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Crystal Field Theory - Octahedral Complexes

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.
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Electric Field of a Non Uniformly Charged Sphere

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Gauss's law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for determining the expressions for the electric field for a particular charge distribution if the electric flux is known.
Consider a non-uniformly charged sphere, for which the density of charge depends only on the distance from a point in space and not on the direction. Such a sphere has a spherically symmetrical charge distribution. Here, the electric...

Crystal Field Theory - Tetrahedral and Square Planar Complexes

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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,...

Magnetic Fields

Magnetic Fields

A moving charge or a current creates a magnetic field in the surrounding space, in addition to its electric field. The magnetic field exerts a force on any other moving charge or current that is present in the field. Like an electric field, the magnetic field is also a vector field. At any position, the direction of the magnetic field is defined as the direction in which the north pole of a compass needle points.
A magnetic field is defined by the force that a charged particle experiences...

Electric Potential Energy in a Uniform Electric Field

Electric Potential Energy in a Uniform Electric Field

When an electric field accelerates a free positive charge, it acquires kinetic energy. This process is analogous to an object being accelerated by a gravitational field as if the charge were going down an electrical hill where its electric potential energy is converted into kinetic energy, although, of course, the sources of the forces are very different. The electrostatic or Coulomb force acting on the positive test charge is conservative, which means that the work done on a test charge is...

Magnetic Field of a Solenoid

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A solenoid is a conducting wire coated with an insulating material, wound tightly in the form of a helical coil. The magnetic field due to a solenoid is the vector sum of the magnetic fields due to its individual turns. Therefore, for an ideal solenoid, the magnetic field within the solenoid is directly proportional to the number of turns per unit length and the current. Conversely, the magnetic field outside the solenoid is zero.
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Multiscale Sampling of a Heterogeneous Water/Metal Catalyst Interface using Density Functional Theory and Force-Field Molecular Dynamics

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Published on: April 12, 2019

Uniform magnetic fields in density-functional theory.

Erik I Tellgren¹, Andre Laestadius¹, Trygve Helgaker¹

¹Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.

The Journal of Chemical Physics

|January 15, 2018

Summary

This summary is machine-generated.

We introduce linear vector potential-DFT (LDFT), an intermediate theory between DFT and Current-Density Functional Theory (CDFT) for uniform magnetic fields. LDFT simplifies theoretical challenges found in CDFT.

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Area of Science:

Quantum Chemistry
Condensed Matter Physics
Theoretical Chemistry

Background:

Density-functional theory (DFT) is a cornerstone for electronic structure calculations.
Current-Density Functional Theory (CDFT) extends DFT to include magnetic fields but presents theoretical complexities.
A need exists for a formalism that bridges DFT and CDFT, offering a simpler yet powerful approach for magnetic field problems.

Purpose of the Study:

To develop a novel density-functional formalism, termed linear vector potential-DFT (LDFT).
To establish LDFT as an intermediate theory between conventional DFT and CDFT for uniform external magnetic fields.
To investigate and simplify theoretical issues inherent in CDFT within the LDFT framework.

Main Methods:

Construction of a density-functional formalism incorporating density, canonical momentum, and paramagnetic moment as basic variables.
Development of both a constrained-search formulation and a convex formulation using Legendre-Fenchel transformations.
Analysis of theoretical properties including N-representability, Hohenberg-Kohn-like mappings, and gauge invariance analogs.

Main Results:

LDFT is established as a viable intermediate theory for uniform magnetic fields.
Theoretical challenges in CDFT, such as N-representability and gauge invariance, are shown to have simplified analogs in LDFT.
Existence of minimizers in the constrained-search formulation is proven.
The distinct nature of energy additivity for non-interacting subsystems in LDFT versus CDFT is discussed.

Conclusions:

LDFT offers a simplified yet rigorous framework for studying systems under uniform magnetic fields.
The developed formalism provides new insights into the theoretical underpinnings of magnetic field effects in density-functional theory.
LDFT serves as a valuable theoretical tool, potentially simplifying complex calculations and analyses in quantum chemistry and condensed matter physics.