As stated, the quadratic Stark effect is described by second-order perturbation theory. The zeroth-order eigenproblem H ( 0 ) ψ ψ k 0 = E k ( 0 ) ψ ψ k 0 , k = 0 , 1 , … … , E 0 ( 0 ) < E 1 ( 0 ) ≤ ≤ E 2 ( 0 ) , … … is assumed to be solved. The perturbation theory gives E k ( 2 ) = ∑ ∑ k ′ ≠ ≠ k ⟨ ⟨ ψ ψ k 0 | V i n t | ψ ψ k ′ ′ 0 ⟩ ⟩ ⟨ ⟨ ψ ψ k ′ 0 | V i n t | ψ ψ k 0 ⟩ ⟩ E k ( 0 ) − − E k ′ ( 0 ) ≡ ≡ − − 1 2 ∑ ∑ i , j = 1 3 α α i j F i F j with the components of the polarizability tensor α defined by α α i j = − − 2 ∑ ∑ k ′ ≠ ≠ k ⟨ ⟨ ψ ψ k 0 | μ μ i | ψ ψ k ′ 0 ⟩ ⟩ ⟨ ⟨ ψ ψ k ′ 0 | μ μ j | ψ ψ k 0 ⟩ ⟩ E k ( 0 ) − − E k ′ ( 0 ) . The energy E gives the quadratic Stark effect.