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Hybridization of Atomic Orbitals II03:35

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sp3d and sp3d 2 Hybridization
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The coupling interactions of nuclei across four or more bonds are usually weak, with J values less than 1 Hz. While these are usually not observed in spectra, the presence of multiple bonds along the coupling pathway can result in observable long-range coupling.
In alkenes, spin information is communicated via σ–π overlap, as seen in allylic (four-bond) and homoallylic (five-bond) couplings. These coupling interactions are stronger when the σ bond is parallel to the alkene π orbitals.
Hybridization of Atomic Orbitals I03:24

Hybridization of Atomic Orbitals I

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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Spin systems where the difference in chemical shifts of the coupled nuclei is greater than ten times J are called first-order spin systems. These nuclei are weakly coupled, and their chemical shifts and coupling constant can generally be estimated from the well-separated signals in the spectrum.
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Vicinal or three-bond coupling is commonly observed between protons attached to adjacent carbons. Here, nuclear spin information is primarily transferred via electron spin interactions between adjacent C‑H bond orbitals. This generally favors the antiparallel arrangement of spins, so 3J values are usually positive.
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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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On the Equivalence of Two-Point Basis-Set Extrapolations and Robust Parameterization for Coupled-Cluster and

Mark A Iron1

  • 1Computational Chemistry Unit, Department of Chemical Research Support, Weizmann Institute of Science, Rehovot 7610001, Israel.

The Journal of Physical Chemistry. A
|May 27, 2026
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Basis-set extrapolation (BSE) to the complete basis set (CBS) limit is crucial for accurate computational chemistry. This study shows common BSE approximations are mathematically equivalent and yield similar results across methods, providing new extrapolation parameters.

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Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Theoretical Chemistry

Background:

  • Basis-set extrapolation (BSE) is essential for achieving high-accuracy thermochemical data computationally.
  • Approximations in BSE are common but their impact on accuracy across different computational methods requires careful assessment.

Purpose of the Study:

  • To evaluate common approximations used in basis-set extrapolation (BSE) to the complete basis set (CBS) limit.
  • To investigate the transferability of BSE parameters between different computational methods, particularly coupled-cluster and density functional theory.
  • To provide updated Schwenke-type extrapolation parameters for practical use.

Main Methods:

  • Analysis of common two-basis-set extrapolation schemes (exponential, exponential-square root, inverse power).
  • Mathematical reduction of these schemes to a Schwenke-style correction.
  • Fitting of Schwenke-type two-point extrapolation parameters for coupled-cluster and double-hybrid density functional theory methods.

Main Results:

  • Three common two-point extrapolation schemes mathematically reduce to a Schwenke-style correction.
  • The performance difference between method-specific and transferable BSE parameters is minimal on benchmark datasets.
  • New Schwenke-type extrapolation parameters are provided for various basis sets and computational methods.

Conclusions:

  • Common BSE approximations are mathematically equivalent, simplifying their application.
  • BSE parameters derived from coupled-cluster methods perform well when applied to other methods, reducing the need for method-specific fits.
  • The provided parameters facilitate accurate CBS limit calculations in computational thermochemistry.