The energy corrections due to the applied field are given by the expectation value of H ^ ^ s in the | n ℓ ℓ m ⟩ ⟩ basis. It can be shown by the selection rules that ⟨ ⟨ n ℓ ℓ m ℓ ℓ | z | n 1 ℓ ℓ 1 m ℓ ℓ 1 ⟩ ⟩ ≠ ≠ 0 when ℓ ℓ = ℓ ℓ 1 ± ± 1 and m ℓ ℓ = m ℓ ℓ 1 .