Talk:Koopmans' theorem

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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)[reply]

I thought that Koopmans' theorem was that the Hartree-Fock energy was equal to the weighted _average_ of photoemission binding energies (including all shake-ups and satellites due to correlation effects). But this is too much work to look up now. /Pieter Kuiper 09:37, 2 June 2007 (UTC)[reply]

Everything named 'theorem' outside mathematics or logic is misleading and confusing, it should be called Koopmans' Relation or so. As far as I know it not applicable to Density functional because even the exact density functional is only exact for the ground state, and knows nothing about the exact energy of excitations, in opposition Hartree Fock can in principle describe excitations (of course under the assumption that correlation is zero). 76.247.111.238 02:40, 12 August 2007 (UTC) Unsigned, 11 August 2007[reply]

It is widely called the Koopman's theorem. In fact it is rigorous mathematically as a derived result from the Hartree-Fock theory and all the assumptions that come with that. It is a rigorous derivation from an approximation. It is not applicable rigorously to DFT but there is an approximate relationship to the KS orbital energies, which strictly are not part of DFT either. --Bduke 03:04, 12 August 2007 (UTC)[reply]

I have rewritten the entire article, removing the obvious crap while taking into account the points brought up in the discussion above. In response to the several attempts to describe Koopmans' theorem in DFT --- all of which are factually wrong, IMO --- I have also included a mention of Janak's theorem, which I believe is the most appropriate analog of Koopmans' theorem in DFT. This should some day become its own article. Meanwhile, more references to the literature are sorely needed! AcidFlask (talk) 04:43, 21 April 2009 (UTC)[reply]

Pretty good work in rewriting it. One thing is now missing, the link between the energy of excited ionization states as observed in photoelectron spectroscopy with the orbital energy of orbitals lower than the HOMO. This follows in exactly the same way from the expression for the total energy of molecule and ion with frozen orbitals. It is a widely used idea. --Bduke (Discussion) 05:18, 21 April 2009 (UTC)[reply]

I have now added this point using H₂O as an example. Dirac66 (talk) 14:58, 19 February 2011 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

I wonder about the current terminology, "Generalization of". I understand why the editor stated it in this way, but does there exist literature which considers the LUMO-EA relationship a generalization of KT? There are generalizations of KT (eg the extended KT), but all of the current references consider KT to be both the HOMO-IE and LUMO-EA relationships. While Koopmans may not have included the EA in his paper, he also didn't call what he did "Koopmans' theorem" (anyone know who first coined the term??) THEN WHO WAS PHONE? (talk) 03:38, 22 April 2009 (UTC)[reply]

My intentions behind the rewrite are (a) to distinguish what is attributed to KT from what Koopmans ACTUALLY said, and (b) that KT is specific to HF theory, since and the derivation of Koopmans clearly doesn't apply to KS-DFT, but nevertheless one can try to generalize it to KS-DFT.

(a) I may be wrong, but my understanding of academic convention is that one usually does not attribute more results to a named result that wasn't in the original presentation(s). Interesting point about nomenclature though - who first called it KT? Who tacked on the part about EAs - is it in a later paper of Koopmans', or is it a result claimed by someone else (in which case it should be more correctly called the Koopmans-______ theorem)? The statement of a theorem by Koopmans is very specific about being (i) restricted to closed-shell HF, and (ii) said nothing about electron affinities. Your statement about EAs always being presented in treatments of "KT" is false. It is true that many secondary sources, for example, Szabo and Ostlund, do this. However, there are prominent counterexamples, such as in the P Chem texts by Berry and Rice and Ross, Wynn, and Levine.

(b) At risk of sounding whiny, it really, really defeats the purpose of the rewrite to not maintain clearly the distinction between KT, a rigorous result of RHF theory, from generalizations of KT that apply in the context of KS-FDT. I am not familiar with this work, but I would be surprised if a version of KT that applies to KS-DFT has been rigorously derived at the same level at KT is. Your claim that KT-HF is inexact is false; it is exact, albeit within the context of finite-basis HF. I would be extremely surprised if KT-DFT is exact regardless of basis set or choice of functional. Also I think also that you might have added enough information to warrant making J's theorem its own article stub. AcidFlask (talk) 05:04, 23 April 2009 (UTC)[reply]

I have deleted the statement about the exactness of KT-DFT versus the inexactness of KT-HF, since this is obviously false. KT-HF is exact within the framework of HF - this is the whole point of why Koopmans' theorem is a theorem, not a rule or approximation. The proof that you have posted of KT-DFT buries some niggling details of what exactly it means to calculate orbital energies for a fractional number of electrons - in fact, if you take Janak's theorem to be true (which it surely must be for exact KS-DFT), then it is also clear to me that ${\displaystyle \epsilon _{\mathrm {HOMO} }(N+\delta )=\epsilon _{\mathrm {LUMO} }(N-\delta )=\epsilon _{\mathrm {LUMO} }(N)}$ and so I don't understand what additional physical value the proof you've posted is trying to demonstrate.AcidFlask (talk) 05:21, 23 April 2009 (UTC)[reply]

The more I stare at the proof you've added, the less I'm sure that I believe it. The fact that Janak's theorem is a statement about the partial derivative means that you can't just integrate over it to get the energy difference - integrating over other changing quantities are needed. One term that is definitely missing is the one that involves Fukui functions, which correspond to relaxation effects. I've taken down this entire section, which is reposted (with minor corrections) below. I've added back an important reference that went missing. Do you have specific pages of Parr and Yang that address this? AcidFlask (talk) 05:39, 23 April 2009 (UTC)[reply]

One final clarification - Janak's theorem is a result that is valid in DFT, whereas KT is a result valid in HF. The proof of KT involves an orbital formulation that could be applied to another orbital-based method, such as Kohn-Sham DFT. Janak's theorem clearly applies to KS-DFT, but is more general than orbital-based DFT methods. This should be made clear in any more detailed presentation of Koopmans' theorem analogs in DFT and Janak's theorem.AcidFlask (talk) 05:44, 23 April 2009 (UTC)[reply]

By definition, the first vertical ionization energy of an N-electron system is E(N − 1) − E(N) where the geometry is defined by the N-electron state. This energy difference can be expressed in terms of the HOMO energy by writing it as an integral over the occupation number of the HOMO, denoted n_HOMO, which is zero for the N − 1 electron state and one for the N electron state:^[1]

${\displaystyle {\begin{aligned}E(N)-E(N-1)&=\int _{0}^{1}\mathrm {d} n_{\mathrm {HOMO} }\ {\frac {\partial E}{\partial n_{\mathrm {HOMO} }}},\\&=\int _{0}^{1}\mathrm {d} n_{\mathrm {HOMO} }\ \epsilon _{\mathrm {HOMO} }(n_{\mathrm {HOMO} }).\end{aligned}}}$

For the electron affinity, an integral over the occupation of the LUMO of the N-electron state is constructed as

${\displaystyle {\begin{aligned}E(N+1)-E(N)&=\int _{0}^{1}\mathrm {d} n_{\mathrm {LUMO} }\ {\frac {\partial E}{\partial n_{\mathrm {LUMO} }}},\\&=\int _{0}^{1}\mathrm {d} n_{\mathrm {LUMO} }\ \epsilon _{\mathrm {LUMO} }(n_{\mathrm {LUMO} }).\end{aligned}}}$

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.^{[2] [3]} 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 < n_HOMO <1, and the LUMO energy is constant for 0 < n_LUMO <1, so that^[2][4]

${\displaystyle {\begin{aligned}\mathrm {IE} &=-(E(N)-E(N-1))&=-\epsilon _{\mathrm {HOMO} }(N+\delta ),\\\mathrm {EA} &=-(E(N+1)-E(N))&=-\epsilon _{\mathrm {LUMO} }(N+\delta ),\end{aligned}}}$

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.^[5]

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,^[6] as shown in the adjacent figure for a carbon atom. In this example, and quite generally,^[6] 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)[reply]

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)

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)[reply]

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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)[reply]

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)[reply]