Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Electronic Structure of Atoms02:28

Electronic Structure of Atoms

22.6K

An atom comprises protons and neutrons, which are contained inside the dense, central core called the nucleus, with electrons present around the nucleus. Taking into account the wave–particle duality of electrons and the uncertainty in position around the nucleus, quantum mechanics provides a more accurate model for the atomic structure. It describes atomic orbitals as the regions around the nucleus where electrons of discrete energy exist, characterized by four quantum...
22.6K
Hybridization of Atomic Orbitals I03:24

Hybridization of Atomic Orbitals I

47.4K
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...
47.4K
Hybridization of Atomic Orbitals II03:35

Hybridization of Atomic Orbitals II

32.6K
sp3d and sp3d 2 Hybridization
32.6K
The Aufbau Principle and Hund's Rule03:02

The Aufbau Principle and Hund's Rule

53.0K
To determine the electron configuration for any particular atom, we can build the structures in the order of atomic numbers. Beginning with hydrogen, and continuing across the periods of the periodic table, we add one proton at a time to the nucleus and one electron to the proper subshell until we have described the electron configurations of all the elements. This procedure is called the aufbau principle, from the German word aufbau (“to build up”). Each added electron occupies the...
53.0K
Molecular Orbital Theory II03:51

Molecular Orbital Theory II

19.4K
Molecular Orbital Energy Diagrams
19.4K
Atomic Orbitals02:44

Atomic Orbitals

33.9K
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.
33.9K

You might also read

Related Articles

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

Sort by
Same author

Reaching precise proton affinities in non-Born-Oppenheimer calculations.

The Journal of chemical physics·2026
Same author

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

Journal of chemical theory and computation·2026
Same author

OpenOrbitalOptimizer─A Reusable Open Source Library for Self-Consistent Field Calculations.

The journal of physical chemistry. A·2025
Same author

Density functional benchmark for quadruple hydrogen bonds.

Physical chemistry chemical physics : PCCP·2025
Same author

Systematic Study of Hard-Wall Confinement-Induced Effects on Atomic Electronic Structure.

The journal of physical chemistry. A·2025
Same author

Ensemble Generalization of the Perdew-Zunger Self-Interaction Correction: A Way Out of Multiple Minima and Symmetry Breaking.

Journal of chemical theory and computation·2024
Same journal

Modeling the Clustering of Fumaric/Maleic Acid with Water and Na<sup>+</sup>, Cl<sup>-</sup> Ions.

The journal of physical chemistry. A·2026
Same journal

Determining Binding Energies of Key Fluorinated Refrigerants 1,1,1,2-Tetrafluoroethane, 2,3,3,3-Tetrafluoropropene, and 3,3,3-Trifluoropropene.

The journal of physical chemistry. A·2026
Same journal

Kinetic and Mechanistic Insights into H-Abstraction and Subsequent Isomerization and Decomposition of Monoglyme and Key Combustion Intermediates.

The journal of physical chemistry. A·2026
Same journal

First-Principles Analysis of Protonation-Induced Electronic Effects in Tetrakis(<i>p</i>-aminophenyl)porphyrin (TAPP).

The journal of physical chemistry. A·2026
Same journal

Exploring the Reactivity of the CH Radical toward Nitrous Oxide in the Context of the Interstellar Medium.

The journal of physical chemistry. A·2026
Same journal

Infrared Photodissociation Spectroscopy of Benzene-V<sup>+</sup>(CO)<sub>n</sub> "Piano Stool" Cations.

The journal of physical chemistry. A·2026
See all related articles

Related Experiment Video

Updated: Jul 31, 2025

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids
08:04

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

Published on: May 27, 2020

8.5K

Atomic Electronic Structure Calculations with Hermite Interpolating Polynomials.

Susi Lehtola1,2

  • 1Molecular Sciences Software Institute, Blacksburg, Virginia 24061, United States.

The Journal of Physical Chemistry. A
|May 2, 2023
PubMed
Summary
This summary is machine-generated.

We introduce Hermite interpolating polynomials (HIPs) for atomic electronic structure calculations, showing they offer stable and accurate results, outperforming Lagrange interpolating polynomials (LIPs) in certain scenarios. HIPs provide reliable calculations even with large node counts and non-uniform grids.

More Related Videos

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
12:11

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

8.3K
Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
13:56

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

Published on: October 12, 2019

7.7K

Related Experiment Videos

Last Updated: Jul 31, 2025

Excitonic Hamiltonians for Calculating Optical Absorption Spectra and Optoelectronic Properties of Molecular Aggregates and Solids
08:04

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

Published on: May 27, 2020

8.5K
Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
12:11

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

8.3K
Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
13:56

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

Published on: October 12, 2019

7.7K

Area of Science:

  • Computational Quantum Chemistry
  • Materials Science
  • Numerical Analysis

Background:

  • Atomic electronic structure calculations are crucial for understanding molecular and material properties.
  • The finite element method (FEM) with numerical radial basis functions offers a flexible framework for these calculations.
  • Lagrange interpolating polynomials (LIPs) have been used as shape functions, but challenges exist at small radial distances (r).

Purpose of the Study:

  • To develop and evaluate stable evaluation methods for radial basis functions at small r within FEM.
  • To compare the performance of different shape functions, specifically LIPs and Hermite interpolating polynomials (HIPs), in atomic electronic structure calculations.
  • To assess the impact of HIPs on calculations involving meta-generalized gradient approximation (meta-GGA) functionals.

Main Methods:

  • Implementation of atomic electronic structure calculations using the finite element method.
  • Utilized radial basis functions of the form χμ(r) = r⁻¹Bμ(r).
  • Compared three types of shape functions: Lagrange interpolating polynomials (LIPs), analytical first-order Hermite interpolating polynomials (HIPs), and numerically implemented n-th order HIPs.

Main Results:

  • First-order HIPs demonstrate reliability with large node counts and non-uniform grids, yielding superior results compared to LIPs with the same number of basis functions.
  • HIP basis sets avoid discontinuities in the spin-σ local kinetic energy (τσ) observed with small LIP basis sets.
  • Most Minnesota meta-GGA functionals exhibit ill-behaved characteristics when used with HIPs, while HIPs offer explicit derivative control, though confining potentials remain a preferred option for basis set formation.

Conclusions:

  • Hermite interpolating polynomials (HIPs) offer a robust and accurate alternative to Lagrange interpolating polynomials (LIPs) for atomic electronic structure calculations within the finite element method.
  • HIPs provide improved stability and accuracy, particularly with non-uniform grids and in avoiding kinetic energy discontinuities.
  • Further investigation is needed regarding the behavior of specific meta-GGA functionals with HIPs.