[CP2K-user] [CP2K:21930] Re: Real space representation of S and KS matrices with k-points

[Augustin Bussy]

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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