  We complete the leading-log renormalization group scaling of the NRQCD
Lagrangian at $O(1/m^2)$. The next-to-next-to-leading-log renormalization group
scaling of the potential NRQCD Lagrangian (as far as the singlet is concerned)
is also obtained in the situation $m\alpha_s \gg \Lambda_{QCD}$. As a
by-product, we obtain the heavy quarkonium spectrum with the same accuracy in
the situation $m\alpha_s^2 \simg \Lambda_{QCD}$. When $\Lambda_{QCD} \ll
m\alpha_s^2$, this is equivalent to obtain the whole set of
$O(m\alpha_s^{(n+4)} \ln^n \alpha_s)$ terms in the heavy quarkonium spectrum.
The implications of our results in the non-perturbative situation $m\alpha_s
\sim \Lambda_{QCD}$ are also mentioned.
