Category:Forces

Introduction

Forces on particles are a fundamental concept in condensed matter physics and chemistry. These forces describe the interactions that cause particles, such as atoms and molecules, to move and behave in specific ways. In VASP forces result from electromagnetic interactions which can be computed from DFT by the use of the Hellmann-Feynman theorem, the random-phase approximation or by the use of machine learning force fields. Understanding these forces is crucial in many aspects of science, as for example:

• predicting the atomic structure of solids and molecules
• to engineer and design new materials
• predicting and optimizing chemical reactions
• improving and understanding catalysis
• predicting and understanding thermodynamic proerties

Formally the force can be defined as follows. Let ${\displaystyle \mathbf{r}(t)}$ be the position of the particle then the velocity is defined as the change of position with time

${\displaystyle \mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} }$

and the momentum ${\displaystyle \mathbf{p}(t)}$ of the particle is the velocity times the particle mass m

${\displaystyle \mathbf{p}(t) = m\mathbf{v}(t) }$

Newton's second law of motion states that the change of motion of an object is proportional to the force acting on the object and oriented in the same direction as the force vector. Therefore the force is defined as the change of particle momentum with time

${\displaystyle \mathbf{F}(t) = m\frac{d\mathbf{p}(t)}{d t} = m\mathbf{a}(t), }$

where ${\displaystyle \mathbf{a}(t) }$ is the acceleration of the particle. With this equation of motion and the knowledge of some starting conditions ${\displaystyle \mathbf{r}(0)}$ and ${\displaystyle \mathbf{v}(0)}$ the trajectory ${\displaystyle \mathbf{r}(t)}$ of a particle can be predicted for all times.

Theory

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