  In this work, we present a computational scheme for isolating the vibrational
spectrum of a defect in a solid. By quantifying the defect character of the
atom-projected vibrational spectra, the contributing atoms are identified and
the strength of their contribution determined. This method could be used to
systematically improve phonon fragment calculations. More interestingly, using
the atom-projected vibrational spectra of the defect atoms directly, it is
possible to obtain a well-converged defect spectrum at lower computational
cost, which also incorporates the host-lattice interactions. Using diamond as
the host material, four test case defects, each presenting a distinctly
different vibrational behaviour, are considered: a heavy substitutional dopant
(Eu), two intrinsic defects (neutral vacancy and split interstitial), and the
negatively charged N-vacancy center. The heavy dopant and split interstitial
present localized modes at low and high frequencies, respectively, showing
little overlap with the host spectrum. In contrast, the neutral vacancy and the
N-vacancy center show a broad contribution to the upper spectral range of the
host spectrum, making them challenging to extract. Independent of the
vibrational behaviour, the main atoms contributing to the defect spectrum can
be clearly identified. Recombination of their atom-projected spectra results in
the isolated defect spectrum.
