  In the present work, we investigate the scalar, axialvector and tensor
doubly-charmed tetraquark molecular states without strange, with strange and
with doubly-strange via the QCD sum rules, and try to make assignment of the
$T^+_{cc}$ from the LHCb collaboration in the scenario of molecular states. The
predictions favor assigning the $T^+_{cc}$ to be the lighter $DD^{*}$ molecular
state with the spin-parity $J^P=1^+$ and isospin $I=0$, while the heavier
$DD^{*}$ molecular state with the spin-parity $J^P=1^+$ and isospin $I=1$ still
escapes experimental detections, the observation of the heavier $DD^{*}$
molecular state would shed light on the nature of the $T_{cc}^+$. All the
predicted doubly-charmed tetraquark molecular states can be confronted to the
experimental data in the future.
