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π Molecular Orbitals of 1,3-Butadiene01:24

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Conjugated dienes have lower heats of hydrogenation than cumulated and isolated dienes, making them more stable. The enhanced stabilization of conjugated systems can be understood from their π molecular orbitals.
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Allyl radicals are three-carbon conjugated systems. They are readily formed as intermediates in halogenation reactions of alkenes involving the addition of halogen to the allylic carbon instead of the double bond. As seen in allyl cations and anions, each of the three sp2-hybridized carbon atoms in allyl radicals has an unhybridized p orbital. These orbitals combine to give three π molecular orbitals.
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π Electron Effects on Chemical Shift: Overview01:27

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An applied magnetic field causes loosely bound π-electrons in organic molecules to circulate, producing a local or induced diamagnetic field over a large spatial volume. As the molecules tumble in solution, the field generated by π-electrons in spherical substituents results in a zero net field. However, the net field generated by π-electrons in non-spherical substituents is not zero. The effect of this induced field depends on the orientation of the molecule with respect to B0,...
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An allyl group is a three-carbon conjugated system where the sp³-hybridized allylic carbon is bonded to a CH=CH2 group via a single bond. Allyl anions can be obtained by treating propene with a strong base that can deprotonate methyl groups. Allyl cations are formed as intermediates during substitution reactions involving allylic halides. In both cases, the hybridization of the allylic carbon changes from sp3 to sp2, giving rise to a carbon chain with three sp2-hybridized carbons, each with...
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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.
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Finite-Volume Spectrum of π^{+}π^{+} and π^{+}π^{+}π^{+} Systems.

M Mai1, M Döring1,2

  • 1Institute for Nuclear Studies and Department of Physics, The George Washington University, Washington, DC 20052, USA.

Physical Review Letters
|March 2, 2019
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Summary
This summary is machine-generated.

Lattice QCD calculations can now map finite-volume energy spectra to infinite volumes for three-body systems. This study applies the method to three interacting pions, predicting excited states and validating ground state energies.

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Area of Science:

  • Nuclear Physics
  • Quantum Chromodynamics
  • Hadron Spectroscopy

Background:

  • Understanding three-body hadronic systems above threshold is crucial for exotic resonances and baryon spectrum.
  • Mapping finite-volume lattice QCD results to infinite volume is a key challenge.

Purpose of the Study:

  • To apply a formalism for mapping finite-volume lattice QCD spectra to infinite volume for a physical three-body system.
  • To determine three-body scattering amplitudes and resonance properties above threshold.

Main Methods:

  • Utilized ab initio calculations for hadronic three-body systems.
  • Applied formalism to three interacting positively charged pions.
  • Extrapolated results to physical pion masses using effective field theory input.

Main Results:

  • Presented the first application of the formalism to a physical system (three interacting pions).
  • Ground state energies agree with existing lattice QCD results at unphysical pion masses.
  • Predicted the excited energy spectrum for the three-pion system.

Conclusions:

  • Demonstrated the feasibility of determining three-body amplitudes above threshold from lattice QCD.
  • The method allows for the inclusion of resonance properties of axial mesons, exotics, and excited baryons.