  We extend a theory of first order hbar corrections to Gutzwiller's trace
formula for systems with a smooth potential to systems with discrete symmetries
and, as an example, apply the method to the two-dimensional hydrogen atom in a
uniform magnetic field. We exploit the C_{4v}-symmetry of the system in the
calculation of the correction terms. The numerical results for the
semiclassical values will be compared with values extracted from exact quantum
mechanical calculations. The comparison shows an excellent agreement and
demonstrates the power of the hbar expansion method.
