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 + ??(?? ? ???? )
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 ? 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 ? ?????? ? ????? ]
