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