[CUSTOM SOLUTION] Molecular Spectroscopy

1. 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 ofR=Re is given by??(??) = ?? + ??(?? ? ???? ) + ??(?? ? ???? ) 2 + ??(?? ? ???? )3In which a, b, c and d are constants. Treating this molecule as a harmonic oscillator (using ladderoperator), expand dipole moment in Taylor series around R2 and then calculate the relative intensityof v=0->1, v=0->2 and v=0->3 transitions in terms of these constant and harmonic oscillatorconstants ? and ?.3. (McHale chapter10. Problem7) A general harmonic potential function for water is?? = 12 ???? (???1)2 + 12 ???? (???2)2 + 12 ???? (?????)2 + ?????? ???1???2 + ?????? ?????1??? + ?????? ?????2???The last three terms contain off-diagonal force constants, while the first three are diagonal. Inmatrix form, this can be expressed as 2V=RTFR, where R=(???1 ???2 ???) is the vector whoseelements 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 STfS4. For raman spectroscopy, show that the following equation leads to a symmetric tensor, ?????? =??????, in the limit ??0 ? ?????? .(?????? )???? = 1? ?[???|????|??????|???? |?????0 + ?????? + ????? ????|???? |??????|????|?????0 ? ?????? ? ????? ]

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