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, the knowledge of some starting conditions ${\displaystyle \mathbf{r}(0)}$ and ${\displaystyle \mathbf{v}(0)}$ and an algorithm to compute the forces ${\displaystyle \mathbf{F} }$ the trajectory ${\displaystyle \mathbf{r}(t)}$ of a particle can be predicted for all times.

Theory

There is an important relation between forces and the negative gradient of the potential energy which can be computed from the Lagrangian of the particle system of interest. The Lagrangian for a N particle system is

${\displaystyle L= \sum_{i=1}^{N}m_{i}\mathbf{v}^{2}_{i} - V(\{\mathbf{r}_{i}\}), }$

where ${\displaystyle V(\{\mathbf{r}_{i}\})}$ is the potential energy of the system. With Lagrange's equation of the second kind

${\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \mathbf{v}_{i}}=\frac{\partial L}{\partial \mathbf{r}_{i}} }$

the relation

${\displaystyle \mathbf{F}_{i} = -\frac{\partial V(\{\mathbf{r}_{i}\})}{\partial \mathbf{r}_{i}} = -\nabla V(\{\mathbf{r}_{i}\}). }$

Therefore to predict forces and particles trajectories a way to compute the negative gradient of the potential energy has to be established.

DFT Forces

One way to compute the negative gradient of the potential energy is by means of DFT. In DFT there is no classical potential energy function ${\displaystyle V(\{\mathbf{r}_{i}\}) }$ but a Hamiltonian ${\displaystyle \mathcal{H} }$ depending on the ionic positions ${\displaystyle \mathbf{R}_{i}}$ and the electronic positions ${\displaystyle \mathbf{r}_{i}}$. The exact form of the Hamiltonian is given by

${\displaystyle \mathcal{H} = -\frac{1}{2}\sum_{i}\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r}) - \sum_{A}\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n({\bf r}) + \frac{1}{2}\frac{n({\bf r})n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert} + E_{\rm xc} + \frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert} }$

To obtain the force acting on ion I the Hellmann-Feynman theorem has to be used. Assuming </math> \Psi </math> is the ground state wavefunction obtained by DFT the force is obtained by

${\displaystyle \mathbf{F}_{I}=-\nabla E=\nabla \mathcal{H} }$

where E is the total energy of the system.

For more details RPA-forces

Machine-learning forces

How To

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