  In this paper we show that, for a sub-Laplacian $\Delta$ on a $3$-dimensional
manifold $M$, no point interaction centered at a point $q_0\in M$ exists. When
$M$ is complete w.r.t. the associated sub-Riemannian structure, this means that
$\Delta$ acting on $C^\infty_0(M\setminus\{q_0\})$ is essentially self-adjoint.
A particular example is the standard sub-Laplacian on the Heisenberg group.
This is in stark contrast with what happens in a Riemannian manifold $N$, whose
associated Laplace-Beltrami operator is never essentially self-adjoint on
$C^\infty_0(N\setminus\{q_0\})$, if $\dim N\le 3$. We then apply this result to
the Schr\"odinger evolution of a thin molecule, i.e., with a vanishing moment
of inertia, rotating around its center of mass.
