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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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An atomic orbital represents the three-dimensional regions in an atom where an electron has the highest probability to reside. The radial distribution function indicates the total probability of finding an electron within the thin shell at a distance r from the nucleus. The atomic orbitals have distinct shapes which are determined by l, the angular momentum quantum number. The orbitals are often drawn with a boundary surface, enclosing densest regions of the cloud.
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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: Oct 25, 2025

Picometer-Precision Atomic Position Tracking through Electron Microscopy
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Variable-Metric Localization of Occupied and Virtual Orbitals.

Ziling Luo1, Rustam Z Khaliullin1

  • 1Department of Chemistry, McGill University, 801 Sherbrooke St. West, Montreal QC H3A 0B8, Canada.

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PubMed
Summary

The variable-metric approach enables nonorthogonal orbitals without linear dependence, improving localization. New algorithms successfully localize virtual orbitals, showing significant improvements over orthogonal counterparts.

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Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Materials Science

Background:

  • The variable-metric approach allows nonorthogonal orbitals, enhancing the localization of occupied orbitals compared to orthogonal ones.
  • Developing reliable localization procedures for virtual orbitals remains a challenge in computational chemistry.

Purpose of the Study:

  • To design and test novel localization algorithms for virtual orbitals using the variable-metric approach.
  • To establish a straightforward and dependable method for localizing both occupied and virtual nonorthogonal orbitals.

Main Methods:

  • Implementation and testing of various localization algorithms: steepest descent, conjugate gradient (CG), limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), and trust-region (TR) methods.
  • Comparative analysis of algorithm performance for occupied and virtual nonorthogonal molecular orbitals (NLMOs).

Main Results:

  • The CG-based trust-region algorithm demonstrated superior performance for localizing occupied and virtual NLMOs.
  • L-BFGS and CG algorithms also reliably produced NLMOs, though often with higher computational costs.
  • Virtual NLMOs were found to be significantly more localized (13-18%) than their orthogonal counterparts.

Conclusions:

  • The developed variable-metric localization algorithms provide a reliable method for obtaining well-localized occupied and virtual orbitals.
  • The study introduces and validates the localization of virtual nonorthogonal molecular orbitals, a previously undescribed area.
  • The findings offer improved orbital localization for various systems, including molecules and periodic materials.