Rings of carbon with an iconic twist in their structures break the rules for building strongly stable molecules, and now a mathematical proof explains why.
Strongly stable molecules are useful because they hold together long enough to be used in plastics, pipes and polyester clothing. One class of molecules that holds together especially well is called aromatic (though not all of them have odors).
Ring-shaped molecules like benzene, which is the essential precursor in many drugs and plastics, are called aromatic when a particular orbital shell is filled by six electrons, 10 electrons, 14 electrons and so forth — as long as the number of electrons is two plus a multiple of four.
In 1964, Swiss-German chemist Edgar Heilbronner suggested that putting a twist in the ring would change the rules — a Möbius molecule would be aromatic if it contained any multiple of four electrons.
Named for the German mathematician August Ferdinand Möbius, Möbius molecules are shaped like looped bands with a single twist. The same shape can be seen in conveyor belts, M.C. Escher paintings or just by taking a strip of paper, twisting it once and taping the ends together. Now, a theorist at Michigan State University in East Lansing has proven Heilbronner’s conjecture by applying quantum mechanics. Chemist Evangelos Miliordos simulated the quantum mechanics of the Möbius ring, with electrons trapped on the twisted structure. To predict an electron’s behavior, Miliordos relied on a famous calculation called the Schrodinger equation. Students of quantum mechanics often learn the simplest example: How does a particle behave when it’s confined in a box?
“The author is putting electrons not in a molecule, but in a topological box. In a Möbius box,” says Ranier Herges, an organic chemist at the University of Kiel in Germany.
To solve the equation, the key insight by Miliordos was that electrons don’t travel on the surface of the strip, like ants in a famous M.C. Escher painting. The electrons are confined inside the plane of the strip. So instead of traveling twice the distance of the strip to get back to where they started, the electrons travel exactly the length of the ring. Miliordos found that when these electrons are arranged on Möbius molecules, the stability rule is inverted — only when the electron number is a multiple of four will the molecule be aromatic, Miliordos reports in a December 28 Physical Review A paper.
The proof may have surprising implications for fundamental quantum properties of particles. The orbital angular momentum of an electron, a value that is typically a whole number, could take on half-integer values like 1/2, 3/2, and so forth in the Möbius model.
Chemists first created Möbius molecules in the lab as an intellectual challenge. In 2003, Herges and some of his colleagues coaxed a line of molecules to twist before joining the two ends, creating the first Möbius molecule (it was red). Now he’s trying to create molecules with three twists. A molecule with two twists isn’t as interesting, says Herges, because you can tell which side you’re on. An odd number of twists, and you travel the surface in an infinite loop.
“Unfortunately, there are no applications yet,” says Herges.
