Vibrational SFG spectra are produced by overlapping in space and time an infrared (IR) beam and a visible beam at an interfacial sample and measuring the sum frequency output. Under the electric dipole approximation, the response of the system arises solely from the optical electric fields, which are given by (1) Where , , and are unit vectors corresponding to each coordinate axis in the laboratory frame and , , and are the components of the electric field vector. Chiral SFG measures elements of the nonlinear response tensor , a third-order tensor, which is defined by (2) where and are the components of the optical electric fields for the visible and IR beams, respectively, and I, J, and K range over the directions X, Y, and Z. The macroscopic emerges from an ensemble average of the microscopic molecular SFG response, the hyperpolarizability tensor . For uniaxial interfacial assemblies, the orientation distribution in the azimuthal rotation angle f can be assumed to be uniform, leading to