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Hybridization of Atomic Orbitals I

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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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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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Updated: Mar 7, 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

Optimization of numerical orbitals using the Helmholtz kernel.

Eelis Solala¹, Sergio A Losilla¹, Dage Sundholm¹

¹Department of Chemistry, University of Helsinki, P.O. Box 55, A.I. Virtanens plats 1, FIN-00014 Helsinki, Finland.

The Journal of Chemical Physics

|March 3, 2017

Summary

This summary is machine-generated.

This study introduces a novel integration scheme for optimizing molecular orbitals in electronic structure calculations. The method achieves high accuracy for Hartree-Fock calculations, improving computational efficiency.

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Area of Science:

Computational Chemistry
Quantum Chemistry
Theoretical Chemistry

Background:

Accurate electronic structure calculations are crucial for understanding molecular properties.
Orbital optimization is a key step in these calculations, often posing computational challenges.

Purpose of the Study:

To develop and validate a new integration scheme for optimizing molecular orbitals in numerical electronic structure calculations.
To improve the efficiency and accuracy of computational chemistry methods.

Main Methods:

The study employs an integration scheme for the Helmholtz kernel within a double bubble and cube basis.
Numerical integration techniques are applied to both the bubble (near nuclei) and cube (grid-based) regions.
One-center expansions and analytical/numerical integration of spherical harmonics and Bessel functions are utilized.

Main Results:

The implementation was validated using Hartree-Fock self-consistent-field calculations on H2, H2O, and CO.
The developed method achieves high accuracy, with energy errors ranging from 10^-4 to 10^-7 E_h.
The behavior of the integrand in the auxiliary dimension was investigated.

Conclusions:

The presented integration scheme offers an accurate and efficient approach for orbital optimization in electronic structure calculations.
This method has the potential to enhance the performance of computational chemistry software for general molecules.