In Chemistry, is it possible to use the old representation of the atom as a nucleus surrounded by orbiting particles?
When the fixed-orbit particle model is taught in general chemistry classes, it is usually presented as Bohr's fixed circular orbit model.  

However, this was not the culmination of the deterministic-orbit particle model.  In 1916 Sommerfeld expanded the model to include precessing elliptical orbits, and made the theory relativistic. The varying of the eccenticity of the ellipses permits multiple angular momentum values for a given energy. 

The Sommerfeld relativistic model correctly predicts the fine structure of the hydrogen atom.  

In this one aspect, the deterministic-orbit particle model is superior to the non-relativistic Schrodinger model.  The Sommerfeld model correctly predicts the fine structure, whereas the Schrodinger model does not.