The orientational correlation functions measured in the time-resolved second-harmonic generation (TRSHG) and time-resolved sum-frequency generation (TRSFG) experiments are derived. In the laboratory coordinate system, the 〈 Y_l~m (ω_(lab)(t)) Y_2~m (ω_(lab)(0)) 〉 (l=1,3 and m=0,2) correlation functions, where the Y_l~m are spherical harmonics, describe the orientational relaxation observables of molecules at interfaces. A wobbling-in-a-cone model is used to evaluate the correlation functions. The theory demonstrates that the orientational relaxation diffusion constant is not directly obtained from an experimental decay time in contrast to the situation for a bulk liquid. Model calculations of the correlation functions are presented to demonstrate how the diffusion constant and cone half-angle affect the time-dependence of the signals in TRSHG and TRSFG experiments. Calculations for the TRSHG experiments on Coumarin C314 molecules at air-water and air-water-surfactant interfaces are presented and used to examine the implications of published experimental results for these systems.
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