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So regarding Koopman's theorem I found the the explanation that you should not use it to find the ionization potential of localized orbitals because, much unlike canonical orbitals they will not be stationary towards small orbital changes. So any minor perturbation will cause a big difference in energy.

Usually I would simply say that the canonical orbitals are the eigenvalues of the diagonalized fock matrix while localized orbitals are not direct solutions of the fock matrix.

But isn't localization simply another transformation of the matrix that keeps the energy but sort of contracts the density more around bonds or atoms to resemble the bonds or lone pairs we know from organic chemistry? How does this effect the way they behave towards small changes to the orbitals?

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    $\begingroup$ Does this answer your question? $\endgroup$ – hBy2Py Jun 29 '17 at 14:42
  • $\begingroup$ Thank you for the link but I pretty much based my explanation given above from said discussion. The Koopman's theorem here is just an example where it can be used but I was rather referring to the question what this 'stationary towards small orbitals changes' of the energy means for the canonical and localized orbitals. $\endgroup$ – Justanotherchemist Jun 29 '17 at 15:08
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    $\begingroup$ Hm. I don't think I understand your question then. It's not that you shouldn't use localized orbitals to calculate the IE via Koopmans', it's that you can't: localized orbitals have undefined energies. Separately, the electron density field calculated from localized orbitals should be identical to that calculated from canonical orbitals. Can you clarify further what you mean by "the way they behave" and "small changes to the orbitals", e.g., in your last sentence? $\endgroup$ – hBy2Py Jun 29 '17 at 15:21
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    $\begingroup$ Stationary means the orbitals won't react to removing one electron. So this basically refers to the assumption in Koopmans' theorem. For localized orbitals you cannot make such assumptions, they are not even uniquely defined. $\endgroup$ – Feodoran Jun 29 '17 at 16:29
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    $\begingroup$ Well I guess it refers to the orbital energy not being changed, thus being stationary. PES is a whole different story, it is about the nuclear problem, where nuclear coordinates vary. The orbitals belong to the electronic problem, where nuclear coordinates are fixed. Both problems are treated separately due to the Born-Oppenheimer-Approximation. $\endgroup$ – Feodoran Jun 29 '17 at 17:36

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