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When labeling the energies of molecular orbitals, often some anti-bonding MOs are shown with a positive energy. For example, MolCalc has the following energies for methane (the anti-bonding MO #8 at +19.47 eV depicted on the right):

enter image description here

I'm trying to understand what these numbers mean. I get that electrons in MOs #2 through #5 help to keep atoms bound (having them occupied contributes to bonding), whereas electrons in MOs #6 through #9 would lead to an increase in bond lengths or the molecule falling apart. On the other hand, MO #1 does not participate in bonding, right (basically, it is the inner 1s electrons of carbon)?

So what defines the zero reference point of these orbitals? It can't be the free electron because I thought at least for the hydrogen atom, even the excited states are states where the electron is still bound (i.e. energies are negative). It can't be the gradient of increasing all bond lengths (i.e. with the extreme case being separate atoms) because MO #1 would be hardly affected by that, given that it hardly participates in bonding.

This question is not limited to one particular software or author. Here is another example (source: https://www.researchgate.net/publication/234126374_Electron_rectification_through_donor-acceptor-heterocyclics_connected_to_cumulenic_bridge_A_computational_study):

enter image description here

I realize that the language in this question is sloppy, reflecting my poor understanding. I encourage anyone who answers to use the correct terminology.

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    $\begingroup$ The reference point must be the free electron. What you thought for the hydrogen atom is right. But you aren't calculating a hydrogen atom. $\endgroup$
    – Ivan Neretin
    Mar 2, 2020 at 14:34
  • $\begingroup$ I think it will be worth contacting the company to ask for details. I had similar issues with the Matlab wavelet analysis toolbox and after asking around the world, it turned out to be something very arcane and specific to the way Matlab does it. They also had consistent labeling errors too in their toolbox, which are being worked upon. $\endgroup$
    – AChem
    Mar 2, 2020 at 15:03

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The answer technically depends on the method. In most cases, meaning HF or DFT with standard basis sets, it is infinite separation of all bare nuclei and all electrons. One exception known to me is VASP (a periodic, solid-state code), which uses pseudopotentials to avoid having to model the core singularity and core electrons. There, the zero-point is in reference to some atomic state.

Note that the MO energies are basis-set dependent, which is most pronounced and most easily understood for core electrons.

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  • $\begingroup$ If free electrons are defined as zero, what would happen to the methane molecule when you excite an electron in the MO approximation. There is no empty orbital with an energy lower than a free electron, so would it just ionize? What about for aromatics that absorb UV light? I would think the anti-bonding orbitals should have lower energy than the free electron because otherwise, UV light would not lead to absorption and emission of light but rather - again - to ionization. $\endgroup$
    – Karsten Theis
    Mar 2, 2020 at 19:25
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    $\begingroup$ As you excite one electron, the other MOs would not stay the same (unless you are in the frozen orbital approximation, which is poor). There are methods dedicated to the description of such excited states: time-dependent HF (TDHF), CI singles (CIS), time-dependent DFT (TDDFT), see pubs.acs.org/doi/abs/10.1021/cr0505627 $\endgroup$
    – TAR86
    Mar 2, 2020 at 20:06
  • $\begingroup$ So is there any physical interpretation of assigning a positive energy to the unoccupied anti-bonding MOs? $\endgroup$
    – Karsten Theis
    Mar 2, 2020 at 20:18
  • $\begingroup$ Much like Koopman's theorem, they can be used to estimate the electron affinity (in the frozen orbital approximation). But the basis-set dependence is again very high. $\endgroup$
    – TAR86
    Mar 3, 2020 at 8:57

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