Chi-squared Chernoff lower-tail bound ($\mathbb{P}(V\le na)\le e^{-n(a-1-\log a)/2}$)

For $V\sim {\chi }_n^2$ and $0<a<1$, apply the Chernoff bound to $e^{-sV}$ using $\mathbb{E}{e}^{-sV}=(1+2s{)}^{-n/2}$. Optimization at $s=(a^{-1}-1)/2$ proves the displayed estimate. It remains useful when the additive lower threshold in the chi-squared concentration inequality is nonpositive. For $0<r\le 1$, choosing $a=e^{-4r}$ gives a tail at most $e^{-n{r}^2}$ because $e^{-4r}-1+4r\ge 2r^2$.