  An effective hamiltonian for heavy quarkonia is derived from QCD by
separating gluonic fields in background and quantum fields and neglecting
anharmonic contributions. Mesonic states with nonperturbative gluonic
components are constructed. These states are invariant under gauge changes of
the background fields and form an orthogonal basis. The effective hamiltonian
is diagonalized in this basis in a systematic $1/m$- and short distance
expansion. For very heavy quarkonia, we obtain an effective potential similar
to the phenomenological funnel potential. We compare our method to $2^{\rm nd}$
order perturbation theory in the background fields and demonstrate its
applicability even for the relatively light charmonium system. The results to
order $1/m$ for pseudoscalar meson masses and wave functions are shown and
compared with those of the Cornell model.
