  We have calculated the isotropic $C\_6$ coefficients characterizing the
long-range van der Waals interaction between two identical heteronuclear
alkali-metal diatomic molecules in the same arbitrary vibrational level of
their ground electronic state $X^1\Sigma^+$. We consider the ten species made
up of $^7$Li, $^{23}$Na, $^{39}$K, $^{87}$Rb and $^{133}$Cs. Following our
previous work [M.~Lepers \textit{et.~al.}, Phys.~Rev.~A \textbf{88}, 032709
(2013)] we use the sum-over-state formula inherent to the second-order
perturbation theory, composed of the contributions from the transitions within
the ground state levels, from the transition between ground-state and excited
state levels, and from a crossed term. These calculations involve a combination
of experimental and quantum-chemical data for potential energy curves and
transition dipole moments. We also investigate the case where the two molecules
are in different vibrational levels and we show that the Moelwyn-Hughes
approximation is valid provided that it is applied for each of the three
contributions to the sum-over-state formula. Our results are particularly
relevant in the context of inelastic and reactive collisions between ultracold
bialkali molecules, in deeply bound or in Feshbach levels.
