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Coupling interactions are strongest between NMR-active nuclei bonded to each other, where spin information can be transmitted directly through the pair of bonding electrons. While nuclei polarize their electrons to the opposite spins, the bonding electron pair has opposite spins. Configurations with antiparallel nuclear spins are expected to be lower in energy. When coupling makes antiparallel states more favorable, J is considered to have a positive value. The one-bond coupling constant, 1J,...
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When an object is in equilibrium, it is either at rest or moving with a constant velocity. There are two types of equilibrium: static and dynamic. Static equilibrium occurs when an object is at rest, while dynamic equilibrium occurs when an object is moving with a constant velocity. In both cases, there must be a balance of forces acting on the object.
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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.
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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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In bromoethane, the three methyl protons are coupled to the two methylene protons that are three bonds away. In accordance with the n+1 rule, the signal from the methyl protons is split into three peaks with 1:2:1 relative intensities. The methylene protons appear as a quartet, with the relative intensities of 1:3:3:1.
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Configuration Weights in Coupled-Cluster Theory.

Håkon Emil Kristiansen1, Håkon Kvernmoen2, Simen Kvaal1

  • 1Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.

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|March 3, 2025
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Summary
This summary is machine-generated.

We define coupled-cluster (CC) weights as expectation values of projection operators, enabling wave function analysis across various CC theories. Extended CC theory resolves issues with noninteracting subsystems, improving weight behavior.

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Area of Science:

  • Quantum Chemistry
  • Computational Chemistry
  • Theoretical Chemistry

Background:

  • Coupled-cluster (CC) theory is a powerful quantum chemistry method for electronic structure calculations.
  • Analyzing the weight of Slater determinants within CC wave functions is crucial for understanding electron correlation.
  • Existing methods for determinant weighting in CC theory have limitations, especially for complex systems.

Purpose of the Study:

  • To introduce a universally applicable definition for the weight of Slater determinants in coupled-cluster states.
  • To enable wave function analysis in various coupled-cluster formulations comparable to configuration-interaction methods.
  • To investigate the behavior and implications of these weights, particularly concerning orbital basis sets and subsystem interactions.

Main Methods:

  • Definition of determinant weight as the expectation value of a projection operator.
  • Application across diverse CC formalisms: conventional, perturbative, nonorthogonal orbital-optimized, and extended CC.
  • Numerical experiments on single-reference systems and systems with noninteracting subsystems.
  • Comparison with full configuration-interaction (FCI) wave functions and analysis in different determinant bases.

Main Results:

  • The proposed CC weights show excellent agreement with FCI weights for single-reference systems.
  • Orbital basis set insensitivity of truncated CC energies is reflected in the computed determinant weights.
  • Conventional CC parametrization can lead to unphysical weights for noninteracting subsystems.
  • Extended CC theory and quadratic CC theory demonstrate significant improvements in handling such systems.

Conclusions:

  • The introduced definition provides a robust tool for wave function analysis in a wide range of CC theories.
  • Extended CC theory is essential for accurately describing systems with noninteracting subsystems and avoiding ill-behaved weights.
  • Quadratic CC theory offers a promising avenue for improved accuracy and reliability in determinant weighting and wave function analysis.