A symmetry measuring tape shows how skewed atoms and molecules really are

In theory, the world is beautifully ordered. Atoms line up neatly in crystals, molecules have tidy shapes and symmetry is so regular that structures can be drawn as perfectly clean, repeating patterns. On paper, one could rotate or mirror such a structure without anything changing – like a flawless ornamental motif.

In reality, the world looks different. Atoms are slightly off position, bonds are a little stretched or compressed and both universities and industry lack a clear sense of when such distortions are merely cosmetic – and when they fundamentally alter how a material acts. Researchers have so far lacked a precise language for this distinction.

Against this backdrop, Professor of Chemistry Thomas Just Sørensen and his colleague Villads R. M. Nielsen at the Nano-Science Centre and the Department of Chemistry at the University of Copenhagen in Denmark work with materials that can glow, respond to magnetic fields or store information – precisely the kinds of properties for which even very small atomic shifts can have disproportionately large effects.

That work has recently crystallised in an article in Nature Communications that describes a method for quantifying continuously how far a given atomic or molecular structure deviates from ideal symmetry – and for linking these deviations directly to measurable physical properties.

According to Thomas Just Sørensen, the method could change how researchers connect structure and function in materials.

“Our new symmetry measuring tape can open our eyes to why molecules and materials behave the way they do,” he says, describing it as a tool for bridging the gap between abstract, theoretical symmetry and the properties that researchers and companies are trying to control in everything from advanced materials to future quantum computers.

Quantum computers are a special type of computer that exploit quantum mechanics to solve certain computational tasks far faster than even very powerful conventional computers – for example, when simulating advanced materials or breaking highly secure codes.

Small irregularities with major consequences

The idea for the symmetry measuring tape emerged directly from work on advanced materials. Many of the systems studied by the research group contain metals from what chemists call the rare earth elements – such as europium and terbium – that are often used to make materials emit light or respond to magnetic fields.

These elements have a distinctive electronic structure that causes them to emit highly characteristic light and to change noticeably even when their surroundings are altered only slightly. In the materials, they sit as individual atoms inside a kind of cage formed by other atoms, and it is precisely this position that makes them sensitive indicators of their local environment.

Even very small distortions of the cage are directly reflected in the optical and magnetic signals measured by the researchers, creating a rare and unusually clear link between structure, calculation and experiment.

When the researchers began to compare structures and measured properties more systematically, the same pattern kept reappearing. Molecules and materials that ought to belong to the same category according to the classical rules of symmetry acted quite differently. A bond that was just slightly longer than the others, or an angle that was not quite right, could noticeably shift both the wavelength of emitted light and the material’s magnetic behaviour.

When almost identical structures act very differently

In the laboratory, the researchers were confronted with systems that looked almost identical yet exhibited properties that lay far apart. This pointed to a deeper problem: the symmetry concepts chemists normally rely on are too coarse-grained to capture the small differences that, in practice, determine how materials act.

On paper, methods for describing symmetry already existed. They were typically based on fixed categories and idealised reference shapes around a metal atom, which served as a kind of checklist for the symmetry a structure was supposed to have.

Calculations were then used to determine how much the atoms in the real structure would need to be moved to match this ideal answer.

Each method was tailored to a single type of shape, which made it difficult to compare results across materials and measurement techniques. This meant that the calculations usually included only the nearest atoms around the metal, whereas the rest of the molecule or material faded into the background.

Everything comes down to coordinates

At a certain point, Thomas Just Sørensen and colleagues realised that they needed a more general tool. Not simply a better model, but an entirely different way of describing symmetry. Instead of starting by choosing a neat reference shape that ought to fit, they decided to begin with something all structures have in common.

Every atom in a molecule or crystal has a position in space that can be written as a set of coordinates. Collecting all of these positions yields a list of points that describes the entire structure – whether it is a small molecule, a crystal or a more complex material.

The new symmetry measuring tape works from this list. The structure is first oriented in a coordinate system and then subjected to a series of well-defined symmetry operations – such as rotations or reflections – each corresponding to a specific mathematical symmetry.

Symmetry is thus treated as a set of actions that can be performed on the structure, and something that was previously an either–or question suddenly becomes a matter of degree.

For each operation, the software measures how far every point ends up from its ideal position – in the same way that a perfectly round ball produces zero displacement no matter how it is rotated, whereas a slightly flattened ball gradually reveals itself as it is turned.

Molecules and crystals are treated in exactly the same way – only with far more points and symmetry operations.

When symmetry becomes a number

In practice, the new method is at the heart of a piece of software the researchers call CSoM – short for Continuous Symmetry operation Measure. CSoM can be thought of as an elastic measuring tape for symmetry, one that can be laid over everything from small molecules to complex materials.

The programme takes the many small deviations it calculates and condenses them into a single number for each type of symmetry. If the number is close to zero, the structure is almost perfectly symmetric. The higher the number, the further the structure deviates from ideal symmetry.

Thomas Just Sørensen admits that turning the theory into something a computer could work with reliably has been a demanding task.

“We have spent years getting all the possible symmetry operations written correctly, testing them on real structures and making sure the method behaves sensibly – even when the data are not perfect,” he says.

From water to heavy metals

To test the method, the researchers applied it to a series of well-known systems with very different chemical structures and expected symmetries, precisely to see whether the same yardstick produced consistent results across the board.

These range from water molecules in gas, liquid and ice to flat, ring-shaped molecules such as benzene – a basic building block in petrol and many dyes – and on to compounds containing heavy metals and rare earth elements, which appear in screens, magnets and other high-tech materials.

In some cases, CSoM confirms that the classical idealised pictures provide a good description. In others, the method shows that structures considered almost perfect on paper deviate so much in practice that the high symmetry is little more than a rough approximation.

The difference is particularly striking in more complex systems involving heavy metals and rare earth elements, Sørensen points out.

Here, the innermost ring of atoms around the metal may look neat and orderly, whereas the rest of the molecule pulls and pushes, leaving the overall structure with a different – and lower – symmetry than one would normally expect.

Using symmetry as a design tool

Although CSoM stems from basic research, its consequences are beginning to emerge in several areas in which the smallest material details can have the greatest impact.

In the field of quantum computing, many groups work with the smallest computational units, known as qubits. In some systems, the qubit resides in two specific quantum states of a single ion from the rare earth elements embedded in a crystal.

The symmetry around the ion helps determine how well these states remain separated – a relationship that can now be quantified in the same way as in entirely different classes of materials. At very high symmetry, the states are almost isolated, like two rooms with the doors closed. A slight break in symmetry corresponds to a door left slightly ajar.

In that situation, the states begin to couple weakly with one another, and the distinction between them becomes less sharp. From the outside, this means that the qubit more quickly loses the state that was meant to carry the information. According to Thomas Just Sørensen, even a modest difference in symmetry can be enough to drastically shorten the time for which a qubit remains usable.

If quantum computers are ever to operate stably for hours at a time, crystal structures will be required in which the ion experiences a very high degree of symmetry. Here, CSoM can act as the symmetry measuring tape that reveals whether one is close enough.

From quantum bits to luminous materials

Another area concerns the materials that make screens and lamps glow. Many flat-panel displays and light-emitting diodes contain thin layers in which metals from the same rare earth elements cause the sharp red, green and blue colours. In Thomas Just Sørensen’s laboratory, the group works with precisely such luminous molecules.

Here, an almost imperceptible shift in the atoms’ surroundings can slightly change the colour or alter how efficiently the material emits light.

With CSoM, researchers can for the first time quantify how skewed the environment is around the active atom and link this directly to subtle changes in the light spectrum. The idea is that this relationship could eventually be used to tailor materials more precisely so that they emit light with exactly the desired colour and efficiency.

A third area of application lies in materials that change structure with temperature or pressure. In certain crystals, atoms gradually slide into more orderly arrangements long before a clear phase transition becomes visible in classical measurements. Here, CSoM can be used to track how symmetry changes gradually in the local environments around selected atoms. According to Thomas Just Sørensen, the resulting number can indicate early that a material is moving towards a new structure – before the change becomes visible in traditional data.

An atlas of symmetry: the next goal

Although the method has now been described and made available as software, the work is far from finished. For every structure that is analysed, the same question arises: how large a deviation from symmetry is needed before anything changes in practice? When is a measure of asymmetry so small that symmetry still effectively governs a material’s properties – and when is the distortion large enough for the material to act differently?

In the Nature Communications article, the group does not merely present the method itself. They also sketch initial estimates of such limits by applying CSoM to a range of well-studied systems. They show that certain complexes traditionally described as nearly perfect octahedra exhibit deviations so large that the assumed high symmetry is no longer an accurate description.

The group is now planning a series of experiments in which structures are altered in small steps, and the symmetry measured with CSoM is linked directly to the relevant properties. The aim is to build a kind of atlas in which specific intervals of symmetry values can be systematically connected to specific material behaviours, based on repeated measurements across different classes of materials – much like a map showing what to expect as symmetry shifts.

“We have gone from not having a measuring tape to having one. Now it is a matter of understanding what the numbers mean,” says Thomas Just Sørensen.

This marks the first step from a world that only appeared orderly on paper to one in which order can also be measured.