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

[Dmitry Ryndyk]

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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