the MQDT functions in the loss channel $C_2(E,\ell)\approx 1$ and $\tan\lambda_2(E,\ell)\approx 0$. The only remaining functions are $C_1$ and $\tan\lambda_1$, so from now on we will drop the index $1$ in the notation and move the angular momentum dependence to the argument of the functions. Using the analytical results for the $S$ matrix and the definitions from the previous section, we obtain a general formula for the complex scattering length in the entrance channel $\tilde{a}_{\ell m}$ $$\begin{aligned}
\label{alm1}
\tilde{a}_{\ell m}(E)  =
- \frac{1}{k}\tan \left[
\xi(E,\ell) - \tan^{-1} \left( \frac{y C^{-2}(E,\ell)}{i + y \tan \lambda(E,\ell)}
\right)\right].\end{aligned}$$ Substituting this into Eqs.  and , one can express the elastic and reactive rate constants directly in terms of the MQDT functions, obtaining $$\label{KreMQDT}
\mathcal{K}^{\mathrm}{re}_{\ell m}= g \frac{h}{2 \mu k} P^{\mathrm}{re} \frac{ C^{-2}(E,\ell) (1+y)^2}{(1+y C^{-2}(E,\ell))^2 +y^2 \tan^2 \lambda (E,\ell)}$$ for the reactive rate constant and

$$\label{KelMQDT}
\mathcal{K}^{\mathrm}{el}_{\ell m}= g \frac{2 h}{\mu k}\frac{\tan^2 \xi(E,\ell) + y^2 \left(\tan \lambda(E,\ell) \tan \xi(E,\ell) - C^{-2}(E,\ell) \right)^2}{\left(1+\tan^2 \xi(E,\ell)\right) \left( y^2 \tan^2 \lambda(E,\ell) +\left( 1+ y C^{-2}(E,\ell)\right)^2 \right)}$$

for the elastic rate constant. We note that in contrast to the reactive rate, the elastic one depends explicitly on the phase shift $\xi(E,\ell)$ of the reference potential.

We notice that the properties 