Chaos theory sheds new light on examining methods for determining electron wave functions

The ability to model the movement of electrons is an essential feature of computational physics and chemistry. As atoms and molecules become larger, however, techniques like the widely used Hartree-Fock method have a harder time keeping up with the many features of electron orbitals. Recent research has shown the value of examining these techniques through the lens of chaos theory in improving our understanding of electronic wave functions.

Examination of the Hartree-Fock method as a dynamical system, using a complex parameter to study its convergence behavior, is published in Chaos. Larger atoms such as argon showed an increase of fragmentation of the self-consistent field sets while smaller atoms exhibited fractal geometry reminiscent of the Mandelbrot set.

To determine electronic wave functions, the Hartree-Fock method adjusts its model over several iterations to eventually determine an approximate atomic structure, which bears resemblance to dynamical systems in mathematics. The authors tapped into this to by controlling the electron interaction strength via a new parameter, the complex coupling strength, and studying the Hartree-Fock approximations.

Hartree-Fock data produced fractal structures when plotted in the complex plane corresponding to the new coupling strength parameter. As the nuclear charge of atoms increased, so did the fragmentation in their Hartree-Fock data, reflecting that more complexity is encountered in the convergence behavior of the orbital structures.

The findings contribute to a growing body of work for determining complex molecules with the goal of finding a criterion that can be used generally to improve the performance of the Hartree-Fock method.

Source: “The fractal geometry of Hartree-Fock,” by Friethjof Theel, Antonia Karamatskou, and Robin Santra, Chaos (2017). The article can be accessed at https://doi.org/10.1063/1.5001681 .