We present an explicit structure for the Baer invariant of the free n th nilpotent group (the n th nilpotent product of the infinite cyclic group, textbf{ Z} st{n}* textbf{ Z} st{n}* ldots st{n}* textbf{ Z} ) with respect to the variety { cal V with the set of words V= {[ ga_{c_1+1}, ga_{c_2+1}] } , for all c_1 geq c_2 and 2c_2-c_1>2n-2$. In particular, the variety { cal V} can be a polynilpotent variety, { cal N}_{c,1} , for all c> 2n-2 , and also the variety of metabelian groups, { cal S}_2 .

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