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Hybridization of Atomic Orbitals II

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sp3d and sp3d 2 Hybridization

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

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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...

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...

Molecular Orbital Theory I

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Application of Deep Learning-Based Medical Image Segmentation via Orbital Computed Tomography

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A Reusable Library for Second-Order Orbital Optimization Using the Trust Region Method.

Jonas Greiner¹, Ida-Marie Høyvik², Susi Lehtola³

¹DTU Chemistry, Technical University of Denmark, Kemitorvet Bldg. 206, 2800 Kgs. Lyngby, Denmark.

Journal of Chemical Theory and Computation

|January 13, 2026

Summary

This summary is machine-generated.

We developed OpenTrustRegion, a new open-source library for trust region optimization. This robust software efficiently optimizes molecular orbitals in electronic-structure theory, benefiting various applications.

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Area of Science:

Computational chemistry
Electronic-structure theory
Software development

Background:

Molecular orbital optimization is crucial for accurate electronic structure calculations.
Existing methods may lack efficiency or broad applicability.
Trust region algorithms offer a robust framework for complex optimization problems.

Purpose of the Study:

To introduce OpenTrustRegion, a novel open-source library for second-order trust region optimization.
To provide a versatile tool for general-purpose molecular orbital optimization.
To facilitate the integration of advanced optimization techniques into electronic-structure software.

Main Methods:

Implementation of a second-order trust region algorithm.
Development of the reusable and permissibly licensed OpenTrustRegion software library.
Extensive benchmarking on self-consistent field calculations, orbital localization, and symmetrization tasks.

Main Results:

Demonstrated robustness and efficiency of the OpenTrustRegion library.
Successful application to challenging and pathological systems in electronic-structure theory.
Validation of the library's performance across various computational chemistry tasks.

Conclusions:

OpenTrustRegion provides a powerful and flexible tool for molecular orbital optimization.
The library's open-source and permissive license encourages widespread adoption.
This implementation advances the capabilities of electronic-structure software.