Talk:Møller-Plesset perturbation theory

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Just added a few lines to begin with. Hope more will come in time. - Karol Langner

This page is very bad. All the equations peresented here are just a basic Rayleigh-Schroedinger perturbation theory expressions. There is NOTHING specific about Moller-Plesset. No definition of perturbation, no solution for MP2 etc ...

One can easily remove all MP2, MP3, replace words Moller-Plesset with Rayleigh-Schroedinger in title in text and the article would be perfectly fine. —Preceding unsigned comment added by 170.140.189.10 (talk • contribs)

Please sign all comments on talk page by adding ~~~~. I agree that this article needs improvement, but you are being a little unfair. It defines the MP2 as the difference between the Hartree-Fock operator and the exact Hamiltonian. It explains that the first order perturbation only converts the sum of orbital energies to the Hartree-Fock energy so correlation energy only starts to be introduced at second order. It is likely to be chemists who come to this page and in general they do not know about Rayleigh-Schroedinger so there is need to describe it. The equations for the MP2. MP3 and MP4 energies are rather complex and I think the editors of this page wanted to keep things as simple as possible. However I do think the MP2 energy should be added and a link given a source for the MP3 and MP4 energies. Why not try to improve it yourself? --Bduke 22:37, 28 June 2007 (UTC)

[edit] MP2 energy and related matters

You have added some great material to the MP article. Well done. I do however have a slight disagreement. Your development is not wrong, but it is not what seems to me standard. Most texts (e.g. Introduction to Computational Chemistry, Frank Jensen) define the perturbation as simply H - F, and this gives MP0 as the sum of orbital energies and MP1 adds the correction to give the HF energy. You have defined the perturbation such that MP0 is is the HF energy and the first order correction zero. We also need to be clear about the difference between the energy to n'th order and the n'th order correction. Jensen for example gives:-

MP0 = E(MP0) = sum of orbital energies

MP1 = MP0 + E(MP1) = E(HF)

Which way should we go with this? --Bduke 23:37, 29 June 2007 (UTC)

Hi Brian, I am aware of the (slight) difference in conventions in defining the MP perturbation. As you may guess, I have a (small) preference for the convention that I have chosen. This is because I like the first-order energy to be zero. But indeed, you pay the price with a slightly more complicated expression for the perturbation. I had in mind to add later today also the explicit zeroth-order energy, which, as you rightly remarked, is the HF energy. If you feel that the "Jensen convention" is more didactic, please feel free to change it (or perhaps both can be mentioned?).

With regard to nth-order correction versus n-th-order energy: I have always been sloppy in that usage, but you're right, one must carefully distinguish the two. --P.wormer 05:59, 30 June 2007 (UTC)

Can you name a common text that does use your convention? My impression is that all the common books use the other convention. I prefer the "Jensen" convention because the zero order energy is the eigenvalues of the Fock operators. I see no virtue in have the first order correction zero. However I am distracted by an error in some CASSCF calculations so my mind is on that method. I'll return to MP sometime. --Bduke 07:10, 30 June 2007 (UTC)

Well, "my" convention is the first-quantized equivalent of the second-quantized approaches where normal products with respect to the Fermi vacuum are introduced. See, for instance, Paldus and Cizek, AQC (1976). In the meantime I adapted the text somewhat, see if you still don't like it. I admit that the perturbation is slightly awkward, but I feel that the elegant zeroth- and first-order contributions make up for it. But again, in my view differences are minor as you can see in the note I added to the text. --P.wormer 08:48, 30 June 2007 (UTC)

Hi Paul, I have a few ideas about this but am rather tied up today to develop them at length. In the meantime, do you mind if we copy this section from your talk page and the one from mine to the MP article talk page, to encourage others to join in? --Bduke 23:15, 30 June 2007 (UTC)

Hi Brian, please go ahead and copy our MP discussion to the MP talk page (it occurred to me too, that this would be useful). I reread the original MP paper and noticed that they have the same partitioning as I have (in a difficult notation). Hence the MP theorem.--P.wormer 07:01, 1 July 2007 (UTC)

The above discussion has been copied from my talk page and User talk:P.wormer and sorted by date and time. We both welcome further comment. --Bduke 08:06, 1 July 2007 (UTC)

[edit] Møller-Plesset theorem

The article states that "This result is the Møller-Plesset theorem: correlation does not contribute in first-order to the exact electronic energy. In the section on the alternative formulation, it states "Obviously, the Møller-Plesset theorem does not hold". In both formulations, correlation only enters at second order, so why does the theorem not hold for the alternative formulation? Paul, can you clarify this? I have to say that I have never heard of this theorem and I suspect that most chemists have not. I can find no mention of it in chemistry texts. I have found more evidence that the alternative formulation is very common in chemistry texts. Indeed I can find no chemistry texts that use the original formation. I am going to edit the article to make this chemistry usage clearer and give references, when I have finally sorted out the references. While MP originated in Physics, it has been hijacked by chemistry. There are thousands of chemistry articles that use MP2. Many chemists will want to start learning about MP2 from this article so I think we should make it more user-friendly to chemists. I'll have a go without displacing the original formulation. --Bduke 08:32, 6 July 2007 (UTC)

Hi Brian, what you could do is give the "chemistry formulation" first and then as a note the "physics formulation". I heard of the Moller-Plesset theorem at Lowdin's summer school in 1968, and I think I have heard it called that way more often since then, but off-hand I couldn't give a source. Since the difference between the two partitionings is close to trivial you seem to miss the point: E⁽¹⁾ = 0 (physics partitioning) and E⁽¹⁾ ≠ 0 (chemistry partitioning). No doubt you are aware of the very large body of physics papers (Dyson, Brueckner, Wick, Goldstone, Hugenholtz, Abrikosov, van Hove, etc.) on many-body theory? As far as I am aware they all use the second-quantized hole-particle formalism, which is equivalent to what I wrote. (I could write the article in 2nd quantization, too, but then surely most chemists will be lost).--P.wormer 12:40, 6 July 2007 (UTC)

Hi Paul, I'll make some changes today, all being well, and see what you think. I'm inclined to leave the original formulation first as, well, it is the original, but then stress the other formulation is in wide use by chemists and give more references. I guess I heard about the Moller-Plesset theorem at Lowdin's Summer School (I forget when I went - early 70s I think) too, but I do not recall it. It is the definition that I am concerned about. Does it say that electron correlation enters only at second order (in which case both formulations fit it) or does it say E⁽¹⁾ = 0 (in which case the alternative chemists formulation does not)? The article appears to define it as the first, but draws the second conclusion. I hope we do not use second quantization here as chemists would indeed be confused. I know some of the papers you mention, but as a chemist, find them hard and not too useful. --Bduke 21:58, 6 July 2007 (UTC)

As far as I am concerned the article is OK now. --P.wormer 07:53, 7 July 2007 (UTC)

Thanks for much improving the wording. I do not want to belabour the point but I'm still concerned about the definition of the MP theorem. The article states it as "correlation does not contribute in first-order to the exact electronic energy". By the definition of correlation energy, correlation does not contribute in either formulation until 2nd order, not withstanding whether the first order correction is zero or not. Are we using "correlation" in a different sense, or should this definition be rewritten to something like "perturbation does not contribute in first-order to the electronic energy"? I would be happier if we could take the definition exactly from some source. --Bduke 08:09, 7 July 2007 (UTC)

In the "physics" partitioning the first-order perturbation correction is zero. However, since the perturbation is known as "correlation potential", I phrased it the way I did. It is a (small) advantage of the "physics" partitioning that Lowdin's definition of correlation (energy beyond HF) coincides with the MP-PT definition (energy beyond E_MP0 ≡ E_HF). In the "chemistry" partitioning one must define energy beyond E_MP1 to get Lowdin's definition. I will change "correlation" to "correlation potential" in the formulation of the MP theorem. (Note that the "correlation potential" is defined a few lines before this statement). If you still don't like it you may change it to perturbation (if you like that better), because from the context it is clear that the perturbation is the correlation potential.--P.wormer 10:20, 7 July 2007 (UTC). PS You and I together have as much knowledge and experience as any source. Would you indeed feel better if somebody else had put this in print before? Is a research article good enough? I may have written it up somewhere in the past, I would have to check.--P.wormer 10:30, 7 July 2007 (UTC)

That is indeed clearer. I would prefer a good source in case someone wades and in and tries to change everything. I guess we do know a lot between us, but we have to avoid original research. I have exhausted my sources at home. I'll try to spend some time in the library this week. Cheers, Bduke 10:59, 7 July 2007 (UTC)

Back to the "Moller-Plesset Theorem". I have never heard this applied to what I have called Brillouin's Theorem. See eg. Epstein's "The variation method...".Ggallup 16:18, 27 August 2007 (UTC)