= Solution
The measure, Gaussian factor, $r^{-1}$, coordinates, and <derivatives> all transform covariantly under rotations, so $T_{ijkl}$ is an <isotropic tensor integral>. Symmetry gives $\beta=\gamma$.
Contracting $i=j$ and using $\nabla^2(1/r)=0$ away from the origin gives
$$
3\alpha+2\beta=0.
$$
Next contract $i=k$ and $j=l$. Since
$$
\sum_i x_i\partial_i\partial_j(r^{-1})=\frac{2x_j}{r^3},
$$
one obtains
$$
3\alpha+12\beta
=2\int_{\mathbb R^3}\frac{e^{-r^2}}r,dV
=4\pi.
$$
Solving yields
$$
\boxed{\alpha=-\frac{4\pi}{15},\qquad
\beta=\gamma=\frac{2\pi}{5}}.
$$
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