# Molecular Spectroscopy Chemistry Homework

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

β¨π£ β²|(π β ππ ) π |π£”β© 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=RTFR, 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 STfS

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[