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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...
Force and Potential Energy in Three Dimensions01:04

Force and Potential Energy in Three Dimensions

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...
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...
Functions of Three or More Variables01:31

Functions of Three or More Variables

A function of three variables assigns a single real number to each point in three-dimensional space. Every point is identified by its Cartesian coordinates, x, y, and z, and the function maps this ordered triple to a scalar value. Such functions are commonly used to describe physical quantities that vary throughout space.A representative example is the electric potential generated by a point charge. In this case, the potential at a given location depends only on the distance from the charge. If...
Interpretations of Partial Derivatives01:14

Interpretations of Partial Derivatives

A surface defined by a function of two variables can be visualized as a vast, uneven terrain, where each point is identified using Cartesian coordinates. The elevation of the terrain at any point is determined by a function that assigns a height value to every pair of horizontal coordinates. This representation allows the surface to be studied in terms of how its height varies across different directions.At a specific point on this terrain, understanding how the height changes requires...
Parametric Surfaces01:30

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...

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Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions.

Sergei Manzhos1, Tucker Carrington

  • 1Département de Chimie, Université de Montréal, CP 6128, succursale Centre-ville, Montréal (Québec) H3C 3J7, Canada. sergei.manzhos@umontreal.ca

The Journal of Chemical Physics
|July 14, 2007
PubMed
Summary
This summary is machine-generated.

This study introduces a new method for fitting potential energy surfaces, simplifying quantum dynamics calculations. The approach uses neural networks to reduce dimensionality, improving accuracy for complex chemical reactions.

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Area of Science:

  • Computational chemistry
  • Quantum mechanics
  • Chemical physics

Background:

  • Accurate potential energy surfaces (PES) are crucial for understanding chemical reactions and molecular dynamics.
  • Traditional methods for constructing PES can be computationally expensive, especially for systems with many degrees of freedom.

Purpose of the Study:

  • To develop an efficient and accurate method for fitting potential energy surfaces.
  • To facilitate quantum dynamics calculations by reducing the dimensionality of component functions.
  • To demonstrate the general applicability of the proposed method.

Main Methods:

  • A novel approach for fitting potential energy surfaces using a sum of lower-dimensional component functions.
  • Introduction of new, redundant coordinates via linear transformations derived from a neural network.
  • Optimization of these new coordinates during the fitting process for each component function.

Main Results:

  • Successfully fitted reference potential surfaces for hydrogen peroxide and the OH+H2 reaction.
  • Demonstrated the ability to reduce the dimensionality of component functions effectively.
  • Validated the quality and generalizability of the fitting method.

Conclusions:

  • The proposed method offers an efficient way to construct potential energy surfaces.
  • The dimensionality reduction technique enhances the feasibility of complex quantum dynamics simulations.
  • This approach provides a robust tool for computational studies in chemistry and physics.