[Solved] Molecular Spectroscopy Chemistry Homework

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 of
R=Re is given by
( ) = + ( −
) + ( −
)
2 + ( −
)
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 μ and ω.
3. (McHale chapter10. Problem7) A general harmonic potential function for water is
=
1
2
(∆ 1)
2 +
1
2
(∆ 2)
2 +
1
2
( ∆ )
2 + ∆ 1∆ 2 + ∆ 1∆ + ∆ 2∆
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=(∆ 1 ∆ 2 ∆ ) 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, =
, in the limit 0 ≪ .
( ) =
1
ℏ
∑[
⟨ | | ⟩⟨ |
| ⟩
0 + + Γ
−
⟨ |
| ⟩⟨ | | ⟩
0 − − Γ
]
�