For an equilibrium potential ⟨ φ ⟩ o of a system with N atoms, the total potential ⟨ φ ⟩ is found by a Taylor series expansion at the equilibrium and this can be approximated by the second derivatives (the harmonic approximation) as ⟨ ⟨ φ φ ⟩ ⟩ = ⟨ ⟨ φ φ ⟩ ⟩ o + ∑ ∑ i ∑ ∑ α α ∂ ∂ ⟨ ⟨ φ φ ⟩ ⟩ ∂ ∂ d i α α | o d i α α + 1 2 ∑ ∑ i , j ∑ ∑ α α , β β ∂ ∂ 2 ⟨ ⟨ φ φ ⟩ ⟩ ∂ ∂ d i α α ∂ ∂ d j β β | o d i α α d j β β + 1 6 ∑ ∑ i , j , k ∑ ∑ α α , β β , γ γ ∂ ∂ 3 ⟨ ⟨ φ φ ⟩ ⟩ ∂ ∂ d i α α ∂ ∂ d j β β ∂ ∂ d k γ γ | o d i α α d j β β d k γ γ + ⋯ ⋯ ≈ ≈ ⟨ ⟨ φ φ ⟩ ⟩ o + 1 2 ∑ ∑ i , j ∑ ∑ α α , β β Γ Γ α α β β d i α α d j β β ,