Uniaxial order parameters associated with surface SFG spectra. II. Distributions with polar and azimuthal ordering

When a molecule or sub-molecular entity is ordered on a surface in a manner that can be described by a tilt- and azimuthal-angle distribution, the projection of the molecular hyperpolarizability into the laboratory frame can be described using spherical harmonics. We illustrate that this lends itself to the construction of ten achiral order parameters associated with vibrational sum-frequency generation. We describe how these order parameters can be extracted from spectral data, first ignoring and then including the dispersion of the local electric fields. We then use these order parameters to determine the most probable orientation distribution without any assumptions about the surface alignment and in the absence of any other type of experimental data. This constitutes a flexible framework for describing a wide array of surface orientation distributions.