Talk:Koopmans' theorem

message: I believe that the part of Koopmans' theorem referring to electron affinities is just a sort of fable which is passed on by textbooks but that is WRONG, at least as far as Hartree-Fock orbitals are concerned.
Virtual orbitals in the HF scheme are a sort of by-product of using a finite basis set but have no physical meaning. In fact, they don't have well-defined orbital energy, while on the contrary the occupied orbitals do.
As we increase our basis set towards the complete basis set-limit the energies of the virtual orbitals keep changing and eventually collapse to zero.
I would like to hear what other people have to say about this, but if it were for me I'd change the part regarding EA completely. —Preceding unsigned comment added by L0rents (talk • contribs) 
You are largely correct. See Introduction to Computational Chemistry, Frank Jensen, Wiley, 1990, Section 3.4, pg 64 - 65. This gives a good discussion. However, formally Koopmans' theorem does give the electron affinity for frozen orbitals, but that assumption of course is more doubtful for the anion and it is doubtful enough for a cation. Think about a rewording. I will too, but I'm tied up today. Note, Koopmans' theorem strictly only applies to Hartree-Fock. --Bduke 21:19, 21 March 2007 (UTC)
replies: text: I have now added this point using H2O as an example. Dirac66 (talk) 14:58, 19 February 2011 (UTC)
replies: text: Excellent. I have made a slight change. The order of the ion states is not always in the order of the orbital energies. In fact it rarely is for all of them. In particular there are many cases where the HOMO Koppman prediction is not for the ground state of the ion. Long ago I recall a calculation on a Nickel carbonyl compound where the ground state ion came from removal of an electron in the 16th MO down from the HOMO. Another example is the N2 molecule where Hartree-Fock calculations (with all basis sets larger than a minimum basis set) predict the wrong order of the pi and sigma states of the ion. On the latter I could cite a paper of mine in J Chem Educ, but that would be COI. --Bduke  (Discussion) 21:04, 19 February 2011 (UTC)
replies: text: Yes, your wording is more accurate. I did deliberately choose H2O as example rather than N2 because Koopmans' thm predicts the correct order for H2O, but I agree that it is better to point out that this is not always the case. Dirac66 (talk) 21:44, 19 February 2011 (UTC)
replies: text: And I have now finally added a mention of N2 as well using the 1995 JCE article as source. I hope I have it right. Mieux vaut tard que jamais. Dirac66 (talk) 23:49, 7 February 2014 (UTC)

message: To simplify these integrals, the piecewise linear behavior of the total energy as a function of the number of electrons is utilized. In general, for a fixed geometry defined by the N-electron state, the ground state energy when the number of electrons is varied on the interval (N − 1,N) is a linear function with slope equal to the ionization energy, and on (N,N + 1) it is linear with slope equal to the electron affinity. At N, the energy has a derivative discontinuity with respect to the number of electrons.
 In DFT with the exact exchange-correlation energy functional, this fact together with Janak's theorem means that the HOMO energy is constant for 0 < nHOMO <1, and the LUMO energy is constant for 0 < nLUMO <1, so that
:
\begin{align}
\mathrm{IE}&=-(E(N)-E(N-1))&=-\epsilon_\mathrm{HOMO}(N+\delta),\\
\mathrm{EA}&=-(E(N+1)-E(N))&=-\epsilon_\mathrm{LUMO}(N+\delta),
\end{align}
for any δ between zero and one.
At δ = 0 the HOMO and LUMO energies change discontinuously. The interpretation of the above equations in this limit has a contentious history. It is now accepted that the vertical ionization energy is equal to the HOMO energy of the N-electron state, and generally believed that the LUMO energy is not directly related to the electron affinity.
In practice, approximate DFT (and HF) calculations do not obey the straight-line behavior for the total energy, and so the HOMO and LUMO energies are in error. Typically in DFT calculations the energy is curved concavely below the exact linear form and the HF energy curved convexly above it, as shown in the adjacent figure for a carbon atom. In this example, and quite generally, the negative of the slope of the DFT energy for N infinitesimally less than six is too small, so that the ionization energy predicted by the HOMO energy is too small. For N infinitesimally more than six, the negative of the slope of the energy, and consequently the electron affinity predicted by the LUMO energy, is too large.
About ROHF koopman's
The idea that there is no Koopman's theorem for open shells is not 100% true.
http://pubs.acs.org/doi/abs/10.1021/jp101758y   (explains pitfall and problems)
http://pubs.acs.org/doi/abs/10.1021/jp9002593   (explains what they did)
I suggest links to these as references.
130.20.228.97 (talk) 18:37, 14 May 2010 (UTC)
thread_title:  Useful for discussing derivative discontinuity 

message: Hi! For me (non-native speaker), this sentence seems to be incomplete: For H2O, minus the near-Hartree-Fock orbital energies of these orbitals in eV, are 1a1 559.5, 2a1 36.7 1b2 19.5, 3a1 15.9 and 1b1 13.8. Could Someone clarify that sentence to make it more understandable? Thanks --PassPort (talk) 17:49, 16 May 2013 (UTC)
replies: text: I have now revised that sentence - is the new version better? I didn't know whether you were confused by the awkward placement of minus or of in eV, so I changed both. Dirac66 (talk) 23:56, 16 May 2013 (UTC)
thread_title:  incomplete sentence? 

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thread_title:  External links modified 

message: Shouldn't this article be "Koopmans's Theorem", not "Koopmans' Theorem." "Koopmans" is his name, so the the singular possessive should have the second "s" as per the Wikipedia Manual of Style and MLA etc..  — Preceding unsigned comment added by Nblewis (talk • contribs) 15:14, 11 December 2018 (UTC)
replies: text: I think we should conform to the usage in the field, here quantum chemistry. I found Koopmans' used in Atkins' Physical Chemistry (which says Atkins' on the cover and not Atkins's), Levine's Quantum Chemistry, and Drago's Physical Methods in Chemistry. Also in the IUPAC Gold Book. Dirac66 (talk) 00:45, 13 December 2018 (UTC)
thread_title:  Koopmans's Theorem