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Band Theory

Band Theory

When two or more atoms come together to form a molecule, their atomic orbitals combine and molecular orbitals of distinct energies result. In a solid, there are a large number of atoms, and therefore a large number of atomic orbitals that may be combined into molecular orbitals. These groups of molecular orbitals are so closely placed together to form continuous regions of energies, known as the bands.
The energy difference between these bands is known as the band gap.
Conductor, Semiconductor,...

Debye–Huckel–Onsager Conductance Equation

Debye–Huckel–Onsager Conductance Equation

The Debye-Hückel-Onsager equation is a cornerstone of physical chemistry, providing a method to determine the molar conductance (Λm) and molar conductance at infinite dilution (Λ°m) for uni-univalent electrolytes.Uni-univalent electrolytes are electrolytes that dissociate in solution to produce one cation with a +1 charge and one anion with a –1 charge per formula unit.This equation addresses two crucial phenomena: the asymmetry effect and the electrophoretic effect.

Imperfections in Crystal Structure: Stoichiometric Point Defects

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...

Fermi Level

Fermi Level

The Fermi-Dirac function is represented by an S-shaped curve indicating the probability of an energy state being occupied by an electron at a given temperature. The Fermi level is the energy level at which there is a fifty percent chance of finding an electron, and it is positioned between the lower-energy valence band and the higher-energy conduction band.
At absolute zero temperature, electrons fill all energy states up to the Fermi level, leaving upper states empty. As the temperature rises,...

Trends in Lattice Energy: Ion Size and Charge

Trends in Lattice Energy: Ion Size and Charge

An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a compound is a measure of the strength of this attraction. The lattice energy (ΔHlattice) of an ionic compound is defined as the energy required to separate one mole of the solid into its component gaseous ions. For the ionic solid sodium chloride, the lattice energy is the enthalpy change of the process:

Linear Approximation in Frequency Domain

Linear Approximation in Frequency Domain

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

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

Published on: May 27, 2020

Deviations from piecewise linearity in the solid-state limit with approximate density functionals.

Vojtěch Vlček¹, Helen R Eisenberg², Gerd Steinle-Neumann¹

¹Bayerisches Geoinstitut, Universität Bayreuth, D-95440 Bayreuth, Germany.

The Journal of Chemical Physics

|January 24, 2015

Summary

This summary is machine-generated.

Deviation from piecewise linearity in density functional theory (DFT) does not reliably predict electronic structure quality. Curvature vanishes in solid-state limits, suggesting new criteria for functional development are needed.

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Published on: October 12, 2019

Area of Science:

Computational Quantum Chemistry
Materials Science
Solid-State Physics

Background:

Exact density functional theory (DFT) predicts total ground-state energy as piecewise linear between integer electron numbers.
Deviations from piecewise linearity indicate poor predictive power for electronic properties like ionization energies and band gaps.
The curvature, a measure of deviation from linearity, is explored in the solid-state limit.

Purpose of the Study:

To investigate the behavior of curvature in the solid-state limit using two distinct approaches: large finite systems and periodic boundary conditions.
To determine if curvature can serve as a diagnostic or constructive tool for density functional approximations in solids.
To propose a new criterion for evaluating and constructing DFT functionals based on curvature's size-dependence.

Main Methods:

Examined the curvature of the ground-state energy with respect to electron number in two limits: large finite systems and periodic crystals with increasing cell sizes.
Analyzed the impact of compensating background charge in the periodic case on the curvature's convergence.
Investigated the contribution of self-interaction error to curvature in large finite systems.

Main Results:

Curvature approaches zero in both the large finite system and periodic limits, irrespective of the functional's predictive accuracy.
The convergence rate of curvature differs between the two limits due to the presence of background charge in periodic systems.
Self-interaction of the highest occupied eigenstate significantly influences curvature in large finite systems.

Conclusions:

Curvature is not a reliable indicator of functional performance for electronic structure in solids.
A novel criterion for functional development, based on the size-dependence of curvature, is proposed.
Computational studies on solids, nanocrystals, and molecular chains validate the findings and the proposed criterion.