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Related Concept Videos

Kinetic Energy for a Rigid Body01:13

Kinetic Energy for a Rigid Body

Imagine a solid object involved in a general planar movement, with its center of mass pinpointed at a spot labeled G. The object's kinetic energy relative to an arbitrary point A can be quantified for each of its particles - the ith particle in this case. This measurement is achieved through the employment of the relative velocity definition. The position vector, known as rA, extends from point A to the mass element i.
Molecular Kinetic Energy01:21

Molecular Kinetic Energy

The word "gas" comes from the Flemish word meaning "chaos," first used to describe vapors by the chemist J. B. van Helmont. Consider a container filled with gas, with a continuous and random motion of molecules. During collisions, the velocity component parallel to the wall is unchanged, and the component perpendicular to the wall reverses direction but does not change in magnitude. If the molecule’s velocity changes in the x-direction, then its momentum is changed. During the short time of the...
Kinetic Energy - I01:18

Kinetic Energy - I

It’s plausible to suppose that the greater the velocity of a body, the greater effect it could have on other bodies. This does not depend on the direction of the velocity, only its magnitude. At the end of the seventeenth century, a quantity was introduced into mechanics to explain collisions between two perfectly elastic bodies, in which one body makes a head-on collision with an identical body at rest. When they collide, the first body stops, and the second body moves off with the initial...
Moment of Inertia01:14

Moment of Inertia

The comparability between linear and angular velocities, linear and angular accelerations, and the kinematic equations of translational and rotational motion can be extended to the concept of inertia.
If a rigid body is rotating about an axis but is not in translational motion, its translational kinetic energy is zero. However, since each particle undergoes rotational motion, it possesses non-zero velocity and kinetic energy. Thus, the kinetic energy of the rigid body, which is the sum of the...
Kinetic Molecular Theory: Molecular Velocities, Temperature, and Kinetic Energy03:07

Kinetic Molecular Theory: Molecular Velocities, Temperature, and Kinetic Energy

The kinetic molecular theory qualitatively explains the behaviors described by the various gas laws. The postulates of this theory may be applied in a more quantitative fashion to derive these individual laws.
Kinetic Energy - II00:56

Kinetic Energy - II

The kinetic energy of a particle is one-half of the product of the particle’s mass and the square of its speed. Note that just as Newton’s second law can be expressed as either the rate of change of momentum or mass multiplied by the rate of change of velocity, so too can the kinetic energy of a particle be expressed in terms of its mass and momentum, instead of its mass and velocity.

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Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
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Published on: April 8, 2020

Numeric kinetic energy operators for molecules in polyspherical coordinates.

Keyvan Sadri1, David Lauvergnat, Fabien Gatti

  • 1Theoretische Chemie, Ruprecht-Karls-Universität, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany. keyvan.sadri@pci.uni-heidelberg.de

The Journal of Chemical Physics
|July 12, 2012
PubMed
Summary

This study introduces a numerical method for generating kinetic energy operators (KEOs) in curvilinear coordinates, simplifying molecular Schrödinger equation calculations. The approach proves reliable for dynamical simulations, yielding accurate eigenenergies for nitrous acid and ethene.

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Area of Science:

  • * Quantum chemistry
  • * Molecular dynamics
  • * Computational spectroscopy

Background:

  • * Generalized curvilinear coordinates, like polyspherical coordinates, offer advantages over rectilinear coordinates for solving the nuclear Schrödinger equation.
  • * Analytical expressions for kinetic energy operators (KEOs) in polyspherical coordinates can be extremely complex for larger molecular systems.

Purpose of the Study:

  • * To develop and validate a numerical method for generating approximate KEOs in curvilinear coordinates.
  • * To create KEOs in a format suitable for practical dynamical calculations.
  • * To assess the accuracy and reliability of the numerical KEO generation method.

Main Methods:

  • * Numerical generation of kinetic energy operators (KEOs) in generalized curvilinear coordinates.
  • * Application of the method to calculate vibrational spectra and eigenenergies for nitrous acid (HONO).
  • * Calculation of the π → π* photoabsorption spectrum and eigenenergies for ethene (C2H4) with dimensionality reduction.

Main Results:

  • * The numerical KEO generation method was successfully applied to HONO and C2H4.
  • * Calculated vibrational spectra and eigenenergies for HONO showed good agreement with results from an exact analytical KEO.
  • * The π → π* photoabsorption spectrum and eigenenergies for C2H4 also matched previous studies.
  • * The error in calculated eigenenergies was consistently below 1 cm⁻¹ for most states.

Conclusions:

  • * The proposed numerical method for approximating KEOs in curvilinear coordinates is reliable for molecular dynamical calculations.
  • * This approach offers a practical alternative to complex analytical KEO derivations.
  • * The method demonstrates high accuracy, making it suitable for spectroscopic and dynamical studies.