Solving for Constants in Perturbed Simple Harmonic Oscillator with HF Potential

[NeedPhysHelp8]

Homework Statement

The potential of a simple harmonic oscillator of HF has the following form
$$\frac{1}{2}kx^2 + bx^3 + cx^4$$
The first part of the problem involved finding expressions for the first-order energy corrections for the first three states, which I found below. Basically the x³ term does not contribute at all only x⁴:
$$E_{0}^{'} = \frac{3}{4}c(\frac{\hbar}{m\omega_0})^2$$
$$E_{1}^{'} = \frac{15}{4}c(\frac{\hbar}{m\omega_0})^2$$
$$E_{2}^{'} = \frac{39}{4}c(\frac{\hbar}{m\omega_0})^2$$
Now the next part of the problem supplied the known transition energies from the 1st and 2nd excited state to the ground state. I was then able to figure out the value for the natural frequency of the unperturbed harmonic oscillator by constructing the following equations knowing that the energy is equal to the unperturbed (E=1/2(n+1)hw) + the perturbed found previously :

$$\Delta E_{1,0} = \hbar\omega_0 + c \frac{9}{8}(\frac{\hbar}{m\omega_0})^2$$
$$\Delta E_{2,0} = 2\hbar\omega_0 + c 9(\frac{\hbar}{m\omega_0})^2$$

I solved these equations and found ω₀ but the next part involves calculating the constants k and c . I know the relation that:
$$k = m\omega_0^2$$
But I can't figure out a way to solve for each individually only for c/k²

Thanks!

edit: should I just assume I need to look up a reduced mass for HF? or can it be solved another way

 

[member 553083]

? Homework EquationsE_{0}^{'} = \frac{3}{4}c(\frac{\hbar}{m\omega_0})^2 E_{1}^{'} = \frac{15}{4}c(\frac{\hbar}{m\omega_0})^2 E_{2}^{'} = \frac{39}{4}c(\frac{\hbar}{m\omega_0})^2 \Delta E_{1,0} = \hbar\omega_0 + c \frac{9}{8}(\frac{\hbar}{m\omega_0})^2 \Delta E_{2,0} = 2\hbar\omega_0 + c 9(\frac{\hbar}{m\omega_0})^2 k = m\omega_0^2 The Attempt at a Solution I assumed that I need to look up a reduced mass for HF but I can't quite figure out how to relate the constants k and c.