The potential energy of the atoms in a molecule can sometimes be approximated by the Morse Function
U=A[(e^((R-r)/S) -1)^2-1]
where r is the distance between the two atoms and A,R and S positive constants with S<
I solved for ro, and got ro = R, and then substituted r=ro+x to get
U=A[(e^(-x/S) -1)^2-1] .
I'm not sure how to do the next step to show for small displacements, U has the approximate form U=const+1/2 kx^2.