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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.

Valence Bond Theory and Hybridized Orbitals

Valence Bond Theory and Hybridized Orbitals

According to valence bond theory, a covalent bond results when: (1) an orbital on one atom overlaps an orbital on a second atom, and (2) the single electrons in each orbital combine to form an electron pair. The strength of a covalent bond depends on the extent of overlap of the orbitals involved. Maximum overlap is possible when the orbitals overlap on a direct line between the two nuclei.
A σ bond (single bond in a Lewis structure) is a covalent bond in which the electron density is...

Crystal Field Theory - Tetrahedral and Square Planar Complexes

Crystal Field Theory - Tetrahedral and Square Planar Complexes

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,...

Molecular Orbital Theory I

Molecular Orbital Theory I

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Hybridization of Atomic Orbitals II

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sp3d and sp3d 2 Hybridization

Hybridization of Atomic Orbitals I

Hybridization of Atomic Orbitals I

The mathematical expression known as the wave function, ψ, contains information about each orbital and the wavelike properties of electrons in an isolated atom. When atoms are bound together in a molecule, the wave functions combine to produce new mathematical descriptions that have different shapes. This process of combining the wave functions for atomic orbitals is called hybridization and is mathematically accomplished by the linear combination of atomic orbitals. The new orbitals that...

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Updated: Jan 17, 2026

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Learning local and semi-local density functionals from exact exchange-correlation potentials and energies.

Bikash Kanungo¹, Jeffrey Hatch², Paul M Zimmerman²

¹Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA.

Science Advances

|September 19, 2025

Summary

Researchers developed a data-driven approach using neural networks to create accurate exchange-correlation (XC) functionals for density functional theory (DFT). This method significantly improves total energies and densities, offering a promising path for future functional development.

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Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics

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Area of Science:

Computational Chemistry
Materials Science
Quantum Mechanics

Background:

Accurate exchange-correlation (XC) functionals are crucial for density functional theory (DFT) but remain a significant challenge.
Existing functionals struggle to achieve general-purpose chemical accuracy despite decades of development.

Purpose of the Study:

To present a novel data-driven pathway for learning XC functionals using exact densities, energies, and potentials.
To demonstrate the efficacy of neural network (NN)-based XC functionals.

Main Methods:

Obtained exact densities from accurate configuration interaction (CI) calculations.
Derived exact XC energies and potentials via inverse DFT on CI densities.
Trained simple NN-based local density approximation (LDA) and generalized gradient approximation (GGA) functionals.

Main Results:

NN-based LDA and GGA functionals showed remarkable improvement in total energies and densities, even when trained on limited data.
The NN-based GGA functional achieved accuracy comparable to the SCAN meta-GGA on thermochemistry datasets.

Conclusions:

The use of XC potential in modeling XC functionals is promising.
This data-driven approach can pave the way for systematically developing more accurate XC functionals for DFT.