Convergence in distribution to N(0, $\sigma^2$)

Question

I have faced with the following problem. Suppose that $X_1,...,X_n,...$ - sequence of independent random variables with the same distribution $P(X_i=0)=P(X_i=-\ln2)=1/2$, and $$Y_n=-\frac{1}{n}\sum\limits_{i=1}^{n}e^{X_i}-\frac{1}{n^2}\left(\sum\limits_{i=1}^{n}e^{X_i/2}\right)^2.$$ I found out that $\displaystyle Y_n \rightarrow a= -\frac{3}{4}-\left(\frac{\sqrt2+1}{2\sqrt2}\right)^2$ almost surely. I want to find constant $\sigma$ such that $\sqrt n(Y_n-a) \rightarrow N(0,\sigma^2)$ in distribution.

Thanks in advance for any help.