= Solution
With $\beta=1/(k_BT)$ and a consistently normalized reactant <partition function> $Q_r$, the one-dimensional channel formula is
$$
k(T)Q_r=\frac1{2\pi\hbar}\int_0^\infty e^{-\beta E}P(E)\,dE.
$$
Classically the integral starts at $V_0$, giving
$$
\boxed{k_{\rm cl}(T)=\frac{e^{-\beta V_0}}{2\pi\hbar\beta Q_r}}.
$$
An infinite free reactant half-line has no finite canonical population, so define a reactant normalization length $L$, for which $Q_r=L\sqrt{m/(2\pi\beta\hbar^2)}$. Then \b[$k_{\rm cl}=L^{-1}\sqrt{k_BT/(2\pi m)}\,e^{-V_0/(k_BT)}$]. Equivalently $kL$ is a one-dimensional thermal flux per unit reactant density, with dimensions of velocity. A <rate constant> without either this normalization or a specified reactant well is not dimensionally complete.
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