  The non-relativistic energy levels of para-positronium are calculated in the
quadrupole approximation of the interaction potential. This approximation
technique takes into account the anisotropy of the electrostatic
electron-positron interaction in the lowest order. The states due to different
values of the quantum number $(l_z)$ of angular momentum are found to be no
longer degenerate as is the case in the conventional theory. The physical
origin of this elimination of the conventional degeneracy may intuitively be
attributed to the state-dependent inertial \emph{broadening} of the rotating
charge clouds; the corresponding \emph{anisotropic} deformation (in the
quadrupole approximation) lowers then the negative electrostatic interaction
energy. The result of this influence of anisotropy is that the states with
$l_z=0$ adopt smaller binding energy whereas the states with maximal value of
$|l_z|$ (for fixed principal quantum number $n$) have the largest binding
energy within the angular momentum multiplet $(-|l_{z,\mathrm{max}}| \le l_z
\le |l_{z,\mathrm{max}}|)$. This yields a certain kind of electric
fine-structure splitting with the splitted RST levels being placed in a
relatively narrow band around the (highly degenerated) conventional levels.
