Checking my work on a Lennard-Jones potential problem in differential equations

Checking my work on a Lennard-Jones potential problem in differential
equations

The Lennard-Jones potential is $$U(r) = \left[ \left( \frac{\rho}{r}
\right)^{12} - \left( \frac{\rho}{r} \right)^6 \right]$$.
What is the equilibrium distance?
OK, so I know that the equilibrium distance has to be where the force is
zero. So all I need do is take a derivative of $U(r)$. I get:
$$ \frac{dU(r)}{dr} = -12 \rho^{12} r^{-13} + 6 \rho^6 r^{-7} $$
and making that equal to zero we end up with $ r^6 = -2 \rho^6 $
So far so good, that means $r_{min} = \rho 2^{\frac{1}{6}} $
Now to show that it is a simple harmonic oscillator.
$$ F(r) = -12 \rho^{12} r^{-13} + 6 \rho^6 r^{-7} = ma = m \ddot r $$
so
$$ \ddot r = \frac{F(r)}{m} = -12 \frac{\rho^{12}}{r^{13}m} + 6 \frac{
\rho^6} {r^{7}m} $$
When I plug in the value of $r_{min}$ I end up with $ \ddot r (0) =
\frac{-3}{mr} - \frac{3}{mr} = \frac {-6}{mr} $. Which would seem to
indicate that the natural frequency is $\sqrt{\frac{-6}{mr}}$ which is
$\sqrt{\frac{-6}{m} \frac{1}{2^{1/6}\rho}}$
But that's an imaginary quantity so I think I messed up somewhere. If
anyone can point out a mistake that would be most appreciated.