In a series of papers in the early 1980s, Michael Springborg explored an interpretation of the Wigner phase space function as an electron density in a six-dimensional $(q,p)$ phase space. He applied it with some success to several simple compounds.

Is the Springborg 6D phase model model used in modern molecular orbital modeling? If not, are there specific weaknesses that make it unsuitable for current computational approaches?

ADDENDUM 2012-05-14:

In a world dominated by Hilbert spaces, the orbitals of chemistry are interesting because they are one of the few places in physics where a 3D spatial image of the underlying quantum situation is consistently retained. (My recollection is that Schrodinger liked such low-dimensionality approaches, since they emphasized waves that could be visualized.)

Alas, the problem of course is that you need both position $q$ and momentum $p$ to get an accurate picture of the overall electron state. Using a space-only 3D $q$ representation forces the analysis of molecules to flip back and forth at rather arbitrary thresholds between localized (position space $q$) and delocalized (momentum space $p$) views of the electrons in the molecules or metal crystals in question.

That bothers me, both from a physics perspective and from a computational integrity perspective.

The problem is that there exists a very deep and profound physics symmetry between $q$ and $p$ in terms of how effects such as Pauli exclusion applies to fermions. I find that simple symmetry nothing short of amazing, because to me there's just nothing intuitive about the idea that half-unit spin would result in such extraordinarily similar behaviors in these quite different spaces. At some level it seems to be one of those cases of "that's just the way it works." So, it seems pretty reasonable that accurate modeling of orbitals of many diverse sizes likely requires this symmetry to be captured accurately.

If that's true, this older work by Springborg (he still heads a research group BTW) strikes me as a possible opportunity to for taming some models, making them smoother and more sane over a much broader range of molecular sizes. That's because Springborg's 6D electron density functions -- if they work OK in broader contexts -- might help eliminate any need for abrupt switches between "mostly position" or $q$ 3D views (classic "electron clouds," whatever those really mean) and "mostly momentum" or $p$ 3D views.

A good place to test whether 6D $(q,p)$ electron distributions might provide higher computational stability would be in modeling a range of lengths for a long-chain carbon polymer such as polyacetylene. (I always pick polyacetylene for its simplicity, but there are other good choices.) Springborg appears not to have explored that domain, since as best I can tell his papers addressed mostly simpler bonds. (I think; I don't have them.) The instabilities of approximate density functional theories may (or may not) be related to $(q,p)$ transition instabilities. That's not a question I can or should try to address, as there are folks out there who know that domain orders of magnitude better than I ever will.

ADDENDUM 2012-05-16

I mentioned "algorithmic opportunity" and wondered what I meant myself by that, since as @AcidFlask reminds the $q$ and $p$ basis sets are fully interchangeable in terms of representing any stationary wavefunction.

The issue is (again, I think) one of accumulating uncertainty. Picture doing some kind of calculation in the $q$ representation. For a highly localized electron, you should get a pretty solid answer, but for something like a delocalized electron in a conductive polymer, a higher level of error would be likely (not a necessity).

Next, Fourier transform the result into $p$, and do the next iteration of whatever it is you are trying to calculate. There the nature of the errors should change, but in the case of a delocalized electron, it's likely to work out a bit more accurately.

You reverse transform that and repeat, with the objective of converging to a solution that looks pretty stable in both spaces. That, I would argue, is the "algorithmic opportunity" part of the 6D argument: that is, that by incorporating both views, you can come to an overall solution that diverges less from reality when applied to a broad range of molecular orbital sizes.

Are there problems with that analysis? Oh my yes. You'd have to be nuts to do complete Fourier transforms twice per full iteration of your convergence, and even then, the devil will be in the details about whether it even works.

But the operative word was "opportunity," that is, such an analysis hints that there may be away to use the dual perspective to create more stable approximation methods. To really work the algorithm would necessarily "live" a lot closer to a continuous 6D representation with constraints than a per-iteration transform back and forth. Can that be done? I don't know; as I said, folks like AcidFlux know this area orders of magnitude better than I ever will.

And finally, for any non-mathematical readers (AcidFlux, shoo, go away now!) who may have wandered in, the section below provides a simple, easy-to-imagine visual analogy to explain why location and momentum both play a major role at the level of molecules.

Imagine a very flexible membrane with many embedded holes (but the holes don't stretch). Label one side of the membrane "ordinary space" (abbreviated $q$, don't ask why), and the other side "momentum space" (labeled $p$, and I told you not to ask). Next, put water balloons with fixed quantities of water in them through each hole, glued to the holes.

The $q$ side of each balloon now represents the "real space" representation of the water balloon. If you squeeze that part very tightly, it will shrink down almost to a point, so it looks more like a particle than a balloon. You think you have "captured" that balloon n a small space, but beware! On the other side, the $p$ side of the membrane, the balloon has gotten comparatively huge! So your point-like "capture" of the balloon on the $q$ side was more illusion than reality, since in the broader view that includes both sides, all you really did is shift a lot of the balloon from the $q$ world into the hidden $p$ side.

Note also that you don't really need to see both sides at once, because each side of the balloon fully determines the size of the other side. From the amount of water on one side you can always calculate exactly how much water is in the balloon on the other, hidden side. And the opposite is also true! That is, if you know how big each balloon is on the $p$ side, you also automatically know how much of the balloon is on the $q$ side.

That kind of equivalence between two views that may appear at first to be quite different is an example a "basis set." A basis set has enough "pieces" in its tool kit to allow you to represent something completely, in the sense that you have all of the information you will ever need to know about the system.

Alas, complete is not always the same as convenient. For example, if you try to stack together the balloons as they appear on the $q$ side, you will quickly find that the hidden $p$ side of the balloons can make it very difficult. The balloons just won't pack as easily together as their forms on the $q$ side seem to suggest they would. The problem, of course, is that you have those huge $p$ sides of the balloons interacting in a very different way over in the $p$ or "momentum space" side of the membrane.

It's not that any information is missing! After all, since the $q$ side provides a complete basis set for describing the balloons, you know that when a balloon is small there that it must necessarily be large in $p$. The trick, then, is make sure that you don't have to recalculate the size on the balloons on the other side every single time you try to do something with them on the $q$ side. That's inefficient at best, and could easily turn into a significant source of errors.

So, it's a bit simpler and certainly a lot less calculation if you preserve a simultaneous image of both sides of the membrane, and change them only when you have to, e.g. after squeezing some of the balloons on either (or both) the $q$ and $p$ sides. Doing that is inherently redundant, sure, since you only need to have one side to recreate the other.

However, if you've ever done some serious algorithm design, you know that "caching" mostly persistent results can be a great way to reduce calculation times, sometimes hugely. More subtly, it can also help stabilize the problem if it turns out that you have complementary error modes across your different "views" (basis sets) of the final result.

So, put that all together, and that's what I'm trying to suggest may also be true for the physics of electrons, which similarly like to slosh around (as in Planck's Theory of Slosh," and yes of course I just now made that up!) and spend some of their time point-like in one space (called $q$ or position space; ordinary, real space), or alternatively in another more abstract space (called $p$ or momentum space). Interestingly, just as with the balloons, they bump into and push each each other in almost exactly the same way in both of these two spaces. Particles that do that are called fermions.

So why are fermions so turf-protective in both of these quite different spaces? I really don't think anyone really knows, at least not with the kind of depth that is as satisfying as when chemistry started coming together and making deep sense. It's just one of those little mysteries of how the universe works, one (there are several actually) that's still sitting there thumbing its nose at us and saying "ha ha, gotcha!"

And that's cool! What's a universe without a few mysteries still hanging around, after all?


1 Answer 1


Alas, the problem of course is that you need both position q and momentum p to get an accurate picture of the overall electron state.

This is not true in quantum mechanics; it is sufficient to characterize the wavefunction $\left\langle x |\psi\right\rangle$ in position space or $\left\langle k|\psi\right\rangle$ in momentum space. Which is chosen is entirely a matter of convenience: most molecules have the most compact description in position space, while most crystals have compact descriptions in momentum space. It can be proven that position alone, or momentum alone, form a complete basis separately. In fact, position-momentum uncertainly says that an electron state cannot be characterized completely in both position and momentum simultaneously. (This, by the way, has nothing to do with fermionic symmetries and Pauli exclusion, which is another quantum-mechanical axiom unto itself. The same would have held for looking at bosonic particles also.)

It is, however, true that a probability density in phase space $(q,p)$ is needed to completely characterize a classical system. Wigner's function was introduced to address this quantum-classical correspondence and is regarded as the quantum analogue of a classical probability distribution in phase space. However, it is also well known that it can take on negative values and therefore cannot be interpreted in the usual way as a probability distribution. This affects all methods using the Wigner function: in the Springborg paper in the OP, there are regions of negative "probability density" in the lithium hydride molecule being analyzed.

Suffice to say, the question of how to deal with negative probabilities is an extremely controversial subject.

Having said that, the Wigner function has found uses in ab initio molecular dynamics when a classical phase space density is required to sample dynamical trajectories from the potential energy surface generated by quantum electronic states. The regions of negative probability are usually neglected in such procedures with seemingly no loss of accuracy (to my knowledge). I am not aware of other common usages in electronic structure theory. (The other problem brought up in the OP about the inaccuracies in DFT are conventionally attributed to incorrect modeling of electron correlation effects, which is a matter of incorrect treatments of fermion symmetries and its consequences when projected down into coordinate or momentum space.)

  • $\begingroup$ Thanks! A very readable and nicely pointed answer. And yes, I do know that either $q$ or $p$ provides a complete basis set, bridged by the Fourier transform. I was thinking of how ratty each gets in certain domains, and that molecules span both extremes. But me implying both are needed is just wrong. Hmm. I had no idea negative probabilities were involved; that's an interesting surprise. $\endgroup$ May 15, 2012 at 6:36
  • $\begingroup$ @AcidFlux: What I said was badly phrased, but my intent was this: Because Pauli exclusion applies in both $q$ and $p$, you need all six dimensions to model accurately how sets of fermions will reach an energy minimum. I really did mean it only for fermions, for just that reason. I am essentially viewing this a a topological algorithm in a 6D space, one in which multiple fermions interact until they reach a good minimum. The resulting forms will of course be fully equivalent whether represented in $q$ or $p$, but I am postulating (right or wrong) that 6D has interesting algorithmic potential. $\endgroup$ May 15, 2012 at 14:53

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