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The Quantum-Mechanical Model of an Atom

The Quantum-Mechanical Model of an Atom

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.

The Energies of Atomic Orbitals

The Energies of Atomic Orbitals

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.

Crystal Field Theory - Octahedral Complexes

Crystal Field Theory - Octahedral Complexes

Crystal Field Theory
To explain the observed behavior of transition metal complexes (such as colors), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It helps to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.
CFT focuses on...

Crystal Field Theory - Tetrahedral and Square Planar Complexes

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Tetrahedral Complexes
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...

Reduced Mass Coordinates: Isolated Two-body Problem

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In classical mechanics, the two-body problem is one of the fundamental problems describing the motion of two interacting bodies under gravity or any other central force. When considering the motion of two bodies, one of the most important concepts is the reduced mass coordinates, a quantity that allows the two-body problem to be solved like a single-body problem. In these circumstances, it is assumed that a single body with reduced mass revolves around another body fixed in a position with an...

Molecular Orbital Theory I

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Related Experiment Video

Updated: Aug 1, 2025

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Published on: May 27, 2020

Complexity reduction in density functional theory: Locality in space and energy.

William Dawson¹, Eisuke Kawashima¹, Laura E Ratcliff²

¹RIKEN Center for Computational Science, Kobe, Hyogo 650-0047, Japan.

The Journal of Chemical Physics

|April 27, 2023

Summary

We optimized the NTChem program for large-scale hybrid density functional theory (DFT) calculations on the Fugaku supercomputer. This enables efficient analysis of spectral properties and system fragmentation for massive atomic systems.

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Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

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Published on: May 27, 2020

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Published on: April 8, 2020

Area of Science:

Computational Chemistry
Materials Science

Background:

Large-scale electronic structure calculations are crucial for understanding material properties.
Existing methods face challenges with computational cost for complex systems.

Purpose of the Study:

To present advancements in the NTChem program for supercomputing environments.
To assess the impact of computational choices on fragment analysis.
To develop efficient algorithms for electronic structure calculations.

Main Methods:

Utilized hybrid density functional theory (DFT) on the Fugaku supercomputer.
Implemented a complexity reduction framework for computational efficiency.
Employed an all-electron representation to study system fragmentation.
Developed and applied novel algorithms for computing Kohn-Sham orbital energies.

Main Results:

Demonstrated efficient large-scale hybrid DFT calculations with NTChem.
Assessed the influence of basis set and functional choice on fragment properties.
Successfully applied new algorithms to systems with thousands of atoms.
Identified spectral properties' origins using the developed algorithms.

Conclusions:

The enhanced NTChem program and new algorithms enable efficient, large-scale electronic structure calculations.
These developments provide powerful tools for analyzing complex systems and spectral properties.
The study facilitates deeper insights into material behavior through advanced computational methods.