An Harmonic

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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. V ( r)= D e 1− e α ( r r 0 ) [ ] 2

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Harmonic oscillators, quantum mechanics

Transcript of An Harmonic

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

  • So we doPut x = r r0 and Taylor expand:Comparing to the harmonic oscillatorwe see thatto keep the force constant the same but change the anharmonicity

  • use De = 40, = 1; then scale by c

  • Energy levels

  • Morse model

  • Harmonic oscillator model

  • Wavefunctions: harmonic oscillator

  • Wavefunctions: Morse oscillator

  • Wavefunctions: harmonic vs. Morse

  • Wavefunctions

  • Wavefunctions

  • Expectation value of position

  • Expectation value of position

  • Expectation value of position

  • Selection rulesFor anharmonicity, can replace the H.O. wavefunctions with Morse wavefunctionsor can keep more terms in the Taylor expansion of the dipole moment

  • Selection rules

  • Where xturn is the maximum value of xCorrespondence principle

  • Correspondence principle

  • Correspondence principle

  • Correspondence principle