39734112631705E-001
>>>        1       4  0.12433452435606E-002
>>>        2       1  0.39126534955779E+000
>>>        2       2  0.10000000000000E+001
>>>        2       3  0.39126534955779E+000
>>>        2       4  0.39734112631705E-001
>>>        3       1  0.39734112631705E-001
>>>        3       2  0.39126534955779E+000
>>>        3       3  0.10000000000000E+001
>>>        3       4  0.39126534955779E+000
>>>        4       1  0.12433452435606E-002
>>>        4       2  0.39734112631705E-001
>>>        4       3  0.39126534955779E+000
>>>        4       4  0.10000000000000E+001
>>>
>>> Here (1,4) and (4,1) matrix elements are the smallest.
>>>
>>> Besides, this "cyclic shift" depends on k-points and the linear shift of 
>>> atom coordinates.
>>> Even if it does not change k-point calculations itself, it can be 
>>> important in other problems.
>>> Actually, I found this problem in "NEGF" transport systems, where it is 
>>> important.
>>> At the moment, I have not found the origin of this problem.
>>>
>>>
>>> Dmitry Ryndyk schrieb am Dienstag, 28. Oktober 2025 um 11:13:35 UTC+1:
>>>
>>> Dear Augustin,
>>>
>>> thank you for the fast answer. It helps me to understand what is going on!
>>>
>>> Best wishes,
>>> Dmitry
>>>
>>> Augustin Bussy schrieb am Dienstag, 28. Oktober 2025 um 11:11:38 UTC+1:
>>>
>>> For clarity: Both the S and KS matrices are Hermitian in *k-space*, but 
>>> in real space, we have: S_ij^*b* = S_ji^-*b*
>>>
>>> On Tuesday, 28 October 2025 at 10:57:55 UTC+1 Augustin Bussy wrote:
>>>
>>> Dear Dimitry,
>>>
>>> Dealing with CP2K's real space matrices in k-point calculations can be 
>>> quite challenging. In principle, it follows equations (10) and (11) of 
>>> https://arxiv.org/pdf/2508.15559. Elements i of the 
>>> qs_env%ks_env%matrix_s_kp array contains real space overlap matrix elements 
>>> between AOs in the main cell, and AOs in periodic image with index i. The 
>>> indexing of periodic images is that imposed by the neighbor lists.
>>>
>>> For historical reasons, the KP overlap and KS matrices are stored as 
>>> DBCSR *symmetric* types, even though they are not symmetric. That's 
>>> where it gets complicated. Both the S and KS matrices are Hermitian, and 
>>> they have the following symmetry: S_ij^*b* = S_ji^-*b*, where *b* 
>>> denotes the translation from the main cell to a given periodic image.
>>> If you have access to the upper diagonal of a real space matrix with one 
>>> AO in a periodic cell shifted by *b*, and that of a real space matrix 
>>> with an AO shifted by -*b*, then you can reconstruct the full, 
>>> asymmetric, real space matrix at *b*.
>>>
>>> In the code, this operation is done when performing Fourier transforms 
>>> real space to k-space. For example here: 
>>> https://github.com/cp2k/cp2k/blob/5f3bc36082e75c975caee6a92073f395a2af7674/src/kpoint_methods.F#L855-L864
>>> .
>>>
>>> I hope that helps.
>>> Best,
>>> Augustin
>>> On Tuesday, 28 October 2025 at 01:38:16 UTC+1 Dmitry Ryndyk wrote:
>>>
>>> Dear developers,
>>>
>>> I would greatly appreciate it if you could provide me with the reference 
>>> to the exact description of the real space S and KS matrices, stored 
>>> in qs_env%ks_env%matrix_ks_kp and qs_env%ks_env%matrix_s_kp.
>>> I mean not the details of DBCSR matrices, but the way the matrix elements 
>>> are placed inside the matrices.
>>> My test calculations, as well as some code investigation, show that these 
>>> matrix elements are mixed between space replicas ("images") used at k-point 
>>> calculations, and some rearrangement is required to get normal symmetrical 
>>> matrices, which depend on the atomic indices. 
>>>
>>> Thank you,
>>> Dmitry
>>>
>>>
>
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