 Solution

The hydrogen ground-state charge cloud is spherically symmetric and has even parity, so it has no permanent electric dipole moment. A uniform electric field couples through a perturbation proportional to $\mathbf{E}\cdot \mathbf{r}$, which is odd under parity. Its expectation in the ground state therefore vanishes, giving no first-order energy correction.

The field can nevertheless polarize the atom: it admixes odd-parity excited states and creates an induced dipole. Such mixing contributes from second order onward, producing the quadratic Stark effect of the hydrogen ground state.