quantum mechanics - Negative probabilities with Wigner quasi-probability distributions

I was toying with Wigner corrections to thermodynamic equilibrium. The semiclassical correction for the position probability density to second order in $\hbar$ is: $$P(x)= \text{e}^{-\beta V(x)}\left(1-\dfrac{\hbar^2 \beta^2}{12m}V^{\prime \prime}(x)+\dfrac{\hbar^2 \beta^3}{24m}\vert V^{\prime}(x) \vert ^2 \right).$$ (nice derivation of this formula are found in Landau's statistical mechanics or in the appendix to Density Functional Theory of Atoms and Molecules by Parr and Yang. The idea is that this is obtained by plugging $e^{-\beta \hat{H}}$ in the Wigner transform).

I took this expression and applied to typical potential energies of chemical reactions involving protons, in which there are barriers of around 0.5 eV. I am confused because for a lot of situations I obtain negative values for the correction factor (and thus for the probability density), even for high temperatures of T=300. I don't believe higher order terms can be important at such temperatures. I even think that such corrections should be extremely small in such high temperatures. What can be the issue here? Formally, it is clear that this is a probability.

edit: I wrote a relative probability, of course. A convenient normalization factor is missing above.