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I am reading 'Fundamentals of Molecular spectroscopy' by Banwell-McCash and in different sections, one come across these selection rules for the allowed transitions but the book doesn't derive or provide proper explanations for some of them.

For example, in infra-red spectroscopy, the vibrating diatomic molecule, described by a simple harmonic oscillator allows transitions with the selection rule Δn=±1, whereas when described by an anharmonic osicllator, the selection rule is given as Δn=±1,±2,±3,.... This is just stated without any proper explanation to as why this model allows the larger jumps but harmonic one doesn't.

Also, in case of raman scattering in linear molecules, the selection rule for allowed transitions is given as ΔJ=±2, also stated without explanation. Can somebody explain these or can anyone suggest a source/book where these selection rules are explained or derived in detail?

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  • $\begingroup$ Do you know about parity? $\endgroup$
    – Superfast Jellyfish
    Commented Jan 28, 2020 at 7:37
  • $\begingroup$ @user3518839 yes $\endgroup$
    – Harshdeep Singh
    Commented Jan 28, 2020 at 8:31

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The way light interacts with atoms can be modelled by the electron cloud responding to the EM field of light. The response for low intensity light is predominantly dipolar. The dipole operator is odd under parity.

Since near equilibrium we are assuming a harmonic potential for the atom, the states have well-defined parity. It is known that the states even $n$ states have even $(+)$ parity and odd $n$ having odd parity $(-)$.

Let us assume that our initial state has even parity. Then the absorption of light has to take it to a state with odd parity as $\left(- \otimes +\right)=(-)$. And the nearest odd state is the one above or one below the initial state. And it’s easy to see that this is the case if our initial state is odd as well, because $\left(- \otimes -\right)=(+)$, takes it to an even state.

And so if we start at some state $n$, then due to absorption of light we can go to any state $n\pm k$ where $k=1,3,5...$ but since the intensity is low it means that the atom will most probably absorb only one photon and this means due to energy conservation, the state can only go to $n\pm1$

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  • $\begingroup$ That does explain the first part, but for anharmonic oscillator model selection rule is Δn=±1,±2,±3,... which isn't clear from parity operator and also same is the case with raman spectra. $\endgroup$
    – Harshdeep Singh
    Commented Jan 28, 2020 at 13:04
  • $\begingroup$ Since parity is well defined only for harmonic oscillator, there are no restrictions for a harmonic oscillator. $\endgroup$
    – Superfast Jellyfish
    Commented Jan 28, 2020 at 15:52

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