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In this work, we try to understand the inconsistency reported by [Yaresko,Phys. Rev. B. {\bf 77}, 115106 (2008)] in the theoretically estimated natureand 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. Inunconstrained calculations, the nature and variation of $\arrowvert${\itJ$_{1}$}$\arrowvert$ as a function of $U$ in the present study are notconsistent with the experimental data and not according to the relation, {\itJ$_{1}$}$\propto$$\frac{t^{2}}{U}$ especially for CdCr$_{2}$O$_{4}$ andHgCr$_{2}$O$_{4}$ for $U >$3 eV and U=2-6 eV, respectively. Such aninconsistent behavior of $\arrowvert${\it J$_{1}$}$\arrowvert$ is almostsimilar to that of Yaresko for these two compounds for $U$=2-4 eV. ForZnCr$_{2}$O$_{4}$ and MgCr$_{2}$O$_{4}$, the nature and the variation of$\arrowvert${\it J$_{1}$}$\arrowvert$ in the present work are in accordancewith the experimental data and above mentioned relation for $U$=2-6 eV and aresimilar to that of Yaresko for ZnCr$_{2}$O$_{4}$ for $U$=2-4 eV. However, inconstrained calculations the nature and variation of $\arrowvert${\itJ$_{1}$}$_{c}$$\arrowvert$ in the present work are according to experimentaldata and above above mentioned relation for all four compounds. Hence, thepresent study shows the importance of constrained calculations in understandingthe magnetic behaviour of these spinels. The values of magnitude of Curie-Weisstemperature [$\arrowvert$($\varTheta$$_{CW}$)$_{c}$$\arrowvert$] forZnCr$_{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 observedvalues for these spinels.

Constrained DFT+$U$ approach for understanding the magnetic behaviour of ACr$_{2}$O$_{4}$ (A=Zn, Mg, Cd and Hg) compounds