Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Angular Momentum

Angular Momentum

Angular momentum characterizes an object's rotational motion and is defined as the moment of its linear momentum about a specified point O. When a particle moves along a curved path in the x-y plane, the scalar formulation calculates the magnitude of its angular momentum, utilizing the moment arm (d), representing the perpendicular distance from point O to the line of action of the linear momentum. Despite being scalar in formulation, angular momentum is inherently a vector quantity. Its...

Angular Momentum: Single Particle

Angular Momentum: Single Particle

Angular momentum is directed perpendicular to the plane of the rotation, and its magnitude depends on the choice of the origin. The perpendicular vector joining the linear momentum vector of an object to the origin is called the “lever arm.” If the lever arm and linear momentum are collinear, then the magnitude of the angular momentum is zero. Therefore, in this case, the object rotates about the origin such that it lies on the rim of the circumference defined by the lever arm...

Angular Momentum about an Arbitrary Axis

Angular Momentum about an Arbitrary Axis

Imagine a rigid body with a mass denoted as 'm', which has its center of mass at point G and is rotating around an inertial reference frame. The angular momentum at an arbitrary point P can be calculated by taking the cross product of the position vector and linear momentum vector for each individual mass element.
The velocity of a mass element comprises its translational velocity and the relative velocity instigated by the body's rotation. Substituting the velocity equation into...

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.

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.

Molecular Orbital Theory II

Molecular Orbital Theory II

Molecular Orbital Energy Diagrams

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

Organophosphate ethyl paraoxon on Ag and Au surfaces: a density functional theory perspective.

Physical chemistry chemical physics : PCCP·2026

Same author

Mode-resolved electron-phonon fingerprints map topological protection in lithium beyond born-Oppenheimer.

Journal of physics. Condensed matter : an Institute of Physics journal·2026

Same author

Bidirectional reinforcement learning neural network for constrained molecular design.

Scientific reports·2025

Same author

VNFlow: integration of variational autoencoders and normalizing flows for novel molecular design.

Journal of cheminformatics·2025

Same author

A Polydopamine-Based Molecularly Imprinted Electrochemical Sensor for Fentanyl Determination.

ACS omega·2025

Same author

Comprehensive DFT investigation of small-molecule adsorption on the paradigm M-MOF-74 family of metal-organic frameworks.

Physical chemistry chemical physics : PCCP·2025

Search research articles

Related Experiment Video

Updated: Feb 24, 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

Orbital angular momentum eigenfunctions for fast and numerically stable evaluations of closed-form pseudopotential

Anguang Hu¹, Nora W C Chan¹, Brett I Dunlap²

¹Defence R&D Canada-Suffield Research Centre, P.O. Box 4000, Station Main, Medicine Hat, Alberta T1A 8K6, Canada.

The Journal of Chemical Physics

|August 24, 2017

Summary

This study simplifies calculating Gaussian pseudopotential matrix elements by reducing them to standard integrals. This method improves computational efficiency and removes singularities in pseudopotential calculations.

More Related Videos

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Published on: May 27, 2020

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Related Experiment Videos

Last Updated: Feb 24, 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

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids

Published on: May 27, 2020

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Area of Science:

Computational Chemistry
Quantum Chemistry
Theoretical Physics

Background:

Calculating pseudopotential matrix elements is crucial for electronic structure calculations.
Existing methods for Gaussian pseudopotential matrix elements can be computationally intensive and suffer from singularities.
Gaussian orbitals are widely used in quantum chemistry due to their computational advantages.

Purpose of the Study:

To develop a more efficient and robust method for computing s-type Gaussian pseudopotential matrix elements.
To express these matrix elements in terms of simpler, well-known integrals.
To eliminate singularities encountered in previous computational approaches.

Main Methods:

Reduction of pseudopotential matrix elements to three-center overlap or nuclear integrals.
Application of solid-harmonic differentiation to incorporate orbital angular momentum.
Utilizing the solid-harmonic addition theorem to factor integrals into products of one-dimensional integrals and angular factors.

Main Results:

Developed a method that expresses pseudopotential matrix elements using familiar integrals.
Demonstrated that even powers of radial distance correspond to overlap integrals, while odd powers relate to nuclear integrals.
Showed that solid harmonics eliminate singularities, leading to a more stable computational procedure.

Conclusions:

The proposed method offers a significant simplification for computing Gaussian pseudopotential matrix elements.
Precomputing angular factors allows for efficient looping over Gaussian exponents, enhancing computational speed.
This approach provides a singularity-free and computationally advantageous alternative for quantum chemical calculations.