  We estimate the decay rates of $\eta_c\rightarrow 2\gamma$,
$\eta_c'\rightarrow 2\gamma$, and $J/\psi\rightarrow e^+ e^-$,
$\psi^\prime\rightarrow e^+e^-$, by taking into account both relativistic and
QCD radiative corrections. The decay amplitudes are derived in the
Bethe-Salpeter formalism. The Bethe-Salpeter equation with a QCD-inspired
interquark potential are used to calculate the wave functions and decay widths
for these $c\bar{c}$ states. We find that the relativistic correction to the
ratio $R\equiv \Gamma (\eta_c \rightarrow 2\gamma)/ \Gamma (J/ \psi \rightarrow
e^+ e^-)$ is negative and tends to compensate the positive contribution from
the QCD radiative correction. Our estimate gives $\Gamma(\eta_c \rightarrow
2\gamma)=(6-7) ~keV$ and $\Gamma(\eta_c^\prime \rightarrow 2\gamma)=2 ~keV$,
which are smaller than their nonrelativistic values. The hadronic widths
$\Gamma(\eta_c \rightarrow 2g)=(17-23) ~MeV$ and $\Gamma(\eta_c^\prime
\rightarrow 2g)=(5-7)~MeV$ are then indicated accordingly to the first order
QCD radiative correction, if $\alpha_s(m_c)=0.26-0.29$. The decay widths for
$b\bar b$ states are also estimated. We show that when making the assmption
that the quarks are on their mass shells our expressions for the decay widths
will become identical with that in the NRQCD theory to the next to leading
order of $v^2$ and $\alpha_s$.
