The transition matrix elements for pure vibrational transition are μ μ v , v ′ = ⟨ v ′ | μ μ | v ⟩ , where μ μ is the dipole moment of the diatomic molecule in the electronic state α α . Because the dipole moment depends on the bond length R , its variation with displacement of the nuclei from equilibrium can be expressed as: μ μ = μ μ 0 + ( d μ μ d x ) 0 x + 1 2 ( d 2 μ μ d x 2 ) 0 x 2 + . . . . . . . ; where μ μ 0 is the dipole moment when the displacement is zero. The transition matrix elements are, therefore: ⟨ v ′ | μ μ | v ⟩ = μ μ 0 ⟨ v ′ | v ⟩ + ( d μ μ d x ) 0 ⟨ v ′ | x | v ⟩ + 1 2 ( d 2 μ μ d x 2 ) 0 ⟨ v ′ | x 2 | v ⟩ + . . . . . . . = ( d μ μ d x ) 0 ⟨ v ′ | x | v ⟩ + 1 2 ( d 2 μ μ d x 2 ) 0 ⟨ v ′ | x 2 | v ⟩ + . . . . . . . using orthogonality of the states. So, the transition matrix is non-zero only if the molecular dipole moment varies with displacement, for otherwise the derivatives of μ μ would be zero. The gross selection rule for the vibrational transitions of diatomic molecules is then: To show a vibrational spectrum, a diatomic molecule must have a dipole moment that varies with extension. So, homonuclear diatomic molecules do not undergo electric-dipole vibrational transitions. So, a homonuclear diatomic molecule doesn't show purely vibrational spectra.