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The mathematical expression known as the wave function, ψ, contains information about each orbital and the wavelike properties of electrons in an isolated atom. When atoms are bound together in a molecule, the wave functions combine to produce new mathematical descriptions that have different shapes. This process of combining the wave functions for atomic orbitals is called hybridization and is mathematically accomplished by the linear combination of atomic orbitals. The new orbitals that...
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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.
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Conjugated dienes have lower heats of hydrogenation than cumulated and isolated dienes, making them more stable. The enhanced stabilization of conjugated systems can be understood from their π molecular orbitals.
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According to valence bond theory, a covalent bond results when: (1) an orbital on one atom overlaps an orbital on a second atom, and (2) the single electrons in each orbital combine to form an electron pair. The strength of a covalent bond depends on the extent of overlap of the orbitals involved. Maximum overlap is possible when the orbitals overlap on a direct line between the two nuclei.
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Related Experiment Video

Updated: Jan 5, 2026

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
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Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

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Pseudodiagonalization-based wavefunction optimization with contracted planewave basis functions.

Duncan W Stuart1, Nicholas J Mosey1

  • 1Department of Chemistry, 90 Bader Lane, Queen's University, Kingston, Ontario, Canada, K7L 3N6.

Journal of Computational Chemistry
|October 25, 2019
PubMed
Summary
This summary is machine-generated.

New methods accelerate electronic structure calculations using contracted planewave basis functions (CPWBFs). This approach speeds up wavefunction optimization by 6-8 times, offering significant computational advantages for large basis sets and hybrid functionals.

Keywords:
Hartree-Fockcontracted planewave basis functionsdensity functional theorydirect inversion of the iterative subspaceperiodic electronic structure calculationspseudodiagonalizationwavefunction convergence

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Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Materials Science

Background:

  • Electronic structure calculations are crucial for understanding molecular properties.
  • Contracted planewave basis functions (CPWBFs) offer an alternative to traditional basis sets.
  • Existing methods for CPWBFs can be computationally intensive, especially for large systems.

Purpose of the Study:

  • To develop and report methodological advances for optimizing wavefunctions using CPWBFs.
  • To enable efficient electronic structure calculations with CPWBFs.
  • To investigate the performance of CPWBF-based methods with hybrid exchange-correlation functionals.

Main Methods:

  • Developed an FCo-based wavefunction optimization technique.
  • Combined pseudodiagonalization, approximate virtual orbital energies, and iterative subspace optimization.
  • Utilized density functional theory (DFT) calculations.

Main Results:

  • Achieved wavefunction optimization speed-ups of approximately 6-8 times compared to traditional Fock matrix (F)-based methods.
  • Demonstrated that computational cost is relatively insensitive to basis set size.
  • Enabled the use of hybrid exchange-correlation (XC) functionals with only a small increase in computational effort.

Conclusions:

  • The reported FCo-based optimization method significantly enhances the efficiency of electronic structure calculations using CPWBFs.
  • This approach provides substantial computational benefits for large basis sets and complex molecular systems.
  • The method facilitates the application of advanced DFT functionals in CPWBF calculations.