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I have been reading about the relation of molecular graph spectrum, that is, the set of eigenvalues of the Laplacian matrix of the molecular graph, and molecular properties. Recently, I came across a paper that finds an upper bond for maximum eigenvalue (DOI: 10.1109/ICNC.2012.6234727). To motivate their research, authors state that

The eigenvalues of adjacency matrix of the graph is used to present the energy level of specific electronic. Particularly, spectral radius of graphs is the maximum energy level of molecules.

I was wondering if anyone can elaborate on what maximum energy level of molecules is. Is it the maximum energy level among all the atoms of the molecule?

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    $\begingroup$ This is interesting. I don't have access to the paper, but maximum vibrational level makes some sense in that all vibrations are anharmonic and hence dissociate at some finite vibrational quantum number. I'm not sure how they define maximum energy level for the entire molecule though. This is just one way, but this would depend on the vibrational state. $\endgroup$ – jheindel Feb 10 at 18:59
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    $\begingroup$ Ok I got access to the paper and they don't talk about this at all in the paper. They just derive some theorems which give bounds on the spectral radius of a graph. While it may be possible there is some application for this in chemistry it's hard for me to see the connection between what they're describing and whatever type of energy of a molecule they're talking about. $\endgroup$ – jheindel Feb 11 at 3:52
  • $\begingroup$ Yeah, I even looked up maximum energy level of molecules in quotation and all the links in the first page are from this paper, so it seemed to me that they kinda have coined the term! $\endgroup$ – Blade Feb 11 at 13:54
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In the painting "Dream Caused by the Flight of a Bee" by Salvador Dali, among other strange items, we may see a fish flying in midair. There is also one in the drawing "Big Fish Eat Little Fish" by Bruegel the Elder. One may conclude that there is a flying fish in every picture out there. One may even publish a paper to that effect. But I believe you see the shortcomings of this way of reasoning.

Same thing here. Molecules don't have the maximum energy level. You may think of the energy required to break a molecule, but that's not a level; rather, it is a lower bound of the continuous spectrum. You may think of the energy levels of its constituent atoms, but those are quite irrelevant to the energy levels of the molecule. Moreover, atoms don't have the maximum energy level either.

At this point you may wonder: what were the authors thinking? Were they (together with the editors) just lunatics? Not at all! Then what did they have in mind? Why, they said it loud and clear: the spectra of graphs. How are those related to the spectra of molecules? Is a molecule the same as a graph? Of course it isn't... unless we are looking at it within certain simplistic approximation where it is.

Yes, this is the old good Hückel method once again. You take a conjugated $\pi$ system, draw a graph of the molecule, solve it for eigenvalues, and get the $\pi$ energy levels of the molecule (only upside down). Naturally, there is but a finite number of those, and there inevitably is a maximum. That's what they mean when talking about the "maximum energy level of a molecule".

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Now, these $\pi$ levels typically include the frontier orbitals (HOMO and LUMO), and those next to them, and answer pretty much any chemical question about the molecule. All in all, it is quite natural and easy to forget altogether about all other orbitals and energy levels of the molecule ($\sigma$ bonds and stuff). But they are still there, and what I said before is still true.

So it goes.

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I can think of two reasons that a 'maximum energy' might exist for a molecule:

  • We can rigorously define the atomization energy for a molecule - that is, the enthalpy required to completely separate a given molecule into constituent atoms.
  • As mentioned above, above a certain vibrational energy, the vibration will dissociate (e.g., one can break a C-H bond in methane without atomizing the entire molecule).

I don't believe you can bound this for all possible molecules. The atomization energy will increase with the number of atoms and bonds - the more of each, the greater the enthalpy required to atomize. (Indeed, my group often uses a normalized atomization energy by dividing by the number of atoms.)

For an isolated vibration, this would depend on a particular bond strength so there may be some limit (e.g. a nitrogen-nitrogen triple bond for main group elements or a sextuple bond for metal-metal interactions). Yet, I think people would talk about this in the context of a limit to bond dissociation energies, not a maximum energy level.

The final possibility would come in the context of electronic levels. To bind an electron, there is an energy required, which is why the standard for orbital energies is negative. Above that point, and removing an electron from an electronic state is favorable.

Yet I can't count that as a maximum 'energy level' because of course we can perform spectroscopy for higher-level electronic excited states.

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