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MO Theory and Covalent Bonding

The molecular orbital theory describes the distribution of electrons in molecules in a manner similar to the distribution of electrons in atomic orbitals. The region of space in which a valence electron in a molecule is likely to be found is called a molecular orbital. Mathematically, the linear combination of atomic orbitals (LCAO) generates molecular orbitals. Combinations of in-phase atomic orbital wave functions result in regions with a high probability of electron density, while...
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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 catalyst, high molecular...
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Versatile Technique to Produce a Hierarchical Design in Nanoporous Gold
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Published on: February 10, 2023

Generating-function approach for bond percolation in hierarchical networks.

Takehisa Hasegawa1, Masataka Sato, Koji Nemoto

  • 1Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan. hasegawa@stat.t.u-tokyo.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary
This summary is machine-generated.

This study explores bond percolation on complex networks, revealing a critical phase where cluster properties change continuously with shortcut probability. Findings detail scaling laws for cluster size distribution and critical exponents.

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Area of Science:

  • Complex networks
  • Statistical physics
  • Network science

Background:

  • Percolation theory investigates connectivity in random networks.
  • Scale-free networks exhibit unique topological properties.
  • Hierarchical structures introduce multi-scale connectivity.

Purpose of the Study:

  • To analyze bond percolation on hierarchical scale-free networks.
  • To investigate the impact of shortcut probability on network connectivity.
  • To characterize the critical phase and scaling behavior.

Main Methods:

  • Generating function approach for theoretical calculations.
  • Numerical simulations to confirm theoretical predictions.
  • Analysis of fractal exponents and cluster size distributions.

Main Results:

  • Identification of a critical phase with intermediate percolating probability (0 < P < 1).
  • Continuous variation of the fractal exponent (ψ) with shortcut probability (p).
  • Power-law distribution of cluster sizes (n(s) ∝ s⁻τ) obeying scaling relations (τ=1+ψ⁻¹).
  • Order parameter critical exponent (β) varies with p, from ≈0.164694 to infinity.

Conclusions:

  • The study elucidates the complex interplay between network hierarchy, shortcuts, and percolation phenomena.
  • Fractal properties and critical behavior are modulated by the probability of shortcuts.
  • The findings contribute to understanding emergent properties in complex systems.