where D is the magnitude of the dipole moment. Whereas the eigenvalues of 𝓵 ⃑ , ℓ ℓ 1 , are defined by the ℓ quantum number, which is always integer, the similarly expressed eigenvalues of the composite operator 𝓵 ⃑ 2𝐷 cos 𝜃 are defined by eigenvalues 𝐿 𝐿 1 , where 𝐿 is non-integer, if D  0. The corresponding eigenfunctions, Ω , replace the standard spherical harmonics of eqn ( ). The non-integer 𝐿 values reflect the mixing of the pure angular momentum components of the free electron due to interactions with the point dipole. The corresponding angular functions can therefore be described as superpositions of pure spherical harmonics: