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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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NMR-active nuclei have energy levels called 'spin states' that are associated with the orientations of their nuclear magnetic moments. In the absence of a magnetic field, the nuclear magnetic moments are randomly oriented, and the spin states are degenerate. When an external magnetic field is applied, the spin states have only 2 + 1 orientations available to them. A proton with = ½ has two available orientations. Similarly, for a quadrupolar nucleus with a nuclear spin value of...
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Atomic Nuclei: Nuclear Relaxation Processes01:23

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In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis.
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Quantifying Delocalization and Static Correlation Errors by Imposing (Spin)Population Redistributions through

Xeno De Vriendt1, Laurent Lemmens1, Stijn De Baerdemacker2

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Summary
This summary is machine-generated.

Density functional approximations often fail due to incorrect behavior at long bond distances. This study models these errors using constrained full configuration interaction, providing insights into exact functional requirements and quantifying approximation errors.

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Area of Science:

  • Quantum Chemistry
  • Computational Materials Science

Background:

  • Density functional approximations (DFAs) exhibit failures in the asymptotic limit of infinite bond distances.
  • These failures are linked to fractional (spin)population redistributions and the violation of flat-plane conditions.

Purpose of the Study:

  • To model fractional (spin)population redistributions and their impact on electronic structure calculations.
  • To computationally investigate the flat-plane conditions in the asymptotic limit.
  • To generate reference data for evaluating approximate methods.

Main Methods:

  • Constraining full configuration interaction (FCI) wave functions to impose fractional (spin)populations on atomic domains.
  • Generating N-representable electronic structure descriptions for small hydrogen chains at various bond distances.

Main Results:

  • Demonstrated the ability to computationally model the effects of flat-plane conditions in the asymptotic limit.
  • Provided chemical insight into the behavior of electronic systems at infinite bond distances.
  • Established a methodology to capture the impact of delocalization and static correlation errors.

Conclusions:

  • The proposed method accurately captures the effects of flat-plane conditions.
  • This approach can generate crucial reference data for assessing DFAs.
  • Enables quantification of delocalization and static correlation errors in approximate methods.