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In this paper, we improve a recent result by Li and Peng on products offunctions in $H_L^1(\bR^d)$ and $BMO_L(\bR^d)$, where $L=-\Delta+V$ is aSchr\"odinger operator with $V$ satisfying an appropriate reverse H\"olderinequality. More precisely, we prove that such products may be written as thesum of two continuous bilinear operators, one from $H_L^1(\bR^d)\timesBMO_L(\bR^d) $ into $L^1(\bR^d)$, the other one from $H^1_L(\bR^d)\timesBMO_L(\bR^d) $ into $H^{\log}(\bR^d)$, where the space $H^{\log}(\bR^d)$ is theset of distributions $f$ whose grand maximal function $\mathfrak Mf$ satisfies$$\int_{\mathbb R^d} \frac {|\mathfrak M f(x)|}{\log (e+ |\mathfrak Mf(x)|)+\log(e+|x|)}dx <\infty.$$

Bilinear decompositions for the product space $H^1_L\times BMO_L$