Sum-frequency order parameter ratio determination for molecules with high-symmetry vibrational modes

The structure of molecules at surfaces and buried interfaces can be probed by second-order nonlinear optical techniques that derive their response from environments with broken centrosymmetry, thereby simultaneously enhancing the signal from the molecules of interest and suppressing the responses of the adjacent bulk phases. However, even though these methods are surface-sensitive, extracting orientation information has often relied on significant assumptions about the shape of the orientation distribution and the underlying molecular electronic structure. Here, we illustrate that the measurement of the relative magnitude and the phase of tensor elements that govern the vibrational sum frequency response can be used to construct a ratio of order parameters ⟨P3⟩/⟨P1⟩. This structural parameter does not require knowledge of the hyperpolarizability or even the hyperpolarizability ratio, thereby dramatically increasing the accuracy of its determination. We then show how this parameter can be used to comment on the most probable distribution of tilt angles. This is demonstrated for para-cyanophenol at three different interfaces.