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In this video of benzene orbitals, a lot of different orbitals are presented. Even orbitals that, I assume, could never be occupied. What is the purpose of calculating all these orbitals? Isn't all the chemical information needed (bonding, reactions) coming from the HOMO and LUMO (FMO theory)?

Here is an image of a few of the orbitals shown in the video:

Energy diagram of the molecular orbitals explored for the D6h structure (left) and the D3h distorted geometry (right)
(from David Casanova and Pere Alemany, Phys. Chem. Chem. Phys. 2010, 12, 15523-15529. DOI:10.1039/C0CP01326A)

Energy diagram of the molecular orbitals explored for the D6h structure (left) and the D3h distorted geometry (right)

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    $\begingroup$ Short answer: No, not all information is present in the HOMO and LUMO shapes and energies. I’ll leave it to one of our quantum gurus, though ;) $\endgroup$
    – Jan
    Nov 14, 2015 at 13:38

2 Answers 2

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A practical answer: in most of molecular quantum chemistry, the cost of obtaining unoccupied ("virtual") orbitals is next to nothing because of the algorithms involved. In short, one diagonalizes some matrix (i.e. finding eigenvalues and eigenfunctions, which are the orbital energies and orbitals) and the virtual orbitals are usually delivered anyway.

A more theoretical answer: While a lot of chemistry involves only the ground state, there is UV/vis spectroscopy and photochemistry. For those fields, virtual orbitals are immensely useful, because they will be used for the description of the electronic transitions that happen in those fields. These transitions are not necessarily between HOMO and LUMO because of symmetry, energy scales involved etc.

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  • $\begingroup$ In general, I'd say you are right, but I think your phrasing is not to the point. You don't get virtual orbitals at "next to nothing" cost, that sounds as if you'd make some extra calculation to get them, but because they are part of the solution. They are not "usually delivered anyway" but they are delivered period. For UV-vis, virtual orbitals are not only "immensly useful" but they are absolutely necessary in almost all cases. The HOMO-LUMO transition is usually only one peak but might not be the most interesting one in photophysics/-chemistry. $\endgroup$
    – pH13 - Yet another Philipp
    Nov 3, 2017 at 9:31
  • $\begingroup$ @pH13-YetanotherPhilipp There are useful algorithms that do not yield all eigenpairs, but merely give you all of them up to a set threshold (the case in solid state QC). I have applied them in molecular QC. For UV/vis, some DFT fanatics will tell you about approaches that are not reliant on virtual orbitals per se. $\endgroup$
    – TAR86
    Nov 3, 2017 at 11:53
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Even orbitals that, I assume, could never be occupied.

That's an incorrect way of thinking. Orbitals are not occupied. You need occupation of those orbitals to describe the electronic density of the transformation you are doing.

This is, in my opinion, the biggest flaw of explaining chemistry transformation at the quantum level. It's explained the other way around.

So what is the purpose of calculating all these orbitals?

Two reasons:

  • you always get as many molecular orbitals as initial atomic orbitals you put in, and the atomic orbitals you put in depends on the size of the basis set you choose.

and

Isn't all the chemical information needed (bonding, reactions) coming from the HOMO&LUMO?

  • Not at all, but it generally is a reasonably good approximation for some reactions, or some transformations. But for some transformations you do need to involve more of them.
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  • $\begingroup$ what do you mean by "transformation"? reaction? $\endgroup$
    – Sparkler
    Nov 14, 2015 at 14:39
  • $\begingroup$ @sparkler Could be a reaction, could be excitation, could be application of an external field, or a geometrical distortion. $\endgroup$
    – Stefano Borini
    Nov 14, 2015 at 14:42
  • $\begingroup$ why do you say that orbitals are not occupied? isn't every orbital occupied by 2 electrons? $\endgroup$
    – Sparkler
    Nov 14, 2015 at 14:47
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    $\begingroup$ Ok, first go check my answer here. It's exactly the same idea with molecular orbitals. Orbitals are occupied by electrons because it's a mathematical convenient way of presenting a description of reality, but it's just a mathematical description. $\endgroup$
    – Stefano Borini
    Nov 14, 2015 at 14:49
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    $\begingroup$ That said, it's is a complex topic not because it's complex, but because I would explain it to you so that it's clear in a couple of hours if we had access to a whiteboard. This place is simply not good enough to explain you static and dynamic correlation. $\endgroup$
    – Stefano Borini
    Nov 14, 2015 at 14:58

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