[CP2K-user] [CP2K:21933] Re: Real space representation of S and KS matrices with k-points

[Fabian Ducry]

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.ryndyk 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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