  Electron capture on $^{20}$Ne is critically important for the final stage of
evolution of stars with the initial masses of 8 - 10 $M_{\odot}$. In the
present paper, we evaluate electron capture rates for a forbidden transition
$^{20}$Ne (0$_{g.s.}^{+}$) $\rightarrow$ $^{20}$F (2$_{g.s.}^{+}$) in stellar
environments by the multipole expansion method with the use of shell-model
Hamiltonians. These rates have not been accurately determined in theory as well
as in experiments. Our newly evaluated rates are compared with those obtained
by a prescription that treats the transition as an allowed Gamow-Teller (GT)
transition with the strength determined from a recent $\beta$-decay experiment
for $^{20}$F (2$_{g.s.}^{+}$) $\rightarrow$ $^{20}$Ne (0$_{g.s.}^{+}$)
\citep{Kirsebom}. We find that different electron energy dependence of the
transition strengths between the two methods leads to sizable differences in
the weak rates of the two methods. We also find that the Coulomb effects, that
is, the effects of screening on ions and electrons are non-negligible. We apply
our e-capture rates on $^{20}$Ne to the calculation of the evolution of
high-density O-Ne-Mg cores of 8 - 10 $M_{\odot}$ stars. We find that our new
rates affect the abundance distribution and the central density at the final
stage of evolution.
