Wait, How the Hell Does This Molecule Exist in 126 Dimensions?!

Caroline Delbert
Photo credit: Laguna Design - Getty Images

From Popular Mechanics

  • By using correlated wavefunctions broken into tiles, scientists have identified the electronic structure of benzene.
  • Benzene is a building block molecule found all over the galaxy, with a simple atomic structure, but a complex electronic one.
  • It's not really possible to solve a 126-dimension function without a serious shortcut.

Scientists have used computing power and clever modeling to solve a 200-year-old chemistry and math puzzle. For all these years, the question at play has been how to plot the electron structure of a really complicated molecule. And it turns out the secret is to do what all our math teachers have touted for centuries: simplify first, then solve.

Benzene is an aromatic compound, which is a retrofitted, non-odorous term for a family of molecules with a specific, stable structure in common. In fact, benzene, which scientist Michael Faraday discovered in 1825, is the boss of the aromatics.

“Benzene is the best-known aromatic compound and the parent to which numerous other aromatic compounds are related,” Britannica explains. “The six carbons of benzene are joined in a ring, having the planar geometry of a regular hexagon in which all of the C—C bond distances are equal.”

Each carbon has one hydrogen attached, giving the molecule a starburst kind of shape. Observing and plotting these has been easy, but the molecule’s dozens of electrons have been much more elusive, creating a thorny argument that’s gone on for nearly a century on its own. There are 42 electrons in there, and trying to model them generates a mathematical function with 126 dimensions: three for the full 3D position of each of the 42 electrons.

Now, about those 126 dimensions: They're purely mathematical, although that math corresponds to some impossible-to-imagine shape or form existing in some specific use case of extended spacetime. (Different models of physics say the universe has four dimensions, or 10, and the wildest models top out at 26 at the very most.)

The X-Y plot many of us first learned in pre-algebra class is two dimensional, and the next step up is a three-dimensional space where the third Z axis is added. But where do you add the fourth, fifth, 100th, and 126th axes?

“As creatures who inhabit three spatial dimensions, we find it difficult to picture objects of higher dimensionality than this,” the researchers explain. One of the ways calculus, for example, has helped world knowledge is by giving us a way to examine how things change as we increase their complexity, but everyday calculus is laughing at whoever has to solve this 126-dimension problem.

These scientists seized on wavefunctions, which describe the movements within individual quantum systems, as their hook to begin studying the problem. As the team explains in its paper:

“The electrons being described by the electronic wavefunction are fundamentally indistinguishable particles. That means that in the 3𝑁-dimensional space of the wavefunction there are regions that are equivalent, related through the permutation of electrons. Because these regions are equivalent and together span the space of the wavefunction, they are analogous to tiles.”

Considering the problem as a repetition of tiles immediately turns 126 dimensions into far fewer, allowing these scientists to consider how the electrons are spinning, bonding in and out of pairs, and otherwise behaving in a way that can be tracked and identified. “[W]e have shown that [our complex equations] may be visualised in terms of wavefunction tiles to discern the electronic structure and how this differs from the uncorrelated, single-determinant wavefunction,” they conclude.

Benzene is a fundamental building block in a lot of modern applications, and understanding how its electrons work has implications for, well, electronics. Moreover, as king of the aromatics, benzene is related to graphene and other substances where this wavefunction idea could trickle down in other useful ways. Faraday would be so proud.

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