Gaussians at lattice points

Question

Fix $d \in \mathbb{N}$ and for $x \in \mathbb{R}^d$ let $N(x)$ be the standard Gaussian centered at $x$. Fix a large constant $c$. Let $\mathcal{V}$ be the subspace of $L^2(\mathbb{R}^d)$ spanned by the vectors $\{N(x): x \in \mathbb{Z}^d,||x||_2 \geq c\}$. I would like to obtain an estimate for the $L^2$ norm of the orthogonal projection of $N(0)$ onto $\mathcal{V}$.