Molecular Spectroscopy Chemistry Homework

1. Use the ladder operator formalism for harmonic oscillator to derive the selection rule on

⟨𝑣 ′|(𝑅 − 𝑅𝑒 ) 𝑛 |𝑣”⟩ for arbitrary n.

2. For a heteronuclear diatomic molecule AB, the dipole moment function in the neighborhood of

R=Re is given by

𝜇(𝑅) = 𝑎 + 𝑏(𝑅 − 𝑅𝑒 ) + 𝑐(𝑅 − 𝑅𝑒 ) 2 + 𝑑(𝑅 − 𝑅𝑒 )

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 μ and ω.

3. (McHale chapter10. Problem7) A general harmonic potential function for water is

𝑉 = 1

2 𝑘𝑟 (∆𝑟1)

2 + 1

2 𝑘𝑟 (∆𝑟2)

2 + 1

2 𝑘𝜃 (𝑟∆𝜃)

2 + 𝑘𝑟𝑟 ∆𝑟1∆𝑟2 + 𝑘𝑟𝜃 𝑟∆𝑟1∆𝜃 + 𝑘𝑟𝜃 𝑟∆𝑟2∆𝜃

The last three terms contain off-diagonal force constants, while the first three are diagonal. In

matrix form, this can be expressed as 2V=RTFR, where R=(∆𝑟1 ∆𝑟2 ∆𝜃) 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 STfS

4. For raman spectroscopy, show that the following equation leads to a symmetric tensor, 𝛼𝜌𝜎 =

𝛼𝜎𝜌, in the limit 𝜔0 ≪ 𝜔𝑒𝑔 .

(𝛼𝜌𝜎 )𝑖𝑓 = 1

ℏ ∑[

⟨𝑖|𝜇𝜌|𝑛⟩⟨𝑛|𝜇𝜎 |𝑓⟩

𝜔0 + 𝜔𝑛𝑓 + 𝑖Γ𝑛 −

⟨𝑖|𝜇𝜎 |𝑛⟩⟨𝑛|𝜇𝜌|𝑓⟩

𝜔0 − 𝜔𝑛𝑖 − 𝑖Γ𝑛 ]

𝑛