= 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>.
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