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This is a cross post from physics stackexchange. I hope to get more answers here, because I think chemists have more to do with such things.

So I was playing around with some data and formulas and wanted to calculate the index for the wave number of the vibration of F2 that I got from here, i.e. which allowed energy level n this corresponds to.

enter image description here

Permitted energy levels are given by:

$$\epsilon_n = h\nu\left(n+\frac{1}{2}\right),\quad n = 1,2,3,\ldots \tag{1}$$

The given wave number ($\pu{893.9 cm^-1}$) can be used to calculate $\epsilon$ with $\pu{1.986E-26 kJ cm}$ (however, now that I think about it a second time, I'm not so sure whether I can determine the energy of something like the wavenumber for vibrations of molecules such as electromagnetic waves via $h$ and $c$ so easily, which leads me to the question of how one is supposed to find out at what temperature/energy this wavenumber was determined using the above Web site?):

$$\epsilon_n = \pu{1.775E-20 J}$$

Now I just have to convert (1) to $n$ and insert the values:

$$n = \frac{\pu{1.775E-20 J}} {h \cdot c \cdot \pu{89390 m^-1} } - \frac{1}{2} = 0.499.$$

What am I doing wrong?

Edit: In the end, I can't do much with the value $\pu{893.9 cm-1}$ for F2 without knowing to which energy this frequency belongs. After all, I can generally only calculate an energy for electromagnetic waves via Plank's constant and $c$ with the help of the frequency, but here we are dealing with molecular oscillation; and as long as I don't know at which energy this frequency was measured, this tells me nothing about the nature of F2 (?)

I'm questioning how molecular vibrations can even be expressed in wave numbers $\pu{cm^-1}$, a unit that seems more fitting for spatial measurements. I'm aware that electromagnetic waves have wave numbers tied to spatial propagation due to the relationship $\lambda v = c$. However, I'm struggling to comprehend how a molecule's vibration relates to this. Unlike electromagnetic waves that propagate spatially, how can molecular vibrations be associated with this unit which normally represents energy? Is ν in (1) even the frequency associated with the molecular vibration or the frequency associated with the photon which is measured in IR Spectroscopy?

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    $\begingroup$ I think units conversions is hurting this problem really badly. Make sure at each step, dimensional analysis is done correctly. $\endgroup$
    – ACR
    Commented Aug 30, 2023 at 0:56
  • $\begingroup$ Why do you have units of kJ.cm? $\endgroup$
    – ACR
    Commented Aug 30, 2023 at 0:58
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    $\begingroup$ To answer the first part spectroscopy doesn't measure energy levels, it measures the difference between energy levels, what does this imply? To answer the edit wavenumber isn't an energy, it is a unit that can be related to energy by some fundamental constants, and this scaling is so common people don't bo5 to mention it, leaving it implied. There's a duplicate to this somewhere here. $\endgroup$
    – Ian Bush
    Commented Aug 30, 2023 at 5:49
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    $\begingroup$ Please note that cross-posting is frowned upon in SE. $\endgroup$ Commented Aug 30, 2023 at 10:14
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    $\begingroup$ I’m voting to close this question because it is generating responses on the Physics site as well. $\endgroup$
    – Karsten
    Commented Aug 30, 2023 at 15:54

2 Answers 2

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Firstly the link you have provided unfortunately has already rotted. I suspect it was a link to a database query which disappear very rapidly. In future can you please provide a link to the original data, not a database query? This would preferably be as a doi of the original paper - e.g. if I were referencing one of my own efforts I would give https://doi.org/10.1016/j.cpc.2006.05.001 . I suspect what you want here is https://doi.org/10.1063/1.2436891

So I am guessing that the number given refers to the lowest observed energy transition in the vibrational spectrum. Note the word transition - it is the difference between two states, not the energy of any one given state. In the harmonic oscillator model assuming an allowed transition of $\Delta n=\pm1$ this is particularly easy, all transitions have an energy given by $E=h\nu$ where h is Planck's constant, and $\nu$ is the observed frequency - which is trivially derived from your equation (1). Thus from the data presented you can't actually get the number you want - and as an aside I wonder if in fact what you want really makes any sense? It is only Energy differences that mean anything, all energies are quoted relative to some zero. And where have you defined your zero in your question? Only once you have defined that can you talk about the Energy of a given level.

[Will come back later and edit this to link to/talk about wavenumbers and why they are used. But there is a duplicate somewhere ...]

Finally note this is a question about spectroscopy, not thermodynamics. Thermodynamic quantities may be derived from the vibrational frequency combined with other things, but that's the next stage, not this one.

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You are using the energy equation for a harmonic oscillator, the $893.9$ is most probably the $n=1\to n=2$ transition but this is a difference in energy (as already pointed out by by Ian Bush) and is $\epsilon_{n+1}-\epsilon_n= hv$ so $n$ disappears as all transitions are of the same frequency in the HO. So you cannot get at $n$ so you will need another model, say Morse Potential and more transition energies.

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