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In present work we study the production of ground and excited charmoniumstates in $e^- e^+ \rightarrow \gamma\chi_{cJ}(nP)(J=0,1)$, and $e^- e^+\rightarrow \gamma \eta_c(nS)$, through leading order (LO) diagrams, whichproceed through exchange of a virtual photon that couples to $\gamma$ and$\eta_c/\chi_{cJ}$ through the triangular quark loop diagram, in the frameworkof $4\times 4$ Bethe-Salpeter equation (BSE), at center of mass energies$\sqrt{s}=10.6$ GeV(Belle energy), and 4.6 GeV (BESIII energy). The amplitudesimplifies to a general form required by Lorentz-covariance, in terms of itsform factors. The cross sections for these processes with leading order (treelevel) diagrams alone at $\sqrt{s}$=10.6 GeV provide a sizable contribution,which might be mainly due to the BSE being a fully relativistic approach thatincorporates the relativistic effect of quark spins and can also describeinternal motion of constituent quarks within the hadron in a relativisticallycovariant manner. Our results are compared with recent Belle data at 10.6 GeV,and BESIII data at 4.6 GeV, as well as other models. Plots of cross sectionsversus the center-of mass energy, $\sqrt{s}$ reveal mild fluctuations in theirbehaviour for all the three processes in the low energy ($\sqrt{s}$ = 4 - 6GeV) region, and are analyzed in terms of the form factors.

($\gamma$, $\chi_{cJ}(J=0,1)$) and ($\gamma$, $\eta_c$) production in  electron-positron annihilation at $\sqrt{s}=10.6$ GeV and $4.6$ GeV in the  framework of Bethe-Salpeter equation