# [Solved] Molecular Spectroscopy Chemistry Homework

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

β©π£

β²

|(π
β π
π)

π

|π£

”

βͺ 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

ππ(βπ1)

2 +

1

2

ππ(βπ2)

2 +

1

2

ππ(πβπ)

2 + πππβπ1βπ2 + ππππβπ1βπ + ππππβπ2βπ

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=(βπ1 βπ2 βπ) 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, πΌππ =

πΌππ, in the limit π0 βͺ πππ.

(πΌππ)ππ =

1

β

β[

β¨π|ππ|πβ©β¨π|ππ

|πβ©

π0 + πππ + πΞπ

β

β¨π|ππ

|πβ©β¨π|ππ|πβ©

π0 β πππ β πΞπ

]

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