Molecular Spectroscopy Chemistry Homework

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

βŸ¨π‘£ β€²|(𝑅 βˆ’ 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

πœ‡(𝑅) = π‘Ž + 𝑏(𝑅 βˆ’ 𝑅𝑒 ) + 𝑐(𝑅 βˆ’ 𝑅𝑒 ) 2 + 𝑑(𝑅 βˆ’ 𝑅𝑒 )

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

𝑉 = 1

2 π‘˜π‘Ÿ (βˆ†π‘Ÿ1)

2 + 1

2 π‘˜π‘Ÿ (βˆ†π‘Ÿ2)

2 + 1

2 π‘˜πœƒ (π‘Ÿβˆ†πœƒ)

2 + π‘˜π‘Ÿπ‘Ÿ βˆ†π‘Ÿ1βˆ†π‘Ÿ2 + π‘˜π‘Ÿπœƒ π‘Ÿβˆ†π‘Ÿ1βˆ†πœƒ + π‘˜π‘Ÿπœƒ π‘Ÿβˆ†π‘Ÿ2βˆ†πœƒ

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=(βˆ†π‘Ÿ1 βˆ†π‘Ÿ2 βˆ†πœƒ) 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œŒπœŽ =

π›ΌπœŽπœŒ, in the limit πœ”0 β‰ͺ πœ”π‘’π‘” .

(π›ΌπœŒπœŽ )𝑖𝑓 = 1

ℏ βˆ‘[

βŸ¨π‘–|πœ‡πœŒ|π‘›βŸ©βŸ¨π‘›|πœ‡πœŽ |π‘“βŸ©

πœ”0 + πœ”π‘›π‘“ + 𝑖Γ𝑛 βˆ’

βŸ¨π‘–|πœ‡πœŽ |π‘›βŸ©βŸ¨π‘›|πœ‡πœŒ|π‘“βŸ©

πœ”0 βˆ’ πœ”π‘›π‘– βˆ’ 𝑖Γ𝑛 ]

𝑛

 
"Looking for a Similar Assignment? Get Expert Help at an Amazing Discount!"