I would like to know how to compute the frequency at which nitrogen inversion happens in ammonia gas at STP. The activation energy for such process is 24.2 kJ/mol, therefore I thought that it is enough to plug that number into $E=h\nu$ and solve for $\nu$, but I obtain a value which is very far from the correct one (23.8 GHz). Many sources say that the value can be computed from quantum theory but they are not more specific on that. Also, does the same calculation applies to amines like e.g. piperidine? Thank you.
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2$\begingroup$ E=hν? Dude, you can't just plug values into random equation! $\endgroup$– MithoronCommented May 2 at 16:46
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$\begingroup$ And, for starters, it's obviously dependant on temperature. $\endgroup$– MithoronCommented May 2 at 16:48
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1$\begingroup$ See Feynman's elucidation: feynmanlectures.caltech.edu/III_09.html $\endgroup$– DrMoishe PippikCommented May 3 at 0:42
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The inversion frequency depends on the vibrational energy level, and the shape of the potential which has the form of a double well with a 'hill' in between The lowest transition has a gap of $0.7935\,\mathrm{cm^{-1}}$. The atoms tunnel from one side to the other below the barrier top which is at $2063\,\mathrm{cm^{-1}}$ and levels are doubled below the barrier. To calculate the energy one has to fit spectroscopic data to a potential energy equation you choose and numerically solve the Schroedinger enq. See Swalen, J. & Ibers, J. J. (1962). Chem. Phys. 36: p1914. As many geometries are needed I'm not sure if it can be solved using an ab initio method, I'm not an expert in this.
Ammonia inversion potential showing energy levels and wavefunctions. Below the barrier most levels are so close together that they cannot be distinguished on this scale. In the equation I varied $a,b$ to fit the experimental data. ($1$ hartree = $219474.631\,\mathrm{cm^{-1}}$) See https://applying-maths-book.com chapter 11 section 10.2 for more details and calculations.
