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The Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

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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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Following the work of Ernest Rutherford and his colleagues in the early twentieth century, the picture of atoms consisting of tiny dense nuclei surrounded by lighter and even tinier electrons continually moving about the nucleus was well established. This picture was called the planetary model since it pictured the atom as a miniature “solar system” with the electrons orbiting the nucleus like planets orbiting the sun. The simplest atom is hydrogen, consisting of a single proton as...
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Electron Orbital Model01:18

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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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An atom comprises protons and neutrons, which are contained inside the dense, central core called the nucleus, with electrons present around the nucleus. Taking into account the wave–particle duality of electrons and the uncertainty in position around the nucleus, quantum mechanics provides a more accurate model for the atomic structure. It describes atomic orbitals as the regions around the nucleus where electrons of discrete energy exist, characterized by four quantum...
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Crystal Field Theory
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In-plane structure of the electric double layer in the primitive model using classical density functional theory.

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The primitive model in classical density functional theory: beyond the standard mean-field approximation.

Moritz Bültmann1, Andreas Härtel1

  • 1Physikalisches Institut, Albert-Ludwigs-Universität, 79104 Freiburg, Germany.

Journal of Physics. Condensed Matter : an Institute of Physics Journal
|March 16, 2022
PubMed
Summary

This study enhances classical density-functional theory (DFT) for ion systems by modifying electrostatic contributions. The improved DFT functional better predicts ion behavior, especially at high concentrations.

Keywords:
Barker–Hennderson perturbation theoryclassical density functional theorycorrelation functions and decay lengthselectric double layerelectrolytes and ionic fluidsprimitive modelstatistical physics

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Area of Science:

  • Physical Chemistry
  • Computational Chemistry
  • Statistical Mechanics

Background:

  • Classical density-functional theory (DFT) models ions as charged hard spheres.
  • The mean-field electrostatic contribution in DFT is approximated and can be inaccurate at short distances due to hard-core overlap.
  • Existing DFT functionals require improvement for accurate ion system descriptions.

Purpose of the Study:

  • To modify the mean-field electrostatic contribution in classical DFT for ion systems.
  • To improve the accuracy of DFT in describing ion interactions, particularly at high concentrations.
  • To develop a more robust theoretical framework for electrostatic interactions in ionic matter.

Main Methods:

  • Modified the mean-field electrostatic contribution by enforcing a constant pair potential below the ion contact distance.
  • Incorporated weighted densities, drawing parallels with fundamental measure theory.
  • Evaluated the modified functional against established functionals and simulation data using density profiles, correlation functions, and thermodynamic consistency checks (sum rule, virial pressure).

Main Results:

  • The modified DFT functional demonstrated improved predictions for density profiles and direct correlation functions, especially in high concentration scenarios.
  • Observed enhanced thermodynamic consistency when calculating properties via different routes.
  • The modifications successfully addressed limitations of the standard mean-field approach, particularly regarding layering effects.

Conclusions:

  • Modifications beyond the standard mean-field approximation significantly improve classical DFT for electrostatic interactions in ion systems.
  • The developed formalism provides a foundation for systematic advancements in DFT for ionic matter.
  • This work offers a more accurate computational tool for studying electrolytes and other ionic systems.