In the present context, we investigate to obtain some more results about$2$-nilpotent multiplier $\mathcal{M}^{(2)}(L)$ of a finite dimensionalnilpotent Lie algebra $L$. For instance, we characterize the structure of$\mathcal{M}^{(2)}(H)$ when $H$ is a Heisenberg Lie algebra. Moreover, we givesome inequalities on $ \mathrm{dim}~ \mathcal{M}^{(2)}(L)$ to reduce a wellknown upper bound on $2$-nilpotent multiplier as much as possible. Finally, weshow that $H(m)$ is 2-capable if and only if m=1.
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