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Related Concept Videos

Atomic Nuclei: Nuclear Spin State Overview01:03

Atomic Nuclei: Nuclear Spin State Overview

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 one, the...
The Pauli Exclusion Principle03:06

The Pauli Exclusion Principle

The arrangement of electrons in the orbitals of an atom is called its electron configuration. We describe an electron configuration with a symbol that contains three pieces of information:
¹H NMR: Interpreting Distorted and Overlapping Signals01:02

¹H NMR: Interpreting Distorted and Overlapping Signals

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.
As Δν decreases and the signals move closer, the doublets appear increasingly distorted. The intensities of the inner lines increase at the cost of those of the outer lines as the signals are slanted or...
Atomic Nuclei: Nuclear Spin State Population Distribution01:14

Atomic Nuclei: Nuclear Spin State Population Distribution

Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.
Valence Bond Theory02:42

Valence Bond Theory

Coordination compounds and complexes exhibit different colors, geometries, and magnetic behavior, depending on the metal atom/ion and ligands from which they are composed. In an attempt to explain the bonding and structure of coordination complexes, Linus Pauling proposed the valence bond theory, or VBT, using the concepts of hybridization and the overlapping of the atomic orbitals. According to VBT, the central metal atom or ion (Lewis acid) hybridizes to provide empty orbitals of suitable...
Valence Bond Theory02:45

Valence Bond Theory

Overview of Valence Bond Theory

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Related Experiment Video

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Experimental Methods for Spin- and Angle-Resolved Photoemission Spectroscopy Combined with Polarization-Variable Laser
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Experimental Methods for Spin- and Angle-Resolved Photoemission Spectroscopy Combined with Polarization-Variable Laser

Published on: June 28, 2018

Considerations on describing non-singlet spin states in variational second order density matrix methods.

Helen van Aggelen1, Brecht Verstichel, Patrick Bultinck

  • 1Department of Inorganic and Physical Chemistry, Ghent University, Krijgslaan 281 S3, 9000 Ghent, Belgium. helen.vanaggelen@ugent.be

The Journal of Chemical Physics
|January 14, 2012
PubMed
Summary

This study addresses challenges in calculating spin states for non-singlet molecules using second-order density matrix theory. New constraints improve accuracy for molecular dissociation and spin degeneracy, crucial for accurate chemical bonding descriptions.

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Area of Science:

  • Quantum chemistry
  • Computational chemistry
  • Theoretical chemistry

Background:

  • Most variational second-order density matrix calculations focus on singlet states, neglecting non-singlet molecules.
  • Deriving a second-order density matrix from a proper N-electron spin state is challenging due to its focus on one- and two-particle interactions.

Purpose of the Study:

  • To propose and evaluate two main approaches for a consistent description of spin in second-order density matrix theory.
  • To assess constraints derived from pure and ensemble spin states for molecular calculations.
  • To apply these methods to potential energy surfaces of oxygen and carbon dimers.

Main Methods:

  • Investigated constraints from pure spin states.
  • Investigated constraints from ensembles of spin states.
  • Applied methods to potential energy surfaces of O2 and C2 dimers.

Main Results:

  • Identified two major shortcomings of applied spin constraints: lack of size consistency and failure to reproduce spin multiplet degeneracy.
  • Observed that spin constraints are weaker for dissociated molecules than for their separate dissociation products.
  • Found that energy is a convex function of spin projection under both pure and ensemble spin state conditions, potentially altering bonding descriptions.

Conclusions:

  • Subspace energy constraints can correct issues like nondegeneracy, size-inconsistency, and unphysical dissociation in the dissociation limit.
  • The choice of spin constraint significantly impacts the calculated potential energy surfaces and bonding picture.
  • Maximal spin projection offers the most constrained energy but is computationally more expensive than spin-averaged ensembles.