[CP2K-user] [CP2K:21931] Re: Real space representation of S and KS matrices with k-points

[Dmitry Ryndyk]

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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