  Inspired by the $T^+_{cc}$ signal discovered by the LHCb Collaboration, we
systematically investigate the doubly heavy tetraquark states with the molecule
configuration $[Q_1\bar{q}_2][Q_3\bar{q}_4]$ ($Q=c$ and $b$, $q=u$, $d$ and
$s$) in a nonrelativistic quark model. The model involves a color screening
confinement potential, meson-exchange interactions and one-gluon-exchange
interactions. The $T^+_{cc}$ state with $IJ^P=01^+$ is a very loosely bound
deuteron-like state with a binding energy around 0.34 MeV and a huge size of
4.32 fm. Both the meson exchange force and the coupled channel effect play a
pivotal role. Without the meson exchange force, there does not exist the
$T^+_{cc}$ molecular state. In strong contrast, the QCD valence bond forms
clearly in the $T^+_{bb}$ system when we turn off the meson-exchange force,
which is very similar to the hydrogen molecule in QED. Moreover, the $T^+_{bb}$
becomes a helium-like QCD-atom if we increase the bottom quark mass by a factor
of three. Especially, the $T^+_{bb}$ states with $01^+$, $T^+_{bc}$ with $00^+$
and $01^+$ and the $V$-spin antisymmetric states $T^+_{bbs}$ with
$\frac{1}{2}1^+$, $T^+_{bcs}$ with $\frac{1}{2}0^+$ and $\frac{1}{2}1^+$ can
form a compact, hydrogen molecule-like or deuteron-like bound state with
different binding dynamics. The high-spin states $T^+_{bc}$ with $02^+$ and
$T^+_{bcs}$ with $\frac{1}{2}2^+$ can decay into $D$-wave $\bar{B}D$ and
$\bar{B}_sD$ although they are below the thresholds $\bar{B}^*D^*$ and
$\bar{B}^*_sD^*$, respectively. The isospin and $V$-spin symmetric states are
unbound. We also calculate their magnetic moments and axial charges.
