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

Classification of Systems-I01:26

Classification of Systems-I

168
Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
168
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

85
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
85
Linear Circuits01:17

Linear Circuits

381
A linear circuit is characterized by its output having a direct proportionality to its input, adhering to the linearity property, which encompasses the principles of homogeneity (scaling) and additivity. Homogeneity dictates that when the input, also referred to as the excitation, is multiplied by a constant factor, the output, known as the response, is correspondingly scaled by the same constant factor. For instance, if the current is multiplied by a constant 'k,' the voltage likewise...
381
Linear time-invariant Systems01:23

Linear time-invariant Systems

211
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
211
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

62
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
62
Conservation of Linear Momentum for a System of Particles01:28

Conservation of Linear Momentum for a System of Particles

214
In the dynamic realm of billiards, a fascinating interplay of forces governs the motion of cue balls and stationary balls. When the cue ball collides with a stationary ball, linear momentum is exchanged. The cue ball imparts a fraction of its linear momentum to the stationary ball, causing the cue ball to decelerate while initiating the motion of the stationary ball.
The impulsive force at play during this interaction is of extremely short duration, rendering its impulse negligible. When...
214

You might also read

Related Articles

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

Sort by
Same author

Many-electron systems with fractional electron number and spin: Exact properties above and below the equilibrium total spin value.

The Journal of chemical physics·2026
Same author

Plateaus in the Potentials of Density-Functional Theory: Analytical Derivation and Useful Approximations.

Journal of chemical theory and computation·2025
Same author

Spin migration in density functional theory: Energy, potential, and density perspectives.

The Journal of chemical physics·2025
Same author

Ensemble Ground State of a Many-Electron System with Fractional Electron Number and Spin: Piecewise-Linearity and Flat-Plane Condition Generalized.

The journal of physical chemistry letters·2024
Same author

Ionization potentials and fundamental gaps in atomic systems from the Ensemble-DFT approach.

The Journal of chemical physics·2023
Same author

Charge-Transfer Steps in Density Functional Theory from the Perspective of the Exact Electron Factorization.

The journal of physical chemistry letters·2021

Related Experiment Video

Updated: Jun 5, 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

8.4K

How the Piecewise-Linearity Requirement for the Density Affects Quantities in the Kohn-Sham System.

Eli Kraisler1

  • 1Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, 9190401 Jerusalem, Israel.

Journal of Chemical Theory and Computation
|December 16, 2024
PubMed
Summary

Kohn-Sham density functional theory (KS-DFT) calculations are improved by understanding how electron density changes linearly with electron number. This study reveals constraints on KS quantities, aiding error reduction in open systems.

More Related Videos

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

8.5K
Evaluation of the Curing of Adhesive Systems by Rheological and Thermal Testing
09:06

Evaluation of the Curing of Adhesive Systems by Rheological and Thermal Testing

Published on: July 3, 2020

7.2K

Related Experiment Videos

Last Updated: Jun 5, 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

8.4K
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

8.5K
Evaluation of the Curing of Adhesive Systems by Rheological and Thermal Testing
09:06

Evaluation of the Curing of Adhesive Systems by Rheological and Thermal Testing

Published on: July 3, 2020

7.2K

Area of Science:

  • Computational Quantum Chemistry
  • Materials Science
  • Condensed Matter Physics

Background:

  • Kohn-Sham density functional theory (KS-DFT) is a widely used method for electronic structure calculations.
  • KS-DFT relies on an auxiliary system of non-interacting electrons to model the density of a real, interacting system.
  • The exact density in KS-DFT exhibits piecewise-linearity with respect to the number of electrons (N).

Purpose of the Study:

  • To investigate how the piecewise-linearity of the exact interacting density is manifested in the Kohn-Sham system.
  • To explore the implications of piecewise-linearity for KS quantities, particularly the total electron density, KS subdensities, and highest occupied (HOMO) orbital density.
  • To analyze common approximations for the HOMO, including frozen and linear regimes, in light of piecewise-linearity.

Main Methods:

  • Formulation of KS quantities using a two-point Taylor expansion in the number of electrons (N).
  • Derivation of analytical results based on the piecewise-linearity requirement.
  • Numerical investigation employing various exchange-correlation approximations to validate analytical findings.

Main Results:

  • The study establishes restrictions on KS expansion coefficients imposed by the piecewise-linearity of the exact density.
  • Analytical insights into the behavior of total electron density, KS subdensities, and HOMO orbital density are presented.
  • Numerical results confirm the analytical predictions across different exchange-correlation approximations.

Conclusions:

  • Understanding and enforcing piecewise-linearity in KS-DFT is crucial for accurate calculations.
  • The findings provide a theoretical framework to address and mitigate density-driven errors in KS-DFT, especially for open systems and ensembles.
  • This work contributes to the development of more robust and reliable DFT methods.