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Related Concept Videos

Polar and Cylindrical Coordinates01:22

Polar and Cylindrical Coordinates

The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
Spherical Coordinates01:23

Spherical Coordinates

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
Polar Coordinate System01:30

Polar Coordinate System

The polar coordinate system provides a natural way to describe points in the plane when distances and directions are more meaningful than horizontal and vertical displacements. It is especially useful for modeling non-rectangular regions such as circles and spirals, where symmetry about a center point is easier to express than it is in a rectangular grid. A familiar example is a ship’s plan position indicator, which marks detected targets as dots positioned relative to the ship at the display’s...
Curvilinear Motion: Polar Coordinates01:27

Curvilinear Motion: Polar Coordinates

In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
The particle's location is described using a unit vector along the radial direction. Deriving the particle's position with respect to time...
Polar Coordinates01:24

Polar Coordinates

The polar coordinate system offers an alternative to the Cartesian coordinate system for specifying points in a plane, using a distance and an angle instead of x and y coordinates. This system is particularly advantageous in situations involving circular or rotational symmetry, such as in physics or engineering problems involving waves, oscillations, or orbital paths.Defining Polar CoordinatesIn polar coordinates, a point is represented as P(r, ��), where r is the radial distance from a fixed...
Polar Coordinates: Problem Solving01:27

Polar Coordinates: Problem Solving

Directional radiation patterns are central to antenna analysis, as they illustrate how signal strength varies with direction. These patterns are often modeled using polar plots, where the radial distance from the origin represents signal intensity at a given angle. A commonly used idealized form is the four-lobed rose curve, which captures the concept of directional beams in a simplified mathematical form.The four-lobed rose curve, described by r = cos⁡(2θ), features four symmetric lobes, each...

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Related Experiment Videos

Locally optimized coordinates in modified Shepard interpolation.

Christian R Evenhuis1, Michael A Collins

  • 1Research School of Chemistry, Australian National University.

The Journal of Physical Chemistry. A
|March 17, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces an improved interpolation method for potential energy surfaces, enhancing accuracy and convergence rates for molecular simulations. The new approach utilizes adaptable local coordinate sets for more precise ab initio energy calculations.

Related Experiment Videos

Area of Science:

  • Computational chemistry
  • Theoretical chemistry
  • Quantum mechanics

Background:

  • Accurate potential energy surfaces (PES) are crucial for understanding molecular behavior.
  • Traditional interpolation methods can struggle with complex molecular systems.
  • Ab initio calculations provide high accuracy but are computationally expensive.

Purpose of the Study:

  • To develop an advanced interpolation technique for constructing accurate PES.
  • To improve the convergence rate of interpolated PES towards ab initio data.
  • To enable the use of diverse local coordinate systems within a global interpolation framework.

Main Methods:

  • Extension of the modified Shepard interpolation method.
  • Incorporation of different local coordinate sets for Taylor expansions.
  • Adaptive training set generation based on ab initio potential energy data.
  • Application to the bound state potential energy surface of methanol.

Main Results:

  • Demonstrated significant improvement in the convergence rate of the interpolated PES.
  • Achieved higher accuracy in representing the ab initio potential energy.
  • Successfully applied the method to a realistic molecular system (methanol).

Conclusions:

  • The extended modified Shepard interpolation method offers a more efficient and accurate approach to PES construction.
  • This method facilitates more reliable simulations of molecular dynamics and properties.
  • The adaptive use of local coordinates enhances the fidelity of global potential energy surfaces.