[CP2K-user] [CP2K:21929] Re: Real space representation of S and KS matrices with k-points

[Augustin Bussy]

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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