# Molecule of ammonia

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1. Jan 30, 2015

### Konte

Hello everybody,

I am new in this forum and also new with english langage, so don't be shocking if I write like a baby for you. I am little better for understanding when I read.

My question:
Books always tell us about the only ground state of ammonia molecule but never on its excited state. So, I am thinking to myself, could we try to study the first excited state of this molecule by copying the method used for the ground state?

I mean: for the ground state, there is a pair of eigenstate designed by $\psi_1^+$ and $\psi_1^-$ whith wich we can construct another pair of orthogonal state designed by $\phi_1^L$ and $\phi_1^R$. With those basis, the dynamic of the ground state of the molecule is understood.

My question is about generalizing this approach on the other pair of excited level. From the second pair of eigenstate $\psi_2^+$ and $\psi_2^-$, could we construct a kind of $\phi_2^L$ and $\phi_2^R$ ?

Thank you everybody.

2. Jan 30, 2015

### Quantum Defect

You can actually "see" a transition between the two levels. This is the transition for the ammonia maser. http://quantum.lassp.cornell.edu/lecture/ammonia_maser

For the higher levels, the energy levels are known from infrared spectroscopy. D. M. Dennison, Rev. Mod. Phys. v. 12, p. 175 (1940) https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.12.175 [Broken]

Last edited by a moderator: May 7, 2017
3. Jan 30, 2015

### Konte

Yes, I already know about ammonia maser. It is described by the first pair of state of the ammonia. But my question is about excited ammonia, which include other pair of excited state. Can we describe the "dynamic" of ammonia when it is excited?

4. Jan 30, 2015

### Quantum Defect

I edited the entry above. The higher levels have been seen. A good description of the wavefunctions is available in Herzberg, "Infrared and Raman Spectra"

Looking for cites to the paper by Dennison will likely lead to a lot of newer information.

5. Jan 31, 2015

### Konte

Thank you for the answer. I needed all night to find the Herzberg's book on the internet. I begin to read it for now, and after I will search the paper of Dennison. I'll be back very soon.

6. Jan 31, 2015

### DrDu

I can also recommend warmly Bunker & Jensen, Molecular Symmetry and Spectroscopy. The point which I found most interesting is that the splitting is only observed for "chiral" states of ammonia, where the nuclear spins allow us to distinguish between the two forms of ammonia related by inversion. This isn't the case for all states of ammonia.

7. Feb 2, 2015

### Konte

8. Feb 13, 2015

### Konte

Hello everybody,

After searching, reading and trying to understand all books and letters that you recommended to me, I am back into the forum for ask a question.
In the Herzberg's book (p. 221), about the symmetric double well potential, he present the eigenvalues as a pair of splitting levels. The pair of eigenfunctions are:

$\psi_s = \psi_{\nu}(x-x_0)+ \psi_{\nu}(x+x_0)$
$\psi_a = \psi_{\nu}(x-x_0)- \psi_{\nu}(x+x_0)$

I think , he means:

$\psi_s = \psi_{\nu}^R+ \psi_{\nu}^L$
$\psi_a = \psi_{\nu}^R- \psi_{\nu}^L$

My question:
Usually, authors talk always about the first pair of level ( $\nu=0,1$) for explaining the tunnelling inter-well.
But now, if we consider that the system is excited and occupy the second pair of level, is it possible to describe tunnelling phenomena in this case as only function of the pair of level corresponding to $\nu=2,3$?
Or, instead, we have to use all of the levels corresponding to $\nu=0,1,2,3$?

Thank you everybody.

9. Feb 13, 2015

### DrDu

Forget Herzberg, he's been an experimentalist and those usually don't understand very well what they are doing :-) In the case of Herzberg theoreticians tried desperately to reproduce his value for the binding energy of the H2 molecule until it turned out that he had measured the wrong line.
What about the book by Bunker and Jensen:
especially around page 277? He treats several transitions.

10. Feb 13, 2015

### Quantum Defect

For the next tunneling levels, you can do the same kind of thing, imagining that they are linear combinations of the left and right well v_l,r = 1 wavefunctions. If you wanted to think of writing all of the actual vibrational wavefunctions in terms of the left ,right harmonic oscillator basis set, you would need to include bits of the other basis functions in the wavefunction. In order to avoid wasting time, you would rewrite the l, r set as +/- wavefunctions and use these as your symmetry-adapted basis (only including + basis set functions in the + wavefunctions, and vice versa for the -)

Even though I was trained as an experimentalist, I too would recommend the Bunker & Jensen book. There are a lot of things with spectroscopy that are historical, that tend to obfuscate, and this book does a very good job of clearing away the cobwebs and showing the molecular physics in modern terms, without the barouque curlicues decorating everything.

If you would like to have some fun with the double well, you might look for a famous paper by J. W. Cooley (J.W. Cooley, Math. Comp. 15 (1961) 363) that describes the Numerov-Cooley method for finding wavenfunctions for arbitrary one-dimensional potentials. You can "write" an implementation using Excel (something that I had my students use to investigate the tunneling splitting of levels in a double-well potential). If you can program, even better. Here is a "recent" BS Thesis that discusses some of these methods: http://ssmf.fjfi.cvut.cz/studthes/2008/bac_thesis_Augustovicova.pdf

Cooley also did very early work in computational FFT (the Cooley-Tukey algorithm is partley named after him): http://en.wikipedia.org/wiki/Cooley–Tukey_FFT_algorithm

11. Feb 13, 2015

### DrDu

I also did a little bit of googling and I think the $\nu=2,3$ levels are already on top of the barrier and higher levels definitely above. So this is not tunneling any more. For molecules like ND3 the 3,4 levels should be still deep in the well and only be little split by tunneling. I think they can be treated as isolated (just like the $\nu=0,1$ levels) for the calculation of tunnel splitting.

12. Feb 25, 2015

### Konte

Dr Du, in your very far previous message (31 January), you advised me to read Bunker & Jensen, Molecular Symmetry and Spectroscopy, and at your latest message you gave me a link of Bunker & Jensen, fundamentals of molecular symmetry around page 277. In which of these books I will find my salvation? I precise that currently, I follow your first advice about the first book, and I confess you , it is too hard to find the response of my question, but I keep courage and still read it. :D

13. Feb 25, 2015

### DrDu

I fear I meant the second one, although it may be that the first one also contains information on this topic. I can't localize my copy. As far as I understand it, in NH3, only the lowest two vibrational states are below the barrier for inversion, while the next two states are already at the barrier. So it is no longer precise to speak of a tunnel splitting for these.
This is different in, say, ND3 where the vibrational states are energetically lower. The situation is considerably complicated by the statistics of the nuclei.

14. Feb 25, 2015

### Konte

You are right about the NH3, only the lowest two vibrational states are below the barrier for inversion. But, in actual fact, I only used the NH3 case to illustrate my questions. The real subject of my work are:
- the first one is about a symmetric double-well which have more than one pair of state below the central barrier.
- the second one is about an asymmetric double-well.

I wanted to begin by understanding the simplest case of ammonia before working on my real systems. That is why I talk about ammonia. :D

My ultimate goal is to describe, for each of these double-wells, how their occupation evolve with the time.

15. Feb 25, 2015

### DrDu

The point I found interesting in this context: when and how can you distinguish the inverted ammonia molecule from a rotated ammonia molecule?

16. Feb 26, 2015

### Konte

Rotation around the 3-fold axis you mean?

17. Feb 26, 2015

### DrDu

No, I mean a rotation which brings the inverted molecule into coincidence with the non-inverted one.