  In present work we study the production of ground and excited charmonium
pairs in $e^- e^+ \rightarrow \Psi(nS)+ \chi_{cJ}(nP)$ for $J=0,1$ and $n=1,2$,
through leading order (LO) tree-level diagrams $\sim O(\alpha_{em} \alpha_s)$,
which proceed through exchange of a virtual photon and an internal gluon line
connecting two quark lines (in the triangle quark loop part of the diagram), in
the framework of $4\times 4$ Bethe-Salpeter equation, at center of mass energy,
$\sqrt{s}=10.6 GeV.$ We have tried to show that the cross sections for this
process calculated with use of the four leading order $\sim O(\alpha_{em}
\alpha_s)$ diagrams using BSE approach come close to experimental data, which
might be mainly due to the consistent treatment of motion of quarks inside the
hadrons in the BSE framework.
