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Molecular Orbital Theory I

Molecular Orbital Theory I

Overview of Molecular Orbital Theory

Atomic Orbitals

Atomic Orbitals

An atomic orbital represents the three-dimensional regions in an atom where an electron has the highest probability to reside. The radial distribution function indicates the total probability of finding an electron within the thin shell at a distance r from the nucleus. The atomic orbitals have distinct shapes which are determined by l, the angular momentum quantum number. The orbitals are often drawn with a boundary surface, enclosing densest regions of the cloud.

Molecular Orbital Theory II

Molecular Orbital Theory II

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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.

Electron Orbital Model

Electron Orbital Model

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.
The first shell is closest to the nucleus, and it has only one subshell with a single spherical orbital called the...

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Hybridization of Atomic Orbitals II

sp3d and sp3d 2 Hybridization

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Updated: Mar 21, 2026

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

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Published on: April 8, 2020

Petascale Orbital-Free Density Functional Theory Enabled by Small-Box Algorithms.

Mohan Chen¹, Xiang-Wei Jiang², Houlong Zhuang¹

¹Department of Mechanical and Aerospace Engineering, Princeton University , Princeton, New Jersey 08544, United States.

Journal of Chemical Theory and Computation

|May 5, 2016

Summary

This summary is machine-generated.

A new small-box fast Fourier transform (FFT) algorithm enhances orbital-free density functional theory (OFDFT) simulations. This method enables efficient first-principles calculations for materials with millions of atoms.

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Area of Science:

Computational Physics
Materials Science
Quantum Mechanics

Background:

Orbital-free density functional theory (OFDFT) uses electron density for quantum-mechanical calculations.
Fast Fourier Transform (FFT) is computationally intensive in OFDFT, limiting large-scale simulations.
Existing FFT algorithms face parallelization challenges for massive atomistic systems.

Purpose of the Study:

To develop a more efficient FFT algorithm for OFDFT.
To overcome parallelization limitations in conventional FFT methods.
To enable first-principles simulations of significantly larger material systems.

Main Methods:

Designed and implemented a small-box FFT (SBFFT) algorithm.
Proposed real-space truncation for the nonlocal Wang-Teter kinetic energy density functional (KEDF) kernel.
Tested SBFFT scalability with simulations of 1,024,000 lithium atoms.

Main Results:

Demonstrated efficient simulation of a full optimization step for over a million lithium atoms on 65,536 cores.
Achieved excellent agreement with original OFDFT results in various simulations (bulk, nanocrystalline, liquid Li).
Validated the scalability and accuracy of the SBFFT algorithm.

Conclusions:

The developed OFDFT-SBFFT algorithm significantly improves computational efficiency.
This advancement allows for first-principles simulations of materials with millions of atoms.
Opens new possibilities for studying large-scale material properties and behaviors.