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Band Theory02:35

Band Theory

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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.
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Isolated atoms have discrete energy levels that are well described by the Bohr model. And, it quantifies the energy of an electron in a hydrogen atom as En. Higher quantum numbers 'n' yield less negative, closer electron energy levels.
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The vacuum level denotes the energy threshold required for an electron to escape from a material surface. It is usually positioned above the conduction band of a semiconductor and acts as a benchmark for comparing electron energies within various materials.
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Fermi Level01:18

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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.
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There is variation in the electrical conductivity of materials - metals, semiconductors, and insulators that are showcased with the help of the energy band diagrams.
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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.
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Updated: Mar 22, 2026

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
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Using Wannier functions to improve solid band gap predictions in density functional theory.

Jie Ma1, Lin-Wang Wang1

  • 1Joint Center for Artificial Photosynthesis and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.

Scientific Reports
|April 27, 2016
PubMed
Summary

This study introduces a new method using Wannier functions to accurately calculate band gaps in solids, overcoming limitations of previous techniques for extended electronic states. The approach also accurately predicts ionization potentials and electron affinities for molecules.

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Area of Science:

  • Condensed matter physics
  • Quantum chemistry
  • Materials science

Background:

  • Calculating electronic properties like band gaps in solids is crucial for materials science.
  • Existing methods using the straight-line condition on eigen states fail for solids due to extended orbitals.
  • The self-interaction error is a known issue in density functional theory (DFT) calculations.

Purpose of the Study:

  • To develop a robust method for accurately calculating band gaps in solids.
  • To address the limitations of applying the straight-line condition to extended electronic states in solids.
  • To provide a parameter-free, computationally efficient approach for electronic structure calculations.

Main Methods:

  • Extended the straight-line condition of total energy to fractional electron addition/removal on Wannier functions.
  • Constructed Wannier functions within occupied and unoccupied subspaces.
  • Applied the method to calculate band gaps for solids and ionization potentials/electron affinities for molecules.

Main Results:

  • Accurate band gaps for solids were obtained, showing good agreement with experimental data.
  • The method effectively removes self-interaction energies associated with Wannier functions.
  • Accurate ionization potentials and electron affinities were calculated for molecules, comparable to experimental values.

Conclusions:

  • The novel straight-line condition applied to Wannier functions provides an accurate and efficient method for calculating electronic properties of solids and molecules.
  • This approach offers a viable alternative to standard methods like LDA+U, without adjustable parameters.
  • The method successfully overcomes the self-interaction error in extended systems, paving the way for more reliable materials simulations.