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Related Concept Videos

Stability of Substituted Cyclohexanes02:30

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This lesson discusses the stability of substituted cyclohexanes with a focus on energies of various conformers and the effect of 1,3-diaxial interactions.
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Ziegler–Natta polymerization is another form of addition or chain‐growth polymerization used for synthesizing linear polymers over branched polymers. The catalyst used for polymerization is the Ziegler–Natta catalyst, named after Karl Ziegler and Giulio Natta, who developed it in 1953. This catalyst is an organometallic complex of titanium tetrachloride and triethyl aluminum, with the active form of the catalyst being an alkyl titanium compound. Using the Ziegler–Natta...
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Step-growth or condensation polymerization is a stepwise reaction of bi or multifunctional monomers to form long-chain polymers. As all the monomers are reactive, most of the monomers are consumed at the early stages of the reaction to form small chains of reactive oligomers, which then combine to form long polymer chains in the late stages. Hence, the reaction has to proceed for a long time to achieve high molecular weight polymers.
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Updated: Jan 6, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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Cayley modification for strongly stable path-integral and ring-polymer molecular dynamics.

Roman Korol1, Nawaf Bou-Rabee2, Thomas F Miller1

  • 1Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA.

The Journal of Chemical Physics
|October 3, 2019
PubMed
Summary

Standard harmonic normal modes in path-integral molecular dynamics (MD) simulations create numerical artifacts. Replacing this with a Cayley transform approximation removes artifacts, enhancing stability and ergodicity in simulations like ring-polymer MD (RPMD).

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Area of Science:

  • Computational Chemistry
  • Quantum Mechanics
  • Statistical Mechanics

Background:

  • Path-integral molecular dynamics (MD) simulations are crucial for quantum Boltzmann properties.
  • Existing MD integration schemes commonly use harmonic normal modes for ring-polymer evolution.
  • This standard practice has been found to introduce numerical artifacts.

Purpose of the Study:

  • To identify and demonstrate numerical artifacts in path-integral MD simulations.
  • To propose a novel modification to remove these artifacts.
  • To enhance the stability and ergodicity of path-integral MD methods.

Main Methods:

  • Demonstration of numerical artifacts in conservative and thermostated equations of motion.
  • Numerical analysis for conventional ring-polymer MD (RPMD) and thermostated RPMD (TRPMD).
  • Introduction of a second-order Cayley transform approximation for ring-polymer evolution.

Main Results:

  • Harmonic normal modes lead to numerical instability and nonergodicity in path-integral MD.
  • The Cayley transform modification effectively removes these artifacts.
  • The modification is compatible with various path-integral MD schemes (PIMD, RPMD, TRPMD, centroid MD).

Conclusions:

  • The Cayley modification provides improved symplectic stability and ergodicity without computational cost increase.
  • This approach allows for larger MD time steps, enhancing simulation efficiency.
  • The Cayley modification is expected to be widely applicable in future path-integral MD simulations.