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Related Concept Videos

Reduced Mass Coordinates: Isolated Two-body Problem01:12

Reduced Mass Coordinates: Isolated Two-body Problem

In classical mechanics, the two-body problem is one of the fundamental problems describing the motion of two interacting bodies under gravity or any other central force. When considering the motion of two bodies, one of the most important concepts is the reduced mass coordinates, a quantity that allows the two-body problem to be solved like a single-body problem. In these circumstances, it is assumed that a single body with reduced mass revolves around another body fixed in a position with an...
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Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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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.
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Virtual work is a powerful method used to solve problems involving several connected rigid bodies. When the system is in equilibrium, virtual work is zero. This allows the calculation of the resulting forces when a system undergoes a virtual displacement. When attempting to analyze such a system, first, use a free-body diagram, where an independent coordinate represents the configuration of the links, and mark its deflected position resulting from the positive virtual displacement.
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Control Volume and System Representations01:16

Control Volume and System Representations

Two key frameworks are employed to analyze mass, energy, and momentum transfer: the control volume approach and the system approach. These frameworks offer different perspectives, depending on whether the focus is on a specific region in space (control volume approach) or a defined mass of fluid (system approach).
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The fast decoupled power flow method addresses contingencies in power system operations, such as generator outages or transmission line failures. This method provides quick power flow solutions, essential for real-time system adjustments. Fast decoupled power flow algorithms simplify the Jacobian matrix by neglecting certain elements, leading to two sets of decoupled equations:

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Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry
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Real-space finite-difference approach for multi-body systems: path-integral renormalization group method and direct

Akira Sasaki1, Masashi Kojo, Kikuji Hirose

  • 1Department of Precision Science and Technology, Graduate School of Engineering, Osaka University, Suita, Osaka, Japan.

Journal of Physics. Condensed Matter : an Institute of Physics Journal
|October 15, 2011
PubMed
Summary

New computational methods, path-integral renormalization group and direct energy minimization, accurately calculate electronic structures for multi-body systems. These techniques handle complex quantum systems, achieving high accuracy with fewer Slater determinants (SDs).

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Area of Science:

  • Computational Physics
  • Quantum Chemistry
  • Materials Science

Background:

  • Accurate first-principles electronic structure calculations are crucial for understanding multi-body quantum systems.
  • Existing methods often face limitations in handling correlation effects and higher dimensions.
  • The Born-Oppenheimer approximation can be a significant constraint in studying electron-nuclear systems.

Purpose of the Study:

  • To introduce novel, practical computational methods for first-principles electronic structure calculations.
  • To enable the study of complex, higher-dimensional, and multicomponent quantum systems.
  • To overcome limitations of the Born-Oppenheimer approximation in electronic structure calculations.

Main Methods:

  • Development and application of the path-integral renormalization group and direct energy minimization methods.
  • Utilizing linear combinations of nonorthogonal Slater determinants (SDs) for multi-body wavefunctions.
  • Implementation of auxiliary fields, optimized kinetic operator treatment, and double-grid techniques for efficiency.

Main Results:

  • Achieved accuracy comparable to variational Monte Carlo methods using only a few Slater determinants (SDs).
  • Successfully calculated total energies, atomic configurations, and electronic structures for model systems.
  • Demonstrated the accuracy, availability, and broad applicability of the developed methods.

Conclusions:

  • The introduced path-integral renormalization group and direct energy minimization methods offer a powerful approach for electronic structure calculations.
  • These methods effectively handle correlation effects and can be extended to multicomponent quantum systems.
  • The ability to study systems without the Born-Oppenheimer approximation opens new avenues in quantum mechanics research.