  We describe a method to compute thermodynamic quantities in the harmonic
approximation for identical bosons and fermions in an external confining field.
We use the canonical partition function where only energies and their
degeneracies enter. The number of states of given energy and symmetry is found
by separating the center of mass motion, and counting the remaining states of
given symmetry and excitation energy of the relative motion. The oscillator
frequencies that enter the harmonic Hamiltonian can be derived from realistic
model parameters and the method corresponds to an effective interaction
approach based on harmonic interactions. To demonstrate the method, we apply it
to systems in two dimensions. Numerical calculations are compared to a brute
force method that is considerably more computationally intensive.
