# [Solved] Molecular Spectroscopy Chemistry Homework

1. Use the ladder operator formalism for harmonic oscillator to derive the selection rule on
β²
|(π β ππ)
π
|π£

βͺ for arbitrary n.
2. For a heteronuclear diatomic molecule AB, the dipole moment function in the neighborhood of
R=Re is given by
π(π) = π + π(π β ππ
) + π(π β ππ
)
2 + π(π β ππ
)
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
π =
1
2
ππ(βπ1)
2 +
1
2
ππ(βπ2)
2 +
1
2
ππ(πβπ)
2 + πππβπ1βπ2 + ππππβπ1βπ + ππππβπ2βπ
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=(βπ1 βπ2 βπ) 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π =
πΌππ, in the limit π0 βͺ πππ.
(πΌππ)ππ =
1
β
β[