  While linear molecules in their vibrational ground state cannot carry angular
momentum around their symmetry axis, the presence of vibrational excitations
can induce deformations away from linearity and therefore also allow angular
momentum along the molecular axis. In this work, a recurrence relation is
established for the calculation of the vibrational level densities (densities
of states) of linear molecules, specified with respect to both energy and
angular momentum. The relation is applied to the carbon clusters of sizes
$n=4,6,7$ as a case study.
