Both the vibrational energy relaxation and pure dephasing of a polar solute in polar liquids are theoretically studied and particularly the role of the long-range Coulomb interaction between the solute and the polarization modes of the polar liquid is focused on in this paper. If the linear coupling potential between the vibrational displacement and the solvent polarization mode is taken into account and assuming that the anharmonic contribution;to the pure dephasing is the dominant mechanism, the vibrational relaxation rates are shown to be related to the dielectric friction. However the pure dephasing of a perfect harmonic oscillator is an effect arising from the nonlinear (with respect to the vibrational coordinate) vibrational coupling potential, and the relationship between the pure dephasing rate and the frequency-dependent friction is no longer valid. By expanding the bare electric field of the polar solute in terms of the multipoles, the vibrational coupling potential is obtained as a power series summation with respect to the vibrational displacement. Each expansion coefficient is found to be associated with the interaction of the projected multipole fields with the solvent polarization. A few simple cases are discussed in detail with an emphasis on the connection between the vibrational relaxation rates and solvation dynamics of static multipoles in polar liquids.