Optimal proposal density for acceptance rejection sampling

Question

For some project I have been sampling from the Gamma distribution. I have been using the exponential distribution intensively. One method I have employed is the Acceptance rejection sampling, particularly for the pdf given $f(x) = 8{e^{ - 4x}}x \cdot {1_{\left\{ {x \ge 0} \right\}}}$. Initially I just used ${g_2}(x) = 2{e^{ - 2x}} \cdot {1_{\left\{ {x \ge 0} \right\}}}$ as the distribution I sampled from for it seemed a good fit. Having looked at just ${g_1}(x) = {e^{ - x}} \cdot {1_{\left\{ {x \ge 0} \right\}}}$, I think it more closely envelopes the target distribution. I am now thinking what would be the optimal exponential distribution, ${g_\lambda }(x) = \lambda {e^{ - \lambda x}} \cdot {1_{\left\{ {x \ge 0} \right\}}}$, to be used and if one even exists. Is there a criterion to work that out?