In various papers written in the 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 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 quantum models that retain the 3D spatial image of the system. (My best recollection is that Schrödinger liked such low-dimensionality approaches since they produced comprehensible waves.)

Alas, the problem 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 somewhat 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 physics and computational integrity perspectives.

The problem is that despite the striking dynamic differences of the $p$ and $q$ spaces, Pauli exclusion works the same way in both. There is nothing intuitive about why having a half-unit of spin should induce the same exclusion behavior in both representations. One can say, "that's just the way it works," but accurate modeling of orbitals of diverse sizes likely requires accurate capture of this persistent feature.

If that's true, this older work by Springborg (he still heads a research group, BTW) strikes me as a possible opportunity for taming some models, making them smoother and saner 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 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 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 his papers addressed mostly simpler bonds, as best I can tell. (I think; I don't have all of 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 folks out there know that domain orders of magnitude better than I ever will.

ADDENDUM 2012-05-16

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

The issue, I think, is one of accumulating uncertainty. Picture doing some calculation in the $q$ representation. The estimate should give low error rates for a highly localized electron, but higher ones for a delocalized electron in, say, a conductive polymer

Next, Fourier-transform the result into $p$, and do the next iteration of whatever you are trying to calculate. 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 to converge to a solution that looks pretty stable in both spaces. That, I would argue, is the "algorithmic opportunity" part of the 6D argument: 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 is always in the details about whether it even works.

But the operative word was "opportunity." Such an analysis hints that a dual perspective may lead to stabler approximation methods. The new algorithm would necessarily "live" 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 the details 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 important play a significant 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 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 shrinks until 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 did was shift a lot of the balloon from the $q$ world into the hidden $p$ side.

Note also that you don't 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 precisely how much water is in the balloon on the other, hidden side. And the opposite is also true! If you know the size of the balloon on the $p$ side, you also automatically know its size on the $q$ side.

That kind of equivalence between two views that appear quite different at first glance is an example of 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 needed to predict how the system evolves.

Alas, complete is not always the same as convenient. For example, if you try to stack the balloons together as they appear on the $q$ side, you quickly discover that the hidden $p$ side of the balloons can make the stacking difficult. The balloons won't pack together as readily as their forms on the $q$ side suggest. The problem, of course, is that you have those vast $p$ 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, it must necessarily be large in $p$. The trick, then, is to make sure that you don't have to recalculate the size of the balloons on the other side every time you try to do something with them on the $q$ side. That's inefficient at best and could quickly turn into a significant source of errors.

So, it's simpler and a lot less calculation to keep simultaneous, redundant images of both sides of the membrane and change them only when necessary, e.g., after squeezing some of the balloons on either (or both) the $q$ and $p$ sides. That is inherently redundant since you can always get away with changing and maintaining just one side.

However, if you've ever done much 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, that's what I'm suggesting may also be valid for the physics of electrons. Like those water balloons, electrons, as in Planck's "Theory of Slosh" (and yes, of course, I just now made that up!), like to spend some of their time point-like in one space (called $q$ or position space; ordinary, real space), or in another more abstract space (called $p$ or momentum space). Interestingly, just as with the balloons, they bump into and push each other 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 don't think anyone 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, "haha, gotcha!"

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


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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