The low frequency vibrational spectrum that reflects the intermolecular interaction of an imidazolium cation ionic liquid 1-butyl-3-methyl imidazolium hexafluorophosphate ([bmim][PF6]) was studied by femtosecond optically heterodyne-detected optical Kerr effect (OHD-OKE) spectroscopy. Several explicit time correlation functions of the Brownian oscillator were used to simulate the kinetic nuclear contribution of the femtosecond OHD-OKE data. The time evolution behavior of each individual oscillator in the time domain was obtained. We obtained the frequency spectrum for each oscillator in the frequency domain by the fast Fourier transform theorem. In addition, we also measured the low frequency vibrational spectrum of [bmim][PF6] by low frequency Raman spectroscopy. These low frequency spectra obtained by low frequency Raman spectroscopy and by femtosecond OHD-OKE spectroscopy show very similar characteristics. These results demonstrate that the explicit time correlation function of the Brownian oscillator can be used to resolve the time evolution behavior of the low frequency vibration mode in ionic liquids.