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Functions of Three or More Variables01:31

Functions of Three or More Variables

A function of three variables assigns a single real number to each point in three-dimensional space. Every point is identified by its Cartesian coordinates, x, y, and z, and the function maps this ordered triple to a scalar value. Such functions are commonly used to describe physical quantities that vary throughout space.A representative example is the electric potential generated by a point charge. In this case, the potential at a given location depends only on the distance from the charge. If...
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
Woodward–Hoffmann Selection Rules and Microscopic Reversibility01:34

Woodward–Hoffmann Selection Rules and Microscopic Reversibility

Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...
The Phase Rule01:20

The Phase Rule

The phase rule describes the relationship between the variance (degrees of freedom), the number of components, and the number of phases in a system at equilibrium.Variance is a concept that denotes the number of independent intensive properties (properties are those that do not depend on the amount of material in the system), such as temperature, pressure, and composition, that can be altered without impacting the number of phases in equilibrium.In a single-component system, such as pure water,...
Separable Differential Equations01:20

Separable Differential Equations

A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...

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Related Experiment Video

Updated: Jun 26, 2026

Creating Objects and Object Categories for Studying Perception and Perceptual Learning
14:38

Creating Objects and Object Categories for Studying Perception and Perceptual Learning

Published on: November 2, 2012

Variational, V-representable, and variable-occupation-number perturbation theories.

Brett I Dunlap1

  • 1Theoretical Chemistry Section, US Naval Research Laboratory, Code 6189, Washington, DC 20375-5342, USA. dunlap@nrl.navy.mil

The Journal of Chemical Physics
|January 7, 2009
PubMed
Summary

This study introduces a new method for density-functional perturbation theory using variationally fitted potentials. The approach simplifies calculations and accurately determines molecular properties like hardness and hyperhardness.

Related Experiment Videos

Last Updated: Jun 26, 2026

Creating Objects and Object Categories for Studying Perception and Perceptual Learning
14:38

Creating Objects and Object Categories for Studying Perception and Perceptual Learning

Published on: November 2, 2012

Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Density-Functional Theory

Background:

  • Density-functional perturbation theory (DFPT) is crucial for calculating molecular properties.
  • Existing methods can be computationally intensive, especially for higher-order derivatives.
  • The development of efficient and accurate DFPT methods is an ongoing area of research.

Purpose of the Study:

  • To develop and present a novel formulation of density-functional perturbation theory.
  • To incorporate variationally fitted Kohn-Sham (KS) potentials for enhanced efficiency.
  • To accurately compute molecular properties, including energy derivatives, hardness, and hyperhardness.

Main Methods:

  • Utilized density-functional perturbation theory with variationally fitted Kohn-Sham potentials.
  • Employed a commutation relation between Fock and density matrices to determine density matrix elements.
  • Expanded the KS potential in a finite basis to create the Sambe-Felton (SF) potential, reducing computational complexity.
  • Applied matrix inversion instead of iterative coupled-perturbed equations for efficiency.

Main Results:

  • The proposed method accurately determines the effect of changing occupation numbers in DFPT.
  • Reduced the dimensionality of perturbation theory from N(2) to N using the SF potential.
  • Achieved precise second and third derivatives of energy with respect to occupation number.
  • Computed hardness and hyperhardness, finding them largely independent of potential constraints.

Conclusions:

  • The new DFPT approach with variational fitting offers a computationally efficient and accurate method.
  • Analytic derivatives are accurate to machine precision across various constraints and fitting basis sets.
  • The method provides a robust framework for calculating electronic properties of molecules.