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Related Concept Videos

¹H NMR: Complex Splitting01:13

¹H NMR: Complex Splitting

A proton M that is coupled to a proton X results in doublet signals for M. However, NMR-active nuclei can be simultaneously coupled to more than one nonequivalent nucleus. When M is coupled to a second proton A, such as in styrene oxide, each peak in the doublet is split into another doublet.
Splitting diagrams or splitting tree diagrams are routinely used to depict such complex couplings. While drawing splitting diagrams, the splitting with the larger coupling constant is usually applied first.
Hückel's Rule Diagram of π MOs: Frost Circle01:08

Hückel's Rule Diagram of π MOs: Frost Circle

The Frost circle or the inscribed polygon method is a graphical method for determining the relative energies of π molecular orbitals (MOs) for planar, fully conjugated, and monocyclic compounds. This method was first described by A. A. Frost and Boris Musulin in 1953.
A Frost circle is constructed by drawing a polygon whose number of edges is equal to the number of carbons of the given cyclic system, with one of the vertices pointing down. Then, a circle is drawn enclosing the polygon so that...
Spin–Spin Coupling Constant: Overview01:08

Spin–Spin Coupling Constant: Overview

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.
Qualitatively, any spin plus-half nucleus polarizes the spins of its electrons to the minus-half state. Consequently, the paired electron in the hydrogen–carbon bond must have a...

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Related Experiment Video

Updated: Jun 4, 2026

All-electronic Nanosecond-resolved Scanning Tunneling Microscopy: Facilitating the Investigation of Single Dopant Charge Dynamics
11:33

All-electronic Nanosecond-resolved Scanning Tunneling Microscopy: Facilitating the Investigation of Single Dopant Charge Dynamics

Published on: January 19, 2018

Ring-polymer instanton method for calculating tunneling splittings.

Jeremy O Richardson1, Stuart C Althorpe

  • 1Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom.

The Journal of Chemical Physics
|February 10, 2011
PubMed
Summary
This summary is machine-generated.

This study simplifies semiclassical instanton calculations for quantum tunneling splitting. The new method uses matrix determinants and a linear polymer model, proving computationally practical for complex molecules.

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Area of Science:

  • Quantum mechanics
  • Chemical physics
  • Computational chemistry

Background:

  • Semiclassical instanton theory is crucial for understanding quantum tunneling.
  • Calculating tunneling splitting often involves complex functional determinants.

Purpose of the Study:

  • To rederive the semiclassical instanton expression for tunneling splitting.
  • To simplify the mathematical treatment using matrix determinants.
  • To develop a computationally practical method for tunneling calculations.

Main Methods:

  • Utilized the ring-polymer representation of the quantum partition function.
  • Replaced functional determinants with matrix determinants.
  • Derived an expression for instanton tunneling splitting using a linear polymer model.

Main Results:

  • The new approach simplifies the mathematics of instanton calculations.
  • Identified a minimum on the potential surface of a linear polymer as key.
  • Demonstrated computational practicality through calculations on HO(2) and malonaldehyde.

Conclusions:

  • The derived method offers a more straightforward and computationally efficient way to calculate tunneling splitting.
  • The approach is generalizable to multiple dimensions.
  • Successful application to HO(2) and malonaldehyde validates the method's utility.