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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.

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.

Electron Orbital Model

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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...

Valence Bond Theory and Hybridized Orbitals

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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...

Molecular Orbital Theory II

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The molecular orbital theory describes the distribution of electrons in molecules in a manner similar to the distribution of electrons in atomic orbitals. The region of space in which a valence electron in a molecule is likely to be found is called a molecular orbital. Mathematically, the linear combination of atomic orbitals (LCAO) generates molecular orbitals. Combinations of in-phase atomic orbital wave functions result in regions with a high probability of electron density, while...

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Orbital energies and nuclear forces in DFT: Interpretation and validation.

Rubén Laplaza^1,2, Carlos Cárdenas^3,4, Patrick Chaquin¹

¹Laboratoire de Chimie Théorique, LCT, Sorbonne Université, CNRS, Paris, France.

Journal of Computational Chemistry

|December 10, 2020

Summary

Dynamical orbital forces, derived from conceptual DFT, reveal molecular orbital bonding and antibonding character. This method offers a general approach to interpret ab initio calculations for various chemical systems.

Keywords:

conceptual density functional theory density functional theory dynamic orbital forces nuclear Fukui function nuclear forces

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Area of Science:

Quantum Chemistry
Theoretical Chemistry
Computational Chemistry

Background:

Molecular orbital theory describes chemical bonding.
Orbital energy derivatives, or dynamical orbital forces, relate to orbital character.
Existing methods often rely on Koopmans' theorem.

Purpose of the Study:

To generalize the derivation of dynamical orbital forces.
To demonstrate the applicability of dynamical orbital forces beyond Koopmans' theorem.
To showcase the utility of dynamical orbital forces in interpreting chemical phenomena.

Main Methods:

Derivation of dynamical orbital forces from conceptual Density Functional Theory (DFT).
Numerical validation using Kohn-Sham DFT for valence orbitals.
Application to aromatic, antiaromatic, and excited state systems.

Main Results:

A more general derivation of dynamical orbital forces is presented.
The approach is validated for Kohn-Sham DFT.
Useful insights are gained for aromaticity, antiaromaticity, and excited states.

Conclusions:

Dynamical orbital forces provide a versatile tool for analyzing molecular orbitals.
This method bridges wavefunction and density-based ab initio calculations.
It offers a force and occupation-based interpretation of computational results.