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Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra.
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The word "gas" comes from the Flemish word meaning "chaos," first used to describe vapors by the chemist J. B. van Helmont. Consider a container filled with gas, with a continuous and random motion of molecules. During collisions, the velocity component parallel to the wall is unchanged, and the component perpendicular to the wall reverses direction but does not change in magnitude. If the molecule’s velocity changes in the x-direction, then its momentum is changed.
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Orbitals are the areas outside of the atomic nucleus where electrons are most likely to reside. They are characterized by different energy levels, shapes, and three-dimensional orientations. The location of electrons is described most generally by a shell or principal energy level, then by a subshell within each shell, and finally, by individual orbitals found within the subshells.
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In an atom, the negatively charged electrons are attracted to the positively charged nucleus. In a multielectron atom, electron-electron repulsions are also observed. The attractive and repulsive forces are dependent on the distance between the particles, as well as the sign and magnitude of the charges on the individual particles. When the charges on the particles are opposite, they attract each other. If both particles have the same charge, they repel each other.
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An efficient zero-order evolutionary method for solving the orbital-free density functional theory problem by direct minimization.

The Journal of chemical physics·2023
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Updated: Jun 11, 2025

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
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Physics-Inspired Evolutionary Machine Learning Method: From the Schrödinger Equation to an Orbital-Free-DFT Kinetic

Juan I Rodríguez1, Ulises A Vergara-Beltrán2

  • 1Centro de Investigación en Ciencia Aplicada y Tecnología Avanzada, Unidad Querétaro, Instituto Politécnico Nacional, Cerro Blanco 141 Col. Colinas del Cimatario, Querétaro C.P. 76090, México.

The Journal of Physical Chemistry. A
|September 30, 2024
PubMed
Summary

A new machine learning (ML) method, ML-Ω, derives fundamental physical equations from data. It successfully rediscovers Schrödinger's equation and the Thomas-Fermi functional, and develops a superior orbital-free density functional theory (DFT) functional for electronic structure calculations.

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Area of Science:

  • Computational Physics and Chemistry
  • Machine Learning in Scientific Discovery
  • Quantum Mechanics and Electronic Structure Theory

Background:

  • Deriving fundamental physical laws from observational data is a long-standing scientific challenge.
  • Traditional methods often rely on human intuition and predefined theoretical frameworks.
  • Machine learning offers new paradigms for scientific discovery, but integrating it with established physical principles remains an active research area.

Purpose of the Study:

  • To introduce a novel machine learning (ML)-supervised evolutionary method, ML-Ω, inspired by physics' variational principles.
  • To demonstrate ML-Ω's capability to derive underlying differential equations and functionals directly from data.
  • To apply ML-Ω for developing advanced functionals in density functional theory (DFT).

Main Methods:

  • Developed ML-Ω, a hypothesis evolutionary method that learns model functions (functionals) from data.
  • Trained ML-Ω with minimal datasets: three hydrogen-like atom energies to rediscover Schrödinger's equation and functional.
  • Trained ML-Ω with Thomas-Fermi (TF) energies to derive the exact TF functional.
  • Applied ML-Ω to derive a local orbital-free (OF) kinetic energy functional (Ts) using energies of five atoms.

Main Results:

  • ML-Ω successfully derived Schrödinger's exact functional and equation from limited atomic energy data.
  • The method accurately reproduced the exact Thomas-Fermi functional.
  • An ML-Ω-derived local OF-DFT functional (γTFλvW(0.964,1/4)) outperformed existing OF-DFT functionals.
  • The new functional improved the description of stretched bonds in diatomic molecules compared to LDA and some GGA functionals.

Conclusions:

  • The ML-Ω evolutionary method provides a powerful framework for discovering fundamental equations and functionals from data.
  • This approach successfully bridges machine learning with natural sciences, enabling the derivation of complex physical laws.
  • ML-Ω demonstrates significant potential for advancing electronic structure calculations and other scientific domains by generating accurate and efficient theoretical models.