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Anharmonicity in Molecules: Vibrational Spectroscopy Insights

Explore how highly sensitive vibrational spectroscopy detects overtones in molecules, originating from the n=0 state with Δn=+2, +3, etc. Anharmonicity leads to these overtones, modeled realistically with the Morse potential. Understanding Morse vs. harmonic oscillator models elucidates energy levels and wavefunctions, with implications for expectation values and selection rules. Discover the Correspondence principle and how to apply it effectively.

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Anharmonicity in Molecules: Vibrational Spectroscopy Insights

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  1. In real molecules, highly sensitive vibrational spectroscopy can detect overtones, which are transitions originating from the n = 0 state for which Δn = +2, +3, … Anharmonicity 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. 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

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