The centroid density is a function defined for quantum systems in thermodynamic equilibrium that is readily obtained by path integral simulations. The physical information provided by the centroid density is a static response of the system under isothermal conditions, namely, the change in the expectation value of the position operator of the quantum particles upon application of constant external forces. An interesting application of this function is the study of vibrational properties of atomic nuclei in molecules and solids. In particular, the analysis of the tensor defined by the second cumulants of the centroid density (i.e., the static isothermal susceptibility tensor) leads to the definition of the linear response vibrational modes, which are characterized by a response of the quantum system parallel to the applied force. The eigenvalues of the susceptibility tensor provide the linear response of the system. This response function is the basis for the formulation of two approximations to evaluate vibrational frequencies. The capability of these approximations is tested in the study of the tunneling frequency of a particle in a double-well potential, the anharmonic shift in the vibrational frequencies of molecules (H-2 and HOCl), and the anharmonic shifts of phonon frequencies in diamond.