  In this paper, we consider the symmetric KL-divergence between the sum of
independent variables and a Gaussian distribution, and obtain a convergence
rate of $O\left( \frac{\ln n}{\sqrt{n}}\right)$. The proof is based on Stein's
method. The convergence rate of order $O\left( \frac{1}{\sqrt{n}}\right)$ and
$O\left( \frac{1}{n}\right) $ are also obtained under higher moment condition.
