Understand Infrared Absorption & Vibrational Frequencies

[newlabguy]

I'm confused about infrared spectroscopy. Wikipedia states that "when the frequency of the IR is the same as the vibrational frequency of a bond, absorption occurs." However, if I look at some vibrational transitions of HCl for the anharmonic oscillator, I see:

0-->1 [tex]\omega[/tex] = 2885.7
0-->2 [tex]\omega[/tex] = 5771.8

The observed frequency that you would see on the spectrum is clearly different than the vibrational frequency of the bond. My line of thinking is that in vibration state 0, the bond oscillates with a frequency and that in vibration state 1 it oscillates in a higher frequency. So the frequency absorbed corresponds to the frequency that is appropriate to excite from one energy level to the other, instead of a frequency that matches the vibrational frequency of the lower vibrational state such as 0. Can someone tell me in simple terms where my thinking is wrong?

 

[alxm]

Yeah, that's an incorrect statement. The vibrational frequency of the atoms is of course entirely different from the frequency of the radiation involved. Also, it's not the frequency of the 'bond' (in any sense) that has to be matched, but the frequency that corresponds to the energy of the transition.

 

[DrDu]

In contrast to alxm, I think the statement from wikipedia is as correct as a classical description of a quantum phenomenon can be. Especially, Fourier transformation of the vibrational motion of a anharmonic oscillator with the same reduced mass and force constants and anharmonicity would give you approximately the frequencies you cited.
It does not make sense to assign a vibrational frequency to a single state (like state 0) in qm, as you do. The classically observed frequencies always correspond to transitions. This is related to the fact that a state that will behave as classical as possible will not be an eigenstate of the Hamiltonian but a coherent superposition of these, e.g. of the 0, 1 and 2 state.

 

[newlabguy]

Thank you very much for your replies. I'm still confused about the matter and I think it might be due to my error in interpretation of vibrational states. The set of eigenvalues associated with a one-dimensional harmonic oscillator for a diatomic molecule is

E_v=hf(v+(1/2)) v=1,2,3,...

So the way that I read the equation is that for each vibrational number there is an energy associated with it and also a frequency associated with that energy. So, when I look at a classical harmonic oscillator curve and see a horizontal vibrational line drawn on it that corresponds to v=0,1,2..., I assume that there is a frequency of molecular vibration that is associated with that energy level. DrDu, I think that my problem is that I'm still having trouble understanding why it is wrong to assign a frequency to a vibrational state. Can you explain more about what you mean by a "coherent superposition." Thanks again

 

[DrDu]

Let me ask another question: How would you measure or define the frequency of an eigenstate?
A nice exposition about the quantum-classical correspondence can be found in the first chapters of Landau Lifshetz "Quantum mechanics".