Morse potential:

U  =  U0 ( 1  - Exp[-beta ( r  -  R0)])^2

Plot of potential

Plot[U /. {U0 -> 1, R0 -> 1, beta -> 1}, {r, 0, 4}]

part A:  Find minimum in r

Solve[D[U, r] == 0, r]

Part B:  find spring constant:

	Spring constant is second derivative of potential evaluated at the equilibrium distance.

dd  = Simplify[ D[U, {r, 2}]]

K = dd /. r -> R0

Attempt to find perturbation energy.  Energy = 1/2 hbar w + Int[psi (U-1/2 K (r-R0)^2) psi, {r, -Infinity,Infinity}]

Harmonic oscillator wave function for ground state:

psi  =  A Exp[- m w (r - R0)^2/(2 hbar)]

aa = 
      Solve[Integrate[psi^2, {r, -Infinity, Infinity}][[2]] == 
            1, A][[2]]

psi1 = psi /. aa

perturb  =  
      Simplify[psi1 (U  -  1/2 K  (r - R0)^2) psi1]

perturb1 = perturb /. r -> x + R0

a1  = 
      Integrate[perturb1, {x, -Infinity, Infinity}][[2]]

a2 = a1 /. beta -> Sqrt[m /(2 U0)] w

Simplify[Series[a2, {w, 0, 2}]]

\(Plot[
      Evaluate[{U, 1/2\ K\ \((r - R0)\)^2} /. {R0 -> 1, \ U0 -> 1\ , 
            beta -> 1}], 
		{r, 0, 4}]\)