Here, V denotes the potential energy, r e is the equilibrium distance between atoms, ξ = r -r e is the displacement of the structure from the equilibrium geometry, V n = ∂ n V ∂r n (r e ), V h (r) is the harmonic potential, and W a (r) is the anharmonic correction to the harmonic potential. The quantum harmonic oscillator is one of the simplest quantum mechanical problems with known analytical solution. For a molecule with reduced mass µ, the stationary vibrational states |v⟩ (v = 0, 1, 2, . . .) have energies of E v = hω(v + 1/2), where ω = V 2 /µ is the angular vibrational harmonic frequency. The vibrationally averaged displacement ⟨ξ⟩ = ⟨v|ξ|v⟩ = 0, since the harmonic potential is symmetric with respect to inversion of the displacement, V h (+ξ) = V h (-ξ). On the other hand, the vibrational amplitude l, defined as