  A flexible scheme for decomposing the vibrational density of states in terms
of pair vibrational density of states is presented. This scheme provides the
linkage between the site vibrational density of states and pair vibrational
density of states so that vibrational modes, in particular localized modes, can
be conveniently examined in terms of the correlation between the vibration at a
given site and those at its neighboring sites. Furthermore, within the
framework of a total energy vibrational spectrum study, this scheme allows the
analysis of vibrational modes in terms of their electronic origin. A case study
of the vibrational dynamics of the relaxed Si$_{87}$ cluster is carried out to
demonstrate the flexibility of the scheme in analyzing the properties of
vibrational modes, particularly for complex systems with reduced or no
symmetry.
