Powis Group

Less widely recognised, is that this equation is just a special (though common) sub-case of the more general expression:

The coefficients b_n^p are determined by the photoionization dynamics and depend on the light polarization — p = 0 for linear, p = ±1 for left, right circular polarization (CPL) states. Coupling of the electron and photon angular momenta introduces certain symmetry properties and constraints. For the P₂ terms b₂⁰ = − ½ × b₂^±1 ≡ β . More particularly, b₁⁰ = 0 while b₁^±1 also vanishes for achiral molecules; in such circumstances this general expression simply reduces to the well-known first form.

But, for the specific case of a chiral molecule ionized with CPL, these P₁ (1^st Legendre Polynomial) coefficients are found to be antisymmetric for exchange of handedness of either the molecular enantiomer or the light polarization: b₁⁺¹ = − b₁⁻¹ .

A lab-frame PAD represents an average taken over all possible initial molecular orientations. There is a consequent loss of dynamical information, although experimentally it is a less challenging measurement to make. For a one-photon ionization with linearly polarized light the PAD has a well known form^†:

or, expanding P₂ (the 2^nd Legendre Polynomial function) The single parameter β ( −1 ≤ β ≤ +2 ) specifies the magnitude of the anisotropy which can be seen to range from a pure cos²θ to sin²θ form, where θ is measured from the light polarization vector.

† but as our PECD work now shows an additional term is needed for circular polarizations.