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Angular Momentum01:21

Angular Momentum

492
Angular momentum characterizes an object's rotational motion and is defined as the moment of its linear momentum about a specified point O. When a particle moves along a curved path in the x-y plane, the scalar formulation calculates the magnitude of its angular momentum, utilizing the moment arm (d), representing the perpendicular distance from point O to the line of action of the linear momentum. Despite being scalar in formulation, angular momentum is inherently a vector quantity. Its...
492
Principle of Angular Impulse and Momentum01:23

Principle of Angular Impulse and Momentum

923
The angular impulse and momentum principle provides insights into how forces applied at a distance from an object's rotational axis influence its angular velocity. It builds upon the crucial relationship between the moment of force and angular momentum. By integrating this equation, substituting the limits for the initial and final times, a comprehensive expression representing the angular impulse and momentum principle is derived.
923
Angular Momentum: Single Particle01:10

Angular Momentum: Single Particle

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Angular momentum is directed perpendicular to the plane of the rotation, and its magnitude depends on the choice of the origin. The perpendicular vector joining the linear momentum vector of an object to the origin is called the “lever arm.” If the lever arm and linear momentum are collinear, then the magnitude of the angular momentum is zero. Therefore, in this case, the object rotates about the origin such that it lies on the rim of the circumference defined by the lever arm...
6.9K
Angular Momentum and Principle Axes of Inertia01:09

Angular Momentum and Principle Axes of Inertia

329
The concept of angular momentum for a solid structure is illustrated as the cumulative result of the cross-product of the position vector of the mass element and the cross-product of the body's angular velocity with the position vector.
To put this equation into simpler terms, it can be reconfigured using rectangular coordinates. This involves choosing an alternative set of XYZ axes that are arbitrarily inclined with respect to the reference frame. The process of deriving the rectangular...
329
Angular Momentum about an Arbitrary Axis01:11

Angular Momentum about an Arbitrary Axis

310
Imagine a rigid body with a mass denoted as 'm', which has its center of mass at point G and is rotating around an inertial reference frame. The angular momentum at an arbitrary point P can be calculated by taking the cross product of the position vector and linear momentum vector for each individual mass element.
The velocity of a mass element comprises its translational velocity and the relative velocity instigated by the body's rotation. Substituting the velocity equation into...
310
Principle of Angular Impulse and Momentum: Problem Solving01:19

Principle of Angular Impulse and Momentum: Problem Solving

346
Consider a ball of mass m, attached to a massless rod of known length, subjected to a time-dependent torque. If the initial velocity of the mass is known, then the final velocity of the mass for time t can be determined using the principle of angular impulse and momentum.
Initially, a free-body diagram of the system is drawn to illustrate all the forces acting upon the system, providing a crucial understanding of the dynamics at play. Then, the principle of angular impulse and momentum is...
346

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Updated: Nov 5, 2025

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
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Angular-Momentum Extrapolations to the Complete Basis Set Limit: Why and When They Work.

Jerzy Cioslowski1, Krzysztof Strasburger2

  • 1Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland.

Journal of Chemical Theory and Computation
|May 18, 2021
PubMed
Summary

A rigorous proof shows errors in computed molecular energies decrease with angular momentum (L) for certain molecules. This expands upon previous work, but basis set extrapolations still lack full mathematical rigor.

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Area of Science:

  • Computational Chemistry
  • Quantum Chemistry
  • Theoretical Chemistry

Background:

  • Basis set incompleteness is a major source of error in quantum chemical calculations.
  • Extrapolation techniques are often used to estimate energies at the complete basis set limit.
  • Previous theoretical work, like Hill's asymptotic expression, had limited applicability.

Purpose of the Study:

  • To rigorously prove the leading error term's dependence on angular momentum (L) for specific molecular systems.
  • To expand the theoretical foundation for basis set extrapolation methods.
  • To investigate the reliability of angular-momentum extrapolations.

Main Methods:

  • Development of a formalism centered on off-diagonal cusp conditions for reduced density matrices.
  • Rigorous mathematical proof of the L^{-3} error dependence.
  • Analysis of the variability of prefactors and higher-order terms in the error expansion.

Main Results:

  • The L^{-3} dependence of energy errors is rigorously proven for 1Σ states of linear molecules/ions with an even number of electrons.
  • This proof extends the applicability beyond the helium isoelectronic series.
  • The prefactor's variability and higher-order terms limit the mathematical rigor of extrapolations.

Conclusions:

  • The theoretical underpinnings for angular-momentum-based error extrapolation are strengthened.
  • However, current extrapolation practices may rely more on empirical success than strict mathematical justification.
  • Further research is needed to improve the predictability and rigor of basis set extrapolation methods.