  In this work, we try to understand the inconsistency reported by [Yaresko,
Phys. Rev. B. {\bf 77}, 115106 (2008)] in the theoretically estimated nature
and the variation of magnitude of nearest neighbour exchange coupling constant
($\arrowvert${\it J$_{1}$}$\arrowvert$) with increasing $U$ in ACr$_{2}$O$_{4}$
(A=Zn, Cd, Mg and Hg) compounds by using density functional theory. In
unconstrained calculations, the nature and variation of $\arrowvert${\it
J$_{1}$}$\arrowvert$ as a function of $U$ in the present study are not
consistent with the experimental data and not according to the relation, {\it
J$_{1}$}$\propto$$\frac{t^{2}}{U}$ especially for CdCr$_{2}$O$_{4}$ and
HgCr$_{2}$O$_{4}$ for $U >$3 eV and U=2-6 eV, respectively. Such an
inconsistent behavior of $\arrowvert${\it J$_{1}$}$\arrowvert$ is almost
similar to that of Yaresko for these two compounds for $U$=2-4 eV. For
ZnCr$_{2}$O$_{4}$ and MgCr$_{2}$O$_{4}$, the nature and the variation of
$\arrowvert${\it J$_{1}$}$\arrowvert$ in the present work are in accordance
with the experimental data and above mentioned relation for $U$=2-6 eV and are
similar to that of Yaresko for ZnCr$_{2}$O$_{4}$ for $U$=2-4 eV. However, in
constrained calculations the nature and variation of $\arrowvert${\it
J$_{1}$}$_{c}$$\arrowvert$ in the present work are according to experimental
data and above above mentioned relation for all four compounds. Hence, the
present study shows the importance of constrained calculations in understanding
the magnetic behaviour of these spinels. The values of magnitude of Curie-Weiss
temperature [$\arrowvert$($\varTheta$$_{CW}$)$_{c}$$\arrowvert$] for
ZnCr$_{2}$O$_{4}$$>$MgCr$_{2}$O$_{4}$$>$CdCr$_{2}$O$_{4}$$>$HgCr$_{2}$O$_{4}$
for $U$=2-5 eV, which are according to the order of experimentally observed
values for these spinels.
