T1. 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 + ??(?? ? ???? )

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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 ? and ?.

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=RTFR, 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 STfS

4. For raman spectroscopy, show that the following equation leads to a symmetric tensor, ?????? =

??????, in the limit ??0 ? ?????? .

(?????? )???? = 1

? ?[

???|????|??????|???? |???

??0 + ?????? + ????? ?

???|???? |??????|????|???

??0 ? ?????? ? ????? ]