The theorems, recently published in the journal Physical Review Letters, describe how the energy of particle systems (like atoms and molecules) fluctuates with changes in magnetism and particle count. This work extends a series of critical studies initiated in the early 1980s, solving an open problem significant to matter simulation using computers.

The research was conducted by Andrew Burgess, a PhD candidate in Trinity's School of Physics, along with Dr. Edward Linscott from the Paul Scherrer Institute in Switzerland, and Dr. David O'Regan, Associate Professor in Physics at Trinity.

"Imagine a steep-sided valley, where the ground is not curved but instead made up of angular tiles, like you might see in an old arcade game where the images were made using polygons. We have found that the height profile in fractured valleys like this represents the exact energy of isolated collections of particles, like molecules. Heading straight up the valley corresponds to changing the number of electrons that hold together the molecule, while moving to each side increases its magnetism. This work completes the mapping of this valley up to high magnetic states, finding that the valley walls are steep and tilted," explained Dr. David O'Regan.

Andrew Burgess, the lead author, shared insights on the discovery process: "While working on a different problem, I needed to know the shape of this energy valley for simple systems. Hunting through published research, I could find lots of nice graphs but to my surprise they stopped short of mapping the entire valley. I realized that existing quantum mechanical theorems could be used for systems with one electron, such as the hydrogen atom. However, for systems with two electrons, such as the helium atom, these theorems could tell me little about the sides of the valley. Specifically, a quantum mechanical theorem known as the spin constancy condition was incomplete."

Dr. Edward Linscott, from the Laboratory for Materials Simulations at PSI, highlighted the practical implications: "Understanding the geography of this energy landscape may seem quite abstract and esoteric - but actually, this knowledge can help solve all sorts of real-world problems. When colleagues of ours use computer simulations to try to find next-generation materials for more efficient solar panels, or catalysts for more energy-efficient industrial chemistry, our knowledge of the energy landscape can be baked into the calculations that they perform, making their predictions more accurate and reliable."

Dr. O'Regan added: "The energy differences and slopes of this valley landscape underpin the stability of matter, interactions between materials and light, chemical reactions, and magnetic effects. Knowing what the entire valley surface looks like, including at high magnetization, is already helping us to build better tools for simulating complex materials, even when they are not magnetic.

"Motivating this work is the need to provide improved simulation theory and methods for developing materials for renewable energy and chemistry applications. When a battery is discharging, for example, there are metal atoms that change their particle count and magnetism. Here we see that we're moving in that same valley landscape and it's the drop in height, so to speak, that gives the energy that the battery provides. This is an example of applied simulation and abstract quantum theory being practiced side by side, each motivating and improving the other."

Reflecting on the nature of this kind of research, Mr. Burgess added: "This interplay between theory and practical simulation is what I love most about this area of research. We have already developed a new method for modelling materials based on these theorems and are testing it out on battery cathode materials, so there is plenty of exciting work in the pipeline!"

Research Report:Tilted-Plane Structure of the Energy of Finite Quantum Systems

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