For small displacements, the electric dipole moment of a molecule can be expected to vary linearly with the extension of the bond. This would be the case for a heteronuclear molecule in which the partial charges on the two atoms were independent of the internuclear distance. In such cases (known as harmonic approximation), the quadratic and higher terms in the expansion can be ignored and μ μ v , v ′ = ⟨ v ′ | μ μ | v ⟩ = ( d μ μ d x ) 0 ⟨ v ′ | x | v ⟩ . Now, the matrix elements can be expressed in position basis in terms of the harmonic oscillator wavefunctions: Hermite polynomials. Using the property of Hermite polynomials: 2 ( α α x ) H v ( α α x ) = 2 v H v − − 1 ( α α x ) + H v + 1 ( α α x ) , it is evident that x | v ⟩ which is proportional to x H v ( α α x ) , produces two terms, one proportional to | v + 1 ⟩ and the other to | v − − 1 ⟩ . So, the only non-zero contributions to μ μ v , v ′ comes from v ′ = v ± ± 1 . So, the selection rule for heteronuclear diatomic molecules is: Δ Δ v = ± ± 1