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Related Concept Videos

Bond Energies and Bond Lengths02:49

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Stable molecules exist because covalent bonds hold the atoms together. The strength of a covalent bond is measured by the energy required to break it, that is, the energy necessary to separate the bonded atoms. Separating any pair of bonded atoms requires energy — the stronger a bond, the greater the energy required to break it.
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Stress Concentrations in Circular Shafts01:18

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Consider the elastic torsion formula, which applies to a circular shaft with a consistent cross-section. This formula assumes that the shaft's ends are loaded with rigid plates firmly attached. However, in many cases, torques are applied to the shaft through mechanisms like flange couplings or gears, which are connected by keys inserted into keyways. This application method modifies the stress distribution near the point of torque application, causing it to deviate from the distributions...
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A covalently bonded heteronuclear diatomic molecule can be modeled as two vibrating masses connected by a spring. The vibrational frequency of the bond can be expressed using an equation derived from Hooke's law, which describes how the force applied to stretch or compress a spring is proportional to the displacement of the spring. In this case, the atoms behave like masses, and the bond acts like a spring.
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In analyzing a thin-walled hollow shaft subjected to torsional loading, a segment with width dx is isolated for examination. Despite its equilibrium state, this segment faces torsional shearing forces at its ends. These forces are quantitatively described by the product of the longitudinal shearing stress on the segment's minor surface and the area of this surface, leading to the concept of shear flow. This shear flow is consistent throughout the structure, indicating a uniform distribution of...
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Torsion of Noncircular Members01:16

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Circular shafts undergoing torsional stress maintain their cross-sectional integrity due to their axisymmetric nature. This symmetry ensures an even distribution of stress, allowing the shaft to withstand torsion without distorting. In contrast, square bars, lacking this axial symmetry, experience significant distortion across their cross-sections when subjected to torsion, with the exception of along their diagonals and at lines connecting midpoints. A detailed examination of a cubic element...
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Angle of Twist - Elastic Range01:13

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Consider a cylindrical shaft with a length denoted by L and a consistent cross-sectional radius referred to as r. This shaft undergoes a torque at the free end. The highest shearing strain within the shaft is directly proportional to the twist angle and the radial distance from the shaft axis. When the shaft behaves elastically, this shearing strain can be articulated using variables such as the applied torque, radial distance, the polar moment of inertia, and the modulus of rigidity. By...
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Related Experiment Video

Updated: May 1, 2026

Magnetic Tweezers for the Measurement of Twist and Torque
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Kernel density estimation applied to bond length, bond angle, and torsion angle distributions.

Patrick McCabe1, Oliver Korb, Jason Cole

  • 1Cambridge Crystallographic Data Centre , 12 Union Road, Cambridge CB2 1EZ, United Kingdom.

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|April 22, 2014
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Summary

Kernel density estimation (KDE) offers a powerful, underutilized statistical method for analyzing molecular structure data. This approach generates smooth probability density functions ideal for molecular geometry analysis and optimization.

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Area of Science:

  • Computational Chemistry
  • Statistical Modeling
  • Chemoinformatics

Background:

  • Kernel density estimation (KDE) is a versatile non-parametric statistical technique.
  • KDE is effective for multimodal data and generates smooth probability density functions (PDFs).
  • KDE has broad applications in signal processing and econometrics but is underused in molecular sciences.

Purpose of the Study:

  • To introduce and apply Kernel Density Estimation (KDE) to molecular structure data.
  • To highlight the advantages of KDE over traditional methods like histograms for molecular analysis.
  • To demonstrate KDE's suitability for gradient-based optimization in molecular modeling.

Main Methods:

  • Application of Kernel Density Estimation (KDE) to molecular geometry.
  • Analysis of chemical bond lengths, bond valence angles, and torsion angles using KDE.
  • Modeling of arbitrary torsion angle distributions with KDE.

Main Results:

  • KDE provides smooth and informative probability density functions for molecular data.
  • The method effectively models distributions of chemical bond lengths, angles, and torsion angles.
  • KDE-generated PDFs are advantageous for gradient-based optimization tasks.

Conclusions:

  • Kernel density estimation (KDE) is a valuable and underutilized tool for molecular structure analysis.
  • KDE offers superior data representation compared to histograms for molecular geometry.
  • The method facilitates advanced applications in chemoinformatics and molecular structure optimization.