1. Anharmonicity
In real molecules, highly sensitive vibrational spectroscopy can detect overtones, which are transitions originating from the n = 0 state for which Δn = +2, +3, …
Overtones are due to anharmonicity. A good approximation of realistic anharmonicity is given by the Morse potential.

2. Put x = r – r0 and Taylor expand:
Comparing to the harmonic oscillator
we see that
So we do
to keep the force constant the same but change the anharmonicity

3. use De = 40, α = 1; then scale by c

4. Energy levels

5. Morse model dissociated above this
are the generalized Laguerre polynomials

6. Harmonic oscillator model
are the Hermite polynomials

7. Wavefunctions: harmonic oscillator

8. Wavefunctions: Morse oscillator

9. Wavefunctions: harmonic vs. Morse

10. Wavefunctions

11. Wavefunctions

12. Expectation value of position

13. Expectation value of position

14. Expectation value of position

15. …or can keep more terms in the Taylor expansion of the dipole moment
Selection rules
For anharmonicity, can replace the H.O. wavefunctions with Morse wavefunctions…
…or can keep more terms in the Taylor expansion of the dipole moment

16. Selection rules

17. Correspondence principle
Where xturn is the maximum value of x

18. Correspondence principle

19. Correspondence principle

20. Correspondence principle