[Solved] Molecular Spectroscopy Chemistry Homework

1. Use the ladder operator formalism for harmonic oscillator to derive the selection rule on
βŒ©π‘£
β€²
|(𝑅 βˆ’ 𝑅𝑒)
𝑛
|𝑣

βŒͺ for arbitrary n.
2. For a heteronuclear diatomic molecule AB, the dipole moment function in the neighborhood of
R=Re is given by
πœ‡(𝑅) = π‘Ž + 𝑏(𝑅 βˆ’ 𝑅𝑒
) + 𝑐(𝑅 βˆ’ 𝑅𝑒
)
2 + 𝑑(𝑅 βˆ’ 𝑅𝑒
)
3
In which a, b, c and d are constants. Treating this molecule as a harmonic oscillator (using ladder
operator), expand dipole moment in Taylor series around R2 and then calculate the relative intensity
of v=0->1, v=0->2 and v=0->3 transitions in terms of these constant and harmonic oscillator
constants ΞΌ and Ο‰.
3. (McHale chapter10. Problem7) A general harmonic potential function for water is
𝑉 =
1
2
π‘˜π‘Ÿ(βˆ†π‘Ÿ1)
2 +
1
2
π‘˜π‘Ÿ(βˆ†π‘Ÿ2)
2 +
1
2
π‘˜πœƒ(π‘Ÿβˆ†πœƒ)
2 + π‘˜π‘Ÿπ‘Ÿβˆ†π‘Ÿ1βˆ†π‘Ÿ2 + π‘˜π‘Ÿπœƒπ‘Ÿβˆ†π‘Ÿ1βˆ†πœƒ + π‘˜π‘Ÿπœƒπ‘Ÿβˆ†π‘Ÿ2βˆ†πœƒ
The last three terms contain off-diagonal force constants, while the first three are diagonal. In
matrix form, this can be expressed as 2V=RT
FR, where R=(βˆ†π‘Ÿ1 βˆ†π‘Ÿ2 βˆ†πœƒ) is the vector whose
elements 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 S
T
fS
4. For raman spectroscopy, show that the following equation leads to a symmetric tensor, π›ΌπœŒπœŽ =
π›ΌπœŽπœŒ, in the limit πœ”0 β‰ͺ πœ”π‘’π‘”.
(π›ΌπœŒπœŽ)𝑖𝑓 =
1
ℏ
βˆ‘[
βŸ¨π‘–|πœ‡πœŒ|π‘›βŸ©βŸ¨π‘›|πœ‡πœŽ
|π‘“βŸ©
πœ”0 + πœ”π‘›π‘“ + 𝑖Γ𝑛
βˆ’
βŸ¨π‘–|πœ‡πœŽ
|π‘›βŸ©βŸ¨π‘›|πœ‡πœŒ|π‘“βŸ©
πœ”0 βˆ’ πœ”π‘›π‘– βˆ’ 𝑖Γ𝑛
]
οΏ½

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