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Molecular Orbital Theory I

Molecular Orbital Theory I

Overview of Molecular Orbital Theory

Molecular Orbital Theory II

Molecular Orbital Theory II

Molecular Orbital Energy Diagrams

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.

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...

Hybridization of Atomic Orbitals II

Hybridization of Atomic Orbitals II

sp3d and sp3d 2 Hybridization

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...

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Updated: Jun 6, 2025

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

Rigorous Formalization of Orbital Functionals.

Neil Qiang Su¹

¹Center for Theoretical and Computational Chemistry, Frontiers Science Center for New Organic Matter, State Key Laboratory of Advanced Chemical Power Sources, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education), Department of Chemistry, <a href="https://ror.org/01y1kjr75">Nankai University</a>, Tianjin 300071, China.

Physical Review Letters

|December 3, 2024

Summary

This summary is machine-generated.

This study rigorously formalizes orbital and occupation functionals using Clifford algebras, offering a superior alternative to traditional density functional theory methods for complex electronic systems.

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Area of Science:

Quantum chemistry
Theoretical physics
Computational materials science

Background:

Traditional density functional theory (DFT) methods face limitations with systems where electron densities do not map to non-interacting references.
Existing orbital-dependent functionals require rigorous mathematical formalization for broader applicability.

Purpose of the Study:

To rigorously formalize functionals dependent on occupied, unoccupied, or fractional orbitals.
To establish a variational principle for orbital and occupation optimization.
To present a novel perspective on orbital functionals as rigorous alternatives to DFT.

Main Methods:

Formalization using Clifford algebras.
Establishment of a variational principle for optimization.
Development of orbital (and occupation) optimization as a formal implementation method.

Main Results:

Rigorous mathematical framework for orbital and occupation functionals.
A variational principle enabling direct orbital and occupation optimization.
Demonstration of a method that circumvents limitations of standard Kohn-Sham DFT.

Conclusions:

Orbital and occupation functionals, formalized via Clifford algebras, offer a rigorous and superior alternative to traditional DFT.
These advanced functionals overcome limitations in describing systems with non-standard electron density mappings.
The established variational principle facilitates practical implementation and optimization.