[CP2K-user] [CP2K:21937] Re: Real space representation of S and KS matrices with k-points

[Dmitry Ryndyk]

Hi Fabian,

Unfortunately, it does not work if you need to analyse different parts of a 
bigger system and calculate matrix elements for these parts consistently.

Best,
Dmitry
Fabian Ducry schrieb am Donnerstag, 30. Oktober 2025 um 11:24:03 UTC+1:

> 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... 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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