Deduce the existence of the second-order susceptibility (2) in (7.6.1), which gives a response proportional to the square of the electric field E 2

Deduce the existence of the second-order susceptibility χ⁽²⁾ in (7.6.1), which gives a response proportional to the square of the electric field E², following the procedure of Section 7.4 for P ∝ ⟨ψ_t|x|ψ_t ⟩, but using the second-order perturbation theory method introduced here to obtain ⟨c|ψ(t)⟩, for a transition from state |v⟩ to state |c± via one intermediate state |m±. Assume that transitions between all three states are dipole-allowed (which means there cannot be inversion symmetry), and for simplicity, assume E = E₀e^−iωt.

P = X 0E + 2x (2) E + 4x () E +.... (7.6.1)