  In this article, we construct both the color singlet-singlet type and
octet-octet type currents to interpolate the $X(3872)$, $Z_c(3900)$,
$Z_b(10610)$, and calculate the vacuum condensates up to dimension-10 in the
operator product expansion. Then we study the axial-vector hidden charmed and
hidden bottom molecular states with the QCD sum rules, explore the energy scale
dependence of the QCD sum rules for the heavy molecular states in details, and
use the formula $\mu=\sqrt{M^2_{X/Y/Z}-(2{\mathbb{M}}_Q)^2}$ with the effective
masses ${\mathbb{M}}_Q$ to determine the energy scales. The numerical results
support assigning the $X(3872)$, $Z_c(3900)$, $Z_b(10610)$ as the color
singlet-singlet type molecular states with $J^{PC}=1^{++}$, $1^{+-}$, $1^{+-}$,
respectively, more theoretical and experimental works are still needed to
distinguish the molecule and tetraquark assignments; while there are no
candidates for the color octet-octet type molecular states.
