zero temperature defined in (\[alp\_qq\]) in the limit of small distances. Therefore $\alpha_{qq}(r,T)$ will show the (zero temperature) weakening in the short distance perturbative regime. In the large distance limit, however, the coupling will be dominated by Eq. (\[alp\_rT2\]) and will be suppressed by color screening, $\alpha_{qq}(r,T)\simeq\alpha(T)\exp(-m_D(T)r)$, $rT\gg1$. It thus will exhibit a maximum at some intermediate distance. Although in the large distance regime $\alpha_{qq}(r,T)$ will be suppressed by color screening and thus non-perturbative effects will strongly control the value of $\alpha_{qq}(r,T)$, in this limit the temperature dependence of the coupling, $\alpha(T)$, can be extracted by directly comparing the singlet free energy with the high temperature perturbative relation above deconfinement. Results from such an analysis will be given in Sec. \[colorscreening\].

We calculated the derivative, $dF_1/dr$, of the color singlet free energies with respect to distance by using cubic spline approximations of the $r$-dependence of the free energies for each temperature. We then performed the derivatives on basis of these splines. Our results for $\alpha_{qq}(r,T)$ calculated in this way are shown in Fig. \[couplt\] and are compared to the coupling at zero temperature discussed already in Sec. \[couplt=0\]. Here the thin solid line corresponds to the coupling in the Cornell 