How can I reoptimize pseudo-hydrogen pseudodos/basis sets in CP2K?

[Jun Ye]

I want to set up pseudo-hydrogen in CP2K. I've already looked at previous posts on this topic, such as https://groups.google.com/g/cp2k/c/n8E8a3u5Ous/m/Wxn_aGKUHAAJ

However, I found that simply setting the CORE_CORRECTION parameter still causes intermediate states at the band edges. So I reoptimized the pseudo-hydrogen pseudodos, but existing resources (such as tests/ATOM/regtest-pseudo and ling_basis_pseudo.pdf) seem to only cover optimizing regular atoms, not pseudo atoms. I've tried for a long time, and the reoptimized pseudodos still cause intermediate states at the band edges after DOS calculations.

Could anyone offer some help? Example files would be even better.