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Related Concept Videos

Crystal Field Theory - Octahedral Complexes02:58

Crystal Field Theory - Octahedral Complexes

Crystal Field Theory
To explain the observed behavior of transition metal complexes (such as colors), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It helps to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.
CFT focuses on...
Crystal Density01:19

Crystal Density

The crystal lattice structure of a material allows us to determine how many molecules exist in its unit cell. With this information, alongside the unit-cell parameters - three distance parameters (a, b, c) and three angular parameters (α, β, γ).Density (ρ) = (Z × M) / (a × b × c × NA)where:Z is the number of formula units per unit cellM is the molar mass of the substancea, b, and c are the edge lengths of the unit cellNA is Avogadro’s numberFor a simple cubic lattice, atoms are located only at...
Crystal Field Theory - Tetrahedral and Square Planar Complexes02:46

Crystal Field Theory - Tetrahedral and Square Planar Complexes

Tetrahedral Complexes
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
Law of Rational Indices01:29

Law of Rational Indices

The Law of rational indices is a fundamental principle in the field of crystallography. According to this law, the intercepts of a crystal face along the crystallographic axes (the three-dimensional axes along which a crystal is measured) can be expressed as either equivalent to the unit intercepts (a, b, c) or simple whole number multiples of them. These multiples are typically denoted as na, n'b, and n''c, where n, n', and n'' are simple whole numbers.To illustrate, consider a crystal with...
Van der Waals Equation01:10

Van der Waals Equation

The ideal gas law is an approximation that works well at high temperatures and low pressures. The van der Waals equation of state (named after the Dutch physicist Johannes van der Waals, 1837−1923) improves it by considering two factors.
First, the attractive forces between molecules, which are stronger at higher densities and reduce the pressure, are considered by adding to the pressure a term equal to the square of the molar density multiplied by a positive coefficient a. Second, the volume...
Imperfections in Crystal Structure: Stoichiometric Point Defects01:26

Imperfections in Crystal Structure: Stoichiometric Point Defects

Schottky defects arise when some lattice points in a crystal, such as those in NaCl, remain unoccupied, creating lattice vacancies without disturbing the overall electrical neutrality of the crystal. This defect is common in ionic crystals where the positive and negative ions are similar in size, as seen in sodium chloride and cesium chloride. The presence of Schottky defects enables the crystal to conduct electricity to a small extent through an ionic mechanism. Electric fields cause nearby...

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Related Experiment Video

Updated: May 14, 2026

Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics
10:52

Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics

Published on: April 12, 2019

Classical density functional theory and the phase-field crystal method using a rational function to describe the

N Pisutha-Arnond1, V W L Chan, M Iyer

  • 1Materials Science and Engineering Department, University of Michigan, Ann Arbor, Michigan 48109, USA. nirand@umich.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 16, 2013
PubMed
Summary
This summary is machine-generated.

A new rational function fit (RFF) approximates the two-body direct correlation function (DCF), reducing computational costs for classical density functional theory (CDFT) and improving phase-field crystal (PFC) models for materials science.

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Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package
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Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

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Measurements of Long-range Electronic Correlations During Femtosecond Diffraction Experiments Performed on Nanocrystals of Buckminsterfullerene
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Measurements of Long-range Electronic Correlations During Femtosecond Diffraction Experiments Performed on Nanocrystals of Buckminsterfullerene

Published on: August 22, 2017

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Last Updated: May 14, 2026

Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics
10:52

Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics

Published on: April 12, 2019

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package
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Measurements of Long-range Electronic Correlations During Femtosecond Diffraction Experiments Performed on Nanocrystals of Buckminsterfullerene
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Measurements of Long-range Electronic Correlations During Femtosecond Diffraction Experiments Performed on Nanocrystals of Buckminsterfullerene

Published on: August 22, 2017

Area of Science:

  • Computational Materials Science
  • Statistical Mechanics
  • Condensed Matter Physics

Background:

  • Classical Density Functional Theory (CDFT) and Phase-Field Crystal (PFC) methods are crucial for materials simulations.
  • The computational demand of these methods often limits their application.
  • Accurate representation of the two-body direct correlation function (DCF) is essential for predictive power.

Purpose of the Study:

  • To develop a computationally efficient approach for representing the two-body DCF.
  • To enhance the predictive capabilities of CDFT and PFC methods.
  • To enable real-space reformulations for nonperiodic systems and adaptive grids.

Main Methods:

  • Introduction of a rational function fit (RFF) approximation for the two-body DCF in Fourier space.
  • Empirical parametrization of the two-body DCF using RFF.
  • Application of RFF within CDFT and PFC frameworks.

Main Results:

  • RFF effectively approximates the two-body DCF, capturing essential short-wavelength contributions.
  • Thermodynamic properties of solids and liquids calculated using RFF-parametrized DCF match CDFT results with full DCF at reduced computational cost.
  • RFF improves DCF representation in PFC simulations.

Conclusions:

  • The RFF approach significantly alleviates computational demands in CDFT and enhances PFC predictive power.
  • RFF facilitates real-space reformulation, enabling the study of nonperiodic systems and the use of flexible grid strategies.
  • This method offers a computationally efficient and versatile tool for materials simulations.