Molecular Spectroscopy

1. Use the ladder operator formalism for harmonic oscillator to derive the selection rule on⟨ ′|( − ) | “⟩ for arbitrary n.2. For a heteronuclear diatomic molecule AB, the dipole moment function in the neighborhood ofR=Re is given by ( ) = + ( − ) + ( − ) 2 + ( − )3In which a, b, c and d are constants. Treating this molecule as a harmonic oscillator (using ladderoperator), expand dipole moment in Taylor series around R2 and then calculate the relative intensityof v=0->1, v=0->2 and v=0->3 transitions in terms of these constant and harmonic oscillatorconstants μ and ω.3. (McHale chapter10. Problem7) A general harmonic potential function for water is = 12 (∆ 1)2 + 12 (∆ 2)2 + 12 ( ∆ )2 + ∆ 1∆ 2 + ∆ 1∆ + ∆ 2∆ The last three terms contain off-diagonal force constants, while the first three are diagonal. Inmatrix form, this can be expressed as 2V=RTFR, where R=(∆ 1 ∆ 2 ∆ ) is the vector whoseelements are the internal coordinates. Find the symmetry coordinates S1, S2 and S3 for water,and the diagonal force constant f which permits the potential energy in form written STfS4. For raman spectroscopy, show that the following equation leads to a symmetric tensor, = , in the limit 0 ≪ .( ) = 1ℏ ∑[⟨ | | ⟩⟨ | | ⟩ 0 + + Γ −⟨ | | ⟩⟨ | | ⟩ 0 − − Γ ]

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