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Related Concept Videos

Force and Potential Energy in One Dimension01:13

Force and Potential Energy in One Dimension

Force can be calculated from the expression for potential energy, which is a function of position. The component of a conservative force, in a particular direction, equals the negative of the derivative of the corresponding potential energy with respect to the displacement in that direction. For regions where potential energy changes rapidly with displacement, the work done and force is maximum. Also, when force is applied along the positive coordinate axis, the potential energy decreases with...
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Consider a particle moving under the action of a conservative force that has components along each coordinate axis. Each component of force is a function of the coordinates. The potential energy function U is also a function of all three spatial coordinates. Force in one dimension can be written as the negative ratio of potential energy change to the displacement along that coordinate. For minimal displacement, the ratios become derivatives. If a function has many variables, the derivative only...
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In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...

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Setting Limits on Supersymmetry Using Simplified Models
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Published on: November 16, 2013

Relations between coordinate and potential scaling in the high-density limit.

Takeyce K Whittingham1, Kieron Burke

  • 1Department of Chemistry and Chemical Biology, Rutgers University, 610 Taylor Road, Piscataway, NJ 08854, USA.

The Journal of Chemical Physics
|April 26, 2005
PubMed
Summary

This study derives exact relations for density functional theory (DFT) scaling and high-density limits for atoms. It provides benchmarks for approximate DFT functionals by analyzing kinetic correlation energy contributions.

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Area of Science:

  • Quantum Chemistry
  • Computational Physics
  • Materials Science

Background:

  • Density Functional Theory (DFT) is a cornerstone of modern computational chemistry and physics.
  • Accurate approximations for correlation energy are crucial for predicting molecular and material properties.
  • Scaling properties of DFT functionals provide insights into their behavior at different electron densities.

Purpose of the Study:

  • To derive exact relationships between high-density scaling in DFT and the limit of infinite nuclear charge (Z).
  • To establish rigorous benchmarks for approximate DFT functionals.
  • To investigate the kinetic contribution to correlation energy and its behavior in neutral atoms.

Main Methods:

  • Derivation of exact relations for scaling to the high-density limit.
  • Application of Gorling-Levy perturbation theory to hydrogenic densities.
  • Calculation and estimation of kinetic correlation energy contributions.
  • Benchmarking of popular approximate DFT functionals against derived exact results.

Main Results:

  • Exact relations connecting DFT high-density scaling and Z -> infinity limit established for nondegenerate atoms.
  • Gorling-Levy perturbation results derived for hydrogenic densities.
  • Estimates for the kinetic contribution to correlation energy for neutral atoms provided.
  • Performance of common approximate DFT functionals evaluated against theoretical benchmarks.

Conclusions:

  • The derived exact relations offer valuable theoretical benchmarks for assessing DFT approximations.
  • Understanding kinetic correlation energy contributions is essential for developing accurate DFT functionals.
  • This work provides a foundation for improving the predictive power of DFT in various scientific domains.