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)

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)

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: $$ \hat H=-\sum_{i}{\frac{1}{2}\nabla_i^2}-\sum_{A}{\frac{1}{2M_A}\nabla_A^2}- \sum_{iA}{\frac{Z_A}{r_{iA}}} + \frac{1}{2}\sum_{i>j}{\frac{1}{r_{ij}}}+ \frac{1}{2}\sum_{A > B}{\frac{Z_A Z_B}{R_{AB}}} $$ (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)

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 $$\psi_a(x_e,X_n)=\phi(e_e,X_n)\cdot \chi (X_n)$$, 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)

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)

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

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 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)

Young students
The following text was entered at the top of this discussion page at 17:21, on 14 June 2007 by 132.166.20.146. It was moved to the bottom under the new heading "Young students" by P.wormer 07:18, 18 June 2007 (UTC) :

The present discussion and the article are to highly specialised... Before dicussing anything, first try to define it in a few simple words. Think to a young student that heard about BO approximation for the 1st time. Is 1st question is : What's BO approx ??? and not what are the precise conditions to apply it !

The 1st sentences must answer the 1st question : what it is ? Then, you can write at length highly specialized pages ... or discuss complex things like the discussion below ...


 * Because I am responsible for the major part of this article I like to react to the comment of this anonymous editor. Interestingly enough, I wrote this article with "the" young student in mind. I thought of graduate students who are beginning their research in molecular physics, quantum chemistry, or related fields. Apparently, the anon has different young students in mind, presumably college-level (freshmen, sophomores) chemistry and physics students. These students will hear a few words about the BO approximation in their curriculum and for most of them these few words will suffice. If for some reason they want more extended info, then Wikipedia is the place for them. Clearly, they will only search for the lemma Born-Oppenheimer approximation after they have heard of it, and have a notion what it is. Nobody wakes up one morning with the urge to know all about the BO approximation without ever having heard of it. Very coincidentally I heard that my mission was successful: a few weeks  ago a chemistry professor told me that he used this BO article (before it was changed by 132.166.20.146) as a basis for a graduate lecture. As a final remark: I find it increasingly difficult to keep my patience and good temper with these anonymous arrogant self-centered editors, who put their fingers in articles they admittedly don't understand and don't even bother to read  or make an effort to understand.--P.wormer 07:51, 18 June 2007 (UTC)

Explanation of reversal of 18 June 2007
Two anonymi changed the article. The second change borders on vandalism, so I won't say anything about it.

In general terms I reacted earlier to the change of 132.166.20.146. I will explain why I reverted the edit of this anon.

(S)he writes:


 * The electronic wavefunction depends upon the nuclear positions but not upon their velocities

It is self-evident that the electronic wavefunction does not depend on velocities of nuclei. Not even in an exact formalism. The whole point of departure is the q-representation, not  the p-representation. A "young student" may infer from this sentence that wavefunctions ought to depend on nuclear velocities.


 * The nuclear motion sees a smeared out potential from the speedy electrons.

The nuclear motion sees (?) a potential. A motion is influenced by a potential. A motion does not have eyes. What is a smeared out potential? Potentials can be averaged over something (time, coordinates of particles etc.) Who or what smears out a potential? What are speedy electrons? In quantum mechanics one never mentions the speed of electrons and rarely their velocity. It is usually the momentum that one considers. One can define velocity as momentum over mass. Having done that and having argued that the momentum of a nucleus is of the same order of magnitude as of an electron one can say that the velocity of an electron is large compared to the velocity of a nucleus. A "speedy electron" is children's language and much too sloppy and undefined to be used in a science text.

There is an unmatched remaining from careless editing.


 * Using the second assumption, the nuclear kinetic energy Tn [..] is reintroduced

Who uses the second assumption (the "smeared out" potential), the nuclear kinetic energy? Why would a smeared out potential require reintroduction of Tn? --P.wormer 10:25, 18 June 2007 (UTC)

Easy introduction?
I've heard this stated as the approximation that the inertia of electrons can be neglected in comparison to the atom to which they are bound. Could an explanation like that be added to the introduction? —Ben FrantzDale 05:38, 12 November 2007 (UTC)


 * OK, has been done.--Virginia fried chicken (talk) 13:55, 22 November 2007 (UTC)

This seems so useful that it could be moved closer to the top of the article. Could the second paragraph start out --JonRowlands (talk) 01:36, 26 October 2010 (UTC)

move to ndash
Can someone move the page from Born-Oppenheimer approximation to Born–Oppenheimer approximation, which currently redirects to Molecular Hamiltonian? 66.57.7.153 (talk) 23:39, 8 August 2009 (UTC)

The Benzene Example
The article introduction gives the example of a benzene molecule having 6 nuclei and 36 electrons. This is correct if only working with the carbon atoms/electrons. However, the full benzene molecule (C6H6) has 12 nuclei and 42 electrons. Correction?

64.132.94.34 (talk) 14:40, 30 May 2012 (UTC)

Introduction
In my opinion, the introduction part of this article needs some modification. Way too fast, it gets too specific. I mean what's the piont in discussing that using this ansatz benzene, which has 162 degrees of freedom (totally unsolvable), can be separated into an electronic part with 126 degrees of freedom and an ionic part with 36 degrees of freedom (still unsolvable).

A possibly better introduction would define the Born-Oppenheimer approximation most generally as a tool for the quantum mechanical description of molecules and solids and then give a one-sentence explanation of what it does. This could be followed by a very brief history of its development.

The basic principle (without formulae) should be explained in a separate section which is followed by a more mathematical treatment. Etaijoverlap (talk) 21:45, 6 August 2012 (UTC)

I agree. As I understand it the B-O approx. is now used to treat electronic, vibrational, and rotational energies as separate entieties which can be added. At normal room temprature electronic energies are frozen out, so in those circumstances the Hamiltonian is equal to the product of the rotational and vibrational terms. Describing the vibational and rotational as nuclear seems to me to be anachronistic. A B McDonald (talk) —Preceding undated comment added 21:01, 10 August 2012 (UTC)

History
Technically, the B-O approximation introduced in their paper consists of a "double" perturbation expansion in both electronic and nuclear coordinates. The adiabatic approximation uses a perturbation expansion in nuclear coordinates on top of the adiabatic solution to the electronic problem. While the adiabatic theory converges rapidly for well separated electronic states, the B-O perturbation actually converges slowly, as the fourth root of the ratio of the electronic and nuclear masses. The cause is the difference in the way the motion of the electrons near the nuclei are allowed to "follow" the nuclear motions. As the starting point in the adiabatic approximation, the adiabatic electronic states have the electrons already principally "following" the nuclei. But in the original B-O perturbation expansion, the electrons start out in a fixed reference configuration; they only "follow" the nuclei through a perturbation expansion. In a LCAO picture, the adiabatic approximation can use as basis functions atomic orbitals that a centered on the nuclei; in the B-O perturbation expansion, the electronic functions are expanded in orbitals fixed in space, not on the nuclei as they move. In order to concentrate electronic wave function at the nuclei for large nuclear distortions, a great number of terms would be needed in the perturbation expansion. For this reason, the original B-O double perturbation expansion is not used in practice. Nevertheless, the paper had importance theoretically in pointing out that separation of nuclear and electronic motions was justified, and hence their names are attached to the procedure more commonly used, the adiabatic approximation. pde 207.55.91.196 (talk) 19:36, 1 September 2012 (UTC)


 * Well, as I see it things are a bit different. The power-series expansion is not in any of the coordinates but in the ratio of the masses of the electron and the nuclei. The fourth root of the ratio of the electronic and nuclear masses is not the convergence behavior, its the expansion variable. The electronic hamiltonian is thus the zero-order solution of this expansion. The higher order terms of the expansion show that the energy eigenvalue of the electronic Schrödinger equation acts as a potential to the nuclei. Etaijoverlap (talk) 19:46, 20 October 2012 (UTC)

Equal momentum
According to the article "Classically this statement makes sense only if one assumes in addition that the momentum p of electrons and nuclei is of the same order of magnitude. In that case mnuc >> melec implies p2/(2mnuc) << p2/(2melec). Quantum mechanically it is not unreasonable to assume that the momenta of the electrons and nuclei in a molecule are comparable in magnitude"

But you can make an even more rigorous argument in classical mechanics than this hand-waving quasi-quantum argument. If you have a bound system of two particles, they should have equal momentum since by F=dp/dt and F_12=-F_21, the two particles acting on each other should change each other's momentum by equal amounts. It's easy to show explicitly that two particles orbiting around their center of mass have equal momentum.

The same exact energy/momentum relationships hold in classical and quantum mechanics, as well as the F=ma relationship in a sense (i[H,p]=grad(U)=-[H,[H,x]]) so it doesn't make sense to claim that we need to resort to quantum mechanics to show that the momenta are the same order of magnitude.Mpalenik (talk) 18:07, 1 February 2013 (UTC)

Born-Oppenheimer in Physics
Throughout this article it is repeatedly said how important is the Born-Oppenheimer in quantum chemistry. I think it would be necessary to address, as well, its fundamental rôle in solid-state physics, where it is used to decouple the dynamics of lattice and electrons in crystals.

Also, I fail to understand why this article is ranked as low importance in the physics portal.

176.86.179.162 (talk) 15:38, 16 September 2014 (UTC)

Cultural references
In the unaired pilot episode of The Big Bang Theory there is a scene where Dr. Sheldon Cooper is explaining some equations on a large whiteboard in his living room. He points to one area of the board and declares it to be a mathematical joke, "a spoof of the Born–Oppenheimer approximation." Koala Tea Of Mercy ( KTOM's Articulations &amp; Invigilations ) 06:03, 16 February 2016 (UTC)

Validity of BO
The current version does a good job explaining why BO breaks down if E_k(R)=E_{k'}(R), however, the discussion of when BO is a good approximation is badly wanting. First, "if all surfaces are well separated" is very qualitative and handwavy, particularly given that the matrix element is only argued to be "finite" (but could still be arbitrarily large!). Second, the usual justification for the BO is in terms of the very different masses of electronic and ionic degrees of freedom, and this is not even mentioned (except in introduction, without any further explanation). It would be nice to explain this argument in detail and to give some quantitative estimate that people could use to figure if they are looking at a case where BO might be breaking down...

In passing, I believe the statement that the diagonal elements of P_{A,\alpha} vanish is inaccurate: e.g. the 1/R_{AB} term in H_e would produce non-vanishing multiplicative terms. Only after the summation over A such terms would cancel out. 129.93.224.1 (talk) 20:04, 14 September 2017 (UTC)

Cleaning up some jargon
It is a pet peeve of mine that quantum physics uses a jargon that very subtly shifts the meaning of words and made it much more difficult for me to engange with the subject (e.g., most other approximation methods may "fail" or "become invalid", but BO almost never fails - it "breaks down", bringing it metaphorically close to Wave function collapse). In the spirit of Wikipedia not being a textbook and aiming to be understood by a common audience, I have started to unpack a bit of the jargon, while keeping the textbook jargon as an addition. -- Lpd-Lbr (talk) 14:26, 22 July 2019 (UTC)
 * Also, while working through it, I noticed the redundancy of a "Short Description", which should already be the role of the opening paragraph, which is also way too long. -- Lpd-Lbr (talk) 14:26, 22 July 2019 (UTC)

Grammar
Please consider re-writing a sentence, to make it easier to parse, or else please replace it.

AS-IS: “There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity (it is said to "break down"), but is then often used as a starting point for more refined methods.”

SUGGESTION 1: “There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity (it is said to "break down") but then often be used as a starting point for more refined methods.”

REASON: CASES is the antecedent of the pronoun WHICH, the plural main subject. MAKE is the main verb. [TO] is implied before LOSE and also before BE USED; each of those heads an infinitive phrase. VALIDITY is the object of LOSE. AS A STARTING POINT is an adverbial phrase modifying BE USED. FOR MORE REFINED METHODS is a prepositional phrase modifying BE USED.

Prior to this change, ASSUMPTION is the antecedent of WHICH. WHICH has to be the plural subject of MAKE, so it cannot also be the singular subject of IS.

Alternatively, if ASSUMPTION must be the antecedent, then WHICH becomes the singular subject of MAKES.

SUGGESTION 2: “There are cases where the assumption of separable motion no longer holds, which makes the approximation lose validity (it is said to "break down") but then often be used as a starting point for more refined methods.”

Please choose one of these suggestions, based on whether CASES or ASSUMPTION is the more appropriate antecedent for WHICH, or else re-write using multiple sentences.˜˜˜˜ — Preceding unsigned comment added by 73.166.1.117 (talk) 07:33, 28 January 2020 (UTC)