  The Cornell potential with the best fitted parameters $\alpha_{s}$ and
$\kappa$ are modified by adding terms derived from non-perturbative QCD, which
are characterized by a series of non-vanishing vacuum condensates of quarks and
gluons. In terms of this potential, we study the system of heavy quarkonia. The
results show that the correction caused by the additional terms reduces the
deviation between the data and the values calculated with the pure Cornell
potential and improves the splittings of energy levels. The achievements
indicate that the non-perturbative effects induced by vacuum condensates play
an important role for the correction to $1/q^2$, which in general was
phenomenologically put in by hand. This result would be helpful for
understanding non-perturbative QCD along a parallel direction to the QCD sum
rules.
