If the potential energy between two molecules is given by $$U = \frac{A}{r^6} + \frac{B}{r^{12}}$$, then at equilibrium, separation between molecules, and the potential energy are:

$$U = A r^{-6} + B r^{-12}$$

$$\frac{dU}{dr} = -6A r^{-7} - 12B r^{-13} = 0$$

$$-6A + \frac{-12B}{r^6} = 0 \implies 6A = -\frac{12B}{r^6} \implies r^6 = -\frac{2B}{A}$$

$$\frac{dU}{dr} = \frac{6A}{r^7} - \frac{12B}{r^{13}} = 0 \implies 6A = \frac{12B}{r^6} \implies r^6 = \frac{2B}{A} \implies r = \left(\frac{2B}{A}\right)^{1/6}$$

$$U = -\frac{A}{\left(\frac{2B}{A}\right)} + \frac{B}{\left(\frac{2B}{A}\right)^2} = -\frac{A^2}{2B} + \frac{B A^2}{4B^2} = -\frac{A^2}{2B} + \frac{A^2}{4B} = -\frac{A^2}{4B}$$