Talk:Born-Oppenheimer approximation

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The Born-Oppenheimer approximation does not decouple motions.

THERE IS NO MOTION IN QUANTUM MECHANICS.

Motion is a concept that only makes sense in classical mechanics.

Motion = position + motion vector. In quantum chemistry, both are not accessible simultaneously. Thus motion cannot be defined at the scale of electrons (quantic scale).

It is difficult to use another definition anyway. It is easier to understand when using the word "motion", but this just does not makes any sense...

Quantum chemistry is very abstract. I understand it is difficult to explain the Born-Oppenheimer approximation without using the word "motion". I am not sure to be very rigorous (I am a PhD student in biochemistry...), but I would rather say that at a given time the probability of presence of the nucleus of an atom at the center of mass of its electronic distribution is very close to 1. The Born-Oppenheimer approximation says : it IS 1. It is not a very crude approximation unless you try to model nuclear reactions.

Can someone modify the article? I am new here and I don't want to risk messing everything up...

I have just changed the article in order to answer your question. I agree this is a hand-waving way to describe the Born-Oppenheimer approximation but what you say is simply non sense. I think you have not understood what is the Born-Oppenheimer approximation. Try to read carefully this article and the associated ones like adiabatic process (quantum mechanics) and vibronic coupling. If you still don't understand then I am ready to be more explicit but that's a big job to write down the whole underlying set of equations. Regards, vb.

To say that "there is no motion in quantum mechanics" is misleading. There is motion in the form of the flux of the probability density, which can be calculated using the time-dependent Schroedinger Equation. The only difference between this and classical mechanics is that in quantum mechanics, the motion vector is actual a vector field rather than a single vector. Ed Sanville 16:02, 12 December 2006 (UTC)

Contents 1 Vibrons and phonons 2 A peer review 3 Merge with molecular Hamiltonian 4 Revival of old page 5 Nonadiabtic operator 6 Small linguistic question

[edit] Vibrons and phonons

Is the difference between vibron and phonon important? Can vibrons be created and anhililated? Is it possible to produce glauber states in a two atom molecule, so that the molecules do a nearly classical oscilation? --Arnero 11:49, 10 October 2005 (UTC)

I haven't seen a concrete definition of a "vibron" in the literature, but typically, it is used to mean a quantized unit of vibrational energy. I typically associated vibrons with a single molecule, wheras for phonons, perhaps quanta of vibrations in a solid. And depending on the basis set, yes, you can create and annihilate vibrons. I don't know about producing Glauber states in a 2 atom molecule. Might be more resonable to consider a 2 state atom coupled to an oscillatory field... --HappyCamper 00:55, 9 September 2006 (UTC)

[edit] A peer review

This article is in urgent need of work. Here are some suggestions (starting at the 'hand-waving' bit): Get rid of the "To get an idea.." bit, the numbers don't say anything in themselves (no temperature given!) and the relevant point (the relative magnitude of the velocities) has more or less already been given by pointing out that the nuclei are 2000 times heavier. Referring to the Fermi velocity in the same sentence as a 'typical electron velocity' is misleading since it's actually a quite untypical velocity, as it's the lowest one possible. The other parenthesis is even more misleading. Making reference to the speed of sound implies that you're talking about the translational velocity of the atoms/molecules (the speed relative some stationary point) when the relevant quantity is the vibrational velocity (the atoms speed relative eachother in the molecule).

"allowing the system to remain in its ground state" - plain old wrong. There is no requirement at all for the system to be in a ground state as a whole or in part. It's pretty hard to explain what the Franck-Condon principle is all about otherwise. The point is not that the electrons remain in the ground state, but that they remain in the same state. Or to be even more precise, they remain at the same energy level. (obviously the overall state can be considered different if the nuclei positions and electron motion change). Hence the name "adiabatic approximation": A change in nuclear velocity (vibrational transition) does not transfer any kinetic energy to the electrons. Which in the 'hand-waving' rationale makes sense; how could they transfer kinetic energy if they're 'standing still' as far as the electrons are concerned?

Ok, so on to the next paragraph. First sentence: "The motion of the electrons can therefore be considered decoupled from the motion of the nuclei" says 'therefore', but what it says does not follow from from the previous statement (had it been correct). It is merely a restatement of the same underlying idea: We assume the electrons recieve no energy from the nuclear motion. Next up: "which leads to the elimination of several terms[..]". This is far more hand-waving than the physical justification!

And for no good reason at all I might add, because it's already said which terms are eliminated, in principle. And the main point is not that we 'eliminated some terms', but that we eliminated (or perhaps 'assumed to be zero') all the cross-terms between the electronic and nuclear solutions. Alternately, you could say we eliminated all the nondiagonal matrix elements in a basis of the two solutions. Recalling that nondiagonal elements are the transition probabilities, you come back to the original assumption, since a transition would mean a change in energy. (non-adiabaticity)

Ok, then we get to a bad part. And the sudden change in style to an informal third-person narrative certainly does not help! Anyway, you do not solve the Schrödinger equation for the electrons only. You solve the entire Schrödinger equation. Nor do you, or are required to, treat the nuclei classically. No terms are neglected (apart from the approximation already made). The writer of this did not understand it properly. And must've had a lapse in his knowledge of basic PDE-solving; Separation of variables may be an appropriate reference here, since we're doing nothing fancier than using that method. The whole point of the approximation was to put the Schrödinger equation into a seperable form to enable us to use that method. In it's simplest form, all the BO says is the following: Assume the energy of the system can be written: E = Ee + En. Or equivalently, that the wavefunction can be written: U = Ue * Un Or equivalently, that the Hamiltionian is: H = He + Hn (Where He and Hn are *seperated*) (where 'e' is the electronic and 'n' the nuclear parts)

Again, let me emphasise that nothing further is neglected. What has neglected, however, is the mentioning of the name cross terms: Vibronic coupling. It's mentioned right at the end of the paragraph and doesn't really follow any structure. It reads like an afterthought. Obviously that needs to be worked into the description of the interaction we're neglecting.

What's a "routine foundation stone"? Here's a better sentence: "The Born-Oppenheimer approximation is routinely used as a starting point in the physical study[..]". The next one isn't much better, but I'd suggest "The theory of most quantum-chemical methods make use of it.", in any case you shouldn't say 'computational chemistry' since that also refers to all non-quantum methods, none of which have any use for BO.

The final "beyond" bit needs more fleshing out but is okay at least (although I must admit bias there since I'm a fan of Nick (who isn't?) :)). The last paragraph repeats the same information given before (the name of the coupling) but a bit more elaborately. This too needs to be moved to some more structured place.

Which brings me to my the final criticisms. I saved the worst for last.

The first absolutely critical flaw here is that nowhere is it stated in clear terms under which conditions the approximation is valid. "Avoided crossing" and "conical intersection" indeed. I've a hard time believing there are very many people who know what those two terms mean, yet don't know the Born-Oppenheimer appoximation is. A simple example would be good enough, e.g. "The approximation becomes invalid as the nuclear velocity increases. For isntance, in highly excited vibrational states."

And the worst, which extends beyond this article, is the descriptions of the electronic/molecular/nuclear hamiltonians. It's quite wrong in places and at best extremely misleading.

Let's straighten this out. This is the exact non-relativistic, time-independent molecular Hamiltonian. The whole molecule. No Born-Oppenheimer involved: (Using the variable names I've seen most commonly, in Szabo & Ostlund among others. Note that the Molecular Hamiltonian article itself uses about as inconsistent notation as it possibly could!)

• Factors 1/2 must be erased for restricted (i>j) sums! --P.wormer 10:46, 3 January 2007 (UTC)--P.wormer 15:03, 12 February 2007 (UTC)

While this can be written as a sum of operators, the third one depends on the electron-nuclei distance and is thus causes the inseparability. (This is indeed stated in that article, although it's obfuscated by the notation and inconsistency). So what does the BO approximation do here? What it does is that it pretends it's seperable anyway. So separate into nuclear and electronic and treat the nuclei as fixed when solving the electronic function, and the potential of the electrons as being fixed when solving the nuclear one. Thus they're both moving in the potential fields of the other. Now this is stated repeatedly all over the place here and in the articles, and it's stupid. First, the article Electronic molecular Hamiltonian is misnamed, because it primarily describes the Molecular Hamiltonian proper (the one I just gave). Wheras the article Molecular Hamiltonian is actually about the nuclear Hamiltonian! Then there's references to the "Clamped Hamiltonian". Which is just the same thing as the electronic one.

Then there are these frequent references to 'replacing with potential energy surfaces'. This is not false, but it's confusing. Because you are not actually replacing anything with anything. By assuming adiabaticity you are assuming that the interaction takes that form. (Another case of restating something already implied.) 'Potential', by the way, is the normal term here for 'potential energy surface'. The article Potential energy surface is wholly redundant anyway since there's an article on Potentials already, and there is no difference between the two. (Well, strictly speaking the latter is more general. Energy is scalar so PESes can only be scalar potentials.)

You strongly get the impression someone just picked up this phrase in his textbook and kept sticking it in everywhere (at least 4 articles) without truly understanding it. I mean, come on!! "A potential energy surface is generally used within the adiabatic or Born-Oppenheimer approximation[..]"?!?!? If you have potential energy depending on a coordinate, then you have a potential-energy surface. I'd love to see a justification for why the potentials in the separated B.O. Hamiltionians constitute more "generally used" potentials than the ones present in just about every other Hamiltonian.

But the main point is that the article "Molecular Hamiltonian" (actually the nuclear it seems) should be deleted and merged into this one. Likewise with all references to the electronic one. (And the article names should be accurate!) There is no justification for having seperate articles on those matters, because the separation into electronic and nuclear Hamiltonians simply does not exist outside of the Born-Oppenheimer approximation. (Well strictly speaking you can always write H = He + Hn, but this gets you nowhere if the two terms depend on the same variables.) And there really isn't enough to say about these two Hamiltonians that warrants their own article outside of this. It's just the exact molecular Hamiltonian split into two parts, after all.

Okay, so that's it. I guess my critique turned out to be longer than the article itself. :) But if it leads to the article becoming that much better, then maybe someone will be able to learn something from it one day.

--130.237.179.166 11:51, 9 September 2006 (UTC)

Thanks for the feedback, 130 - check back in a little bit to see if things are better. Let's see what we can do here. --HappyCamper 12:55, 9 September 2006 (UTC)

I disagree with the harsh criticism of the term "potential energy surface." Yes, the name is perhaps redundant, but the term is used constantly in the literature, almost always with reference to a multidimensional chemical reaction coordinate. It's just a matter of a conventionally used term... I can see how it might perplex a non physical chemist/chemical physicist, but hey, that's the way it is. Ed Sanville 16:11, 12 December 2006 (UTC)

[edit] Merge with molecular Hamiltonian

I think this should be merged with Molecular Hamiltonian, there is a big overlap of the two. The Born-Oppenheimer approximation is just a adiabatic variant of the Molecular Hamiltonian so everything here should probably go under that article.

The problem is to fully describe the Born-Oppenheimer approximation involves showing how to solve the approximate Schrödinger equation by spiltting the wavefunctions. That part is not really a part of the molecular Hamiltonian, but som litterature describes this under the subject Born-Oppenheimer approximation and some under molecular Hamiltonian. I personally prefer molecular Hamiltonian but there have been some talk on this page suggesting a merge under the subject Born-Oppenheimer approximation. Martin Hedegaard 13:55, 27 September 2006 (UTC)

Would it be easier to explain what the Born Oppenheimer approximation is on Molecular Hamiltonian? If so, I'd be in favour of the merge as well. The stuff about splitting the Wavefunctions can be put back into this article at a later point. Right now, what is more important is that we have something that looks solid from an academic standpoint. What we have on this page doesn't meet the standard I think. --HappyCamper 15:19, 27 September 2006 (UTC)

The only thing missing in Molecular Hamiltonian is a short description of how to solve the molecular Schrödinger equation that describes that to the first approximation the molecular wavefunction should be , and the expansion in its basis. If the decision is to merge with Molecular Hamiltonian, I will just start adding whats currently missing on Molecular Hamiltonian so this page can be closed. The quality on this page is simply to low as it is now, for a relativly advandced subject as this.-- Martin Hedegaard 09:01, 28 September 2006 (UTC)

Sure, let's merge this as well. After that, we can fix some incoming links. We can split information about this again later. --HappyCamper 15:16, 1 October 2006 (UTC)

Merged the content into Molecular Hamiltonian, but havent fixed incomming links yet, we can always split the articles again. --Martin Hedegaard 17:59, 1 October 2006 (UTC)

[edit] Revival of old page

I finished writing the new version of the BO lemma. It is now up to the cybernauts, out there in cyberspace, to shoot holes in it or to polish it, as the case may be. I am eagerly awaiting the stern comments of our Swedish peer 130.237.179.166. --P.wormer 16:02, 11 December 2006 (UTC)

Looks good, but a couple of references would be nice to Martin Hedegaard 13:14, 12 December 2006 (UTC)

[edit] Nonadiabtic operator

Where would be a good place for this? --HappyCamper 03:37, 5 January 2007 (UTC)

[edit] Small linguistic question

I wrote an MO, because I say "an am oh". Somebody changed it to a MO (a molecular orbital). Who is right? --P.wormer 13:47, 21 February 2007 (UTC)

Depends on the style guide one uses. --HappyCamper 16:24, 22 February 2007 (UTC)

And what is the accepted Wikipedia style guide? Chicago style manual (p. 464) says: "an am oh", or "a moh", depending how one pronounces "MO". Does Wikipedia have a standard?--P.wormer 09:15, 23 February 2007 (UTC)

Generally, Wikipedians choose the style which "best" presents the information in its context. Here, I don't think it matters much, so I suppose you can pick the one you like. I can guarantee you that in a few months, another IP address will come by and change it to the other one. So be mindful: little things like this have caused countless numbers of these! The style guide on Wikipedia is Wikipedia:Manual of Style, but it is explicit in stating that it is fluid. There are cases on Wikipedia where even the IUPAC standard is not followed, in deference to the prevalence of other styles present. HTH. --HappyCamper 16:50, 23 February 2007 (UTC)

Hi HC, don't be afraid, I won't loose a minute of sleep about the indefinite article. Because somebody was finicky enough to change "an MO" to "a MO", I (as relative newbie) wanted to know how the Wikipedians stood to such edits. Second, because I suspect that the change was due to lack of knowledge, I wanted to make clear that the rule is more complicated than just consonants versus vowels. --P.wormer 14:54, 24 February 2007 (UTC)

I think we are thinking on the same wavelength :-) --HappyCamper 19:46, 24 February 2007 (UTC)