An international team of physicists, including Dr. Hubert Jóźwiak from Nicolaus Copernicus University in Toruń, has developed a novel algorithm that allows for significantly faster simulations of molecular collisions. The study, just published in the prestigious journal Science Advances, breaks through a computational barrier that scientists have struggled with for decades and paves the way for, among other things, rigorous research into collisions of water molecules.
Collisions of atoms and molecules are among the most fundamental dynamic processes in the universe. Quantum mechanics provides a very concise "recipe" for predicting the dynamics of such an event in the form of the Schrödinger equation. However, solving this equation exactly poses a formidable numerical challenge. The heavier and more complex the colliding molecules are, the more computational power is required to describe their behavior. The scaling of commonly used methods is brutal: if simulating a collision of simple molecules takes a computer one day, a system ten times more complex would require almost three years of continuous computation. This is due to how widely used algorithms operate, unwinding the solution to the equation (the wave function) step by step, utilizing matrix multiplication and inversion operations.
The new algorithm relies on gradually refining the solution to the Schrödinger equation. Starting with an approximate form of the wave function, corrections are introduced until the function reaches the required accuracy. Previous attempts to apply such an approach to molecular collisions, however, were mathematically unstable—instead of bringing the function closer to the correct solution, the numerical error accumulated with each subsequent correction.
The researchers overcame this using regularization: the algorithm successfully identifies and isolates the "problematic" part of the collision from the one that behaves stably. Crucially, this was achieved using significantly cheaper algebraic operations. As a result, the computation time grows with the second, rather than the third, power of the system's complexity. In practice, this means that solving the previously mentioned example will take three months instead of three years.
Drastically reducing computation time opens up the possibility for physicists to simulate systems that were previously beyond the reach of the best supercomputers. In the near future, the new algorithm will allow for the first rigorous simulations of, among others, collisions of water molecules. Understanding this process is crucial both for the precise modeling of Earth's atmosphere and in studies of the isotopic composition of comets, which seek to explain the origin of water on our planet.
