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Related Concept Videos

Energy Diagrams - II01:10

Energy Diagrams - II

Energy diagrams are important to understand the dynamics of a system. The topology of an energy diagram helps illustrate the equilibrium points of the system.
The point in the energy diagram at which the system’s potential energy is the lowest is known as the local minima. The system tends to stay in this position indefinitely unless acted upon by a net force. The slope of the potential energy diagram at the local minima is zero, indicating that zero net force is acting on the system. The slope...
Potential-Energy Criterion for Equilibrium01:16

Potential-Energy Criterion for Equilibrium

Potential energy or potential function plays an essential role in determining the stability of a mechanical system. If a system is subjected to both gravitational and elastic forces, the potential function of the system can be expressed as the algebraic sum of gravitational and elastic potential energy. If the system is in equilibrium and is displaced by a small amount, then the work done on the system equals the negative of the change in the system's potential energy from the initial to the...
Potential Energy00:52

Potential Energy

The energy stored by a structure and location of matter in space is called potential energy. For instance, raising a kettlebell changes its spatial location and increases its potential energy. Similarly, a stretched rubber band contains potential energy which, under certain conditions, can be converted into other forms of energy, such as kinetic energy.
Chemical bonds that form attractive forces between atoms also contain potential energy, called chemical energy. When a chemical reaction...
Potential Energy01:09

Potential Energy

A conservative force, such as a gravitational or elastic force, gives the body the capacity to do work. This capacity, measured as the potential energy, depends on the body's location or “position” relative to a fixed reference position or datum. The gravitational potential energy is considered zero at the reference point. Suppose a body is located at some vertical distance above a fixed horizontal reference or datum. In that case, the weight of the body has positive gravitational potential...
Energy Diagrams - I01:14

Energy Diagrams - I

The dynamics of a mechanical system can be easily understood by interpreting a potential energy diagram. Since energy is a scalar quantity, the interpretation of the dynamics of the system becomes even simpler.
Take the example of a skater on a parabolic ramp. The potential energy at different points along the ramp will be proportional to the height of the ramp, which varies quadratically with the horizontal position on the ramp. As the skater moves down the ramp from the highest position,...
Types of Potential Energy01:16

Types of Potential Energy

Potential energy is also known as energy at rest or stored energy. Common types of potential energy include the gravitational potential energy stored in an apple hanging from a tree, the electrical potential energy stored in an object due to the attraction or repulsion of electric charges, and the chemical potential energy stored in the bonds between atoms and molecules. Additionally, the nuclear energy stored in an atomic nucleus and the elastic energy stored in a stretched spring due to its...

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Related Experiment Videos

Communication: Certifying the potential energy landscape.

Dhagash Mehta1, Jonathan D Hauenstein, David J Wales

  • 1Department of Physics, Syracuse University, Syracuse, New York 13244, USA. dbmehta@syr.edu

The Journal of Chemical Physics
|May 10, 2013
PubMed
Summary
This summary is machine-generated.

This study introduces Smale's α-theory for certifying stationary points in potential energy landscapes. It mathematically proves numerical approximations correspond to true stationary points, ensuring convergence for atomic cluster optimization.

Related Experiment Videos

Area of Science:

  • Computational chemistry
  • Mathematical physics
  • Numerical analysis

Background:

  • Numerical approximations of stationary points in potential energy landscapes may not always converge.
  • Certification proves that a numerical approximation will quadratically converge to the true stationary point.
  • Smale's α-theory provides a rigorous framework for such certification.

Purpose of the Study:

  • To apply Smale's α-theory for certifying stationary points in potential energy landscapes.
  • To provide a mathematical proof of convergence for numerical approximations.
  • To demonstrate the practical application of α-theory for certifying atomic cluster configurations.

Main Methods:

  • Application of Smale's α-theory to stationary points.
  • Development and implementation of certification algorithms.
  • Analysis of Lennard-Jones atomic clusters (N = 7-14).

Main Results:

  • Smale's α-theory successfully certifies stationary points, irrespective of numerical precision.
  • The methodology guarantees quadratic convergence to the true stationary point.
  • All known minima and transition states for LJ(N) clusters (N=7-14) were certified.

Conclusions:

  • Smale's α-theory offers a robust method for certifying stationary points in computational chemistry.
  • This approach provides reliable mathematical proof of convergence for numerical approximations.
  • The study validates the practical utility of α-theory in analyzing complex molecular systems like atomic clusters.