  As our recent quark model calculation~\cite{Noh:2023fdy} suggests a strong
possibility of a compact $T_{cc}$ that closely reproduces experimental mass, we
have a strong incentive to extend our work to investigate the possible compact
configuration of a pentaquark $udcc\bar{s}$, which is related to the structure
of the doubly charmed tetraquark $T_{cc}$. Since the introduction of a complete
set of 3-dimensional harmonic oscillator bases to a spatial wave function in
solving a quark model-based Hamiltonian with variational method leads to a more
accurate value of the mass, it seems natural that future studies of the
pentaquark should be treated with the same elaborate technical approach. To
attain such precision for the ground state energy, we utilize a complete set of
3-dimensional harmonic oscillator base up to 6th quanta. Before carrying out
this process, one important thing that has to be taken into account is to find
out the color $\otimes$ spin states of the pentaquark for the evaluation of
color and spin interaction most essential to the quark model configuration. To
easily identify the suitable configuration, we make a systematic analysis of
$SU(6)_{CS}$ irreducible representation of the pentaquark, from which we find
that there is a correspondence between the color $\otimes$ spin states obtained
from their coupling scheme and the multiplet of the $SU(6)_{CS}$ irreducible
representation of the pentaquark. We find that the energy of the pentaquark
configuration is +18.5 MeV above the lowest threshold for decay into $\Xi_{cc}$
and $K$, suggesting that this configuration is not stable against its decay.
Nonetheless, while we used a Gaussian hyperfine potential, it was recently
found that a Yukawa form leads to a stronger attraction for the $T_{cc}$
configuration. Therefore it is important to study the same configuration using
the latter potential.
