What is it about?

To understand the properties of matter, we first need to understand its building blocks: atoms. This is done by solving fundamental equations from quantum mechanics. For most atoms, the standard Schrödinger equation works well. But for heavy elements like gold or uranium, electrons move near the speed of light, and we need the more powerful—and much trickier—Dirac equation, which accounts for Einstein's theory of relativity. Solving the Dirac equation is notoriously difficult. Many standard computer methods can become unstable and produce "spurious" results—solutions that are mathematically valid but physically meaningless, which corrupts the science. Our new open-source software, featom, is designed to solve both the Schrödinger and Dirac equations with high accuracy. To tame the difficult Dirac equation, we use a special technique. Instead of solving the equation directly, we solve its square. This elegant mathematical maneuver is guaranteed to produce the correct physical answers while completely eliminating the spurious ones, making our method exceptionally robust and reliable. Our work on featom takes a different path from the norm of slow but "accessible" programming languages like Python, demonstrating the resurgent power of Modern Fortran for creating software that delivers uncompromising performance and can solve problems that others cannot.

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Why is it important?

This work serves as a crucial counterpoint to the idea that all modern scientific software should be built on interpreted languages. featom proves that for high-performance applications, the speed and efficiency of compiled Fortran are essential. More importantly, it solves a problem—the stable solution of the radial Dirac equation—that has remained a persistent challenge. By providing an open-source tool that succeeds where other modern efforts have been stymied, we are setting a new benchmark for both performance and capability. This provides the community with a robust, high-performance foundation for the challenging physics of heavy elements, preventing researchers from hitting the performance and capability walls of simpler tools.

Perspectives

The move from finite difference to finite element methods was, in some sense, inevitable for achieving the next level of accuracy and flexibility in atomic physics. However, the practical challenges, especially for the relativistic Dirac equation, made this transition intractable for many years. This work is the result of a decade of grappling with these very problems. Our solution required a two-pronged attack. On the physics front, the novel squared Hamiltonian approach was our answer to the instability of the Dirac equation. On the software front, we deliberately chose Modern Fortran to build a code that was not only fast but also modular and reusable—a clear break from older, monolithic codes. We believe this fusion of advanced numerical methods, a novel physical formulation, and modern software engineering represents a significant step forward for the field. featom is thus a testament to the fact that to solve the hardest problems, you need both cutting-edge software engineering and a sophisticated understanding of the underlying physics. It showcases exactly why Fortran continues to be the premier language for high-performance scientific computing.

Rohit Goswami
University of Iceland

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This page is a summary of: High-order finite element method for atomic structure calculations, Computer Physics Communications, April 2024, Elsevier,
DOI: 10.1016/j.cpc.2023.109051.
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