Category:Forces

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 interaction forces between atoms 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

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

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 particle trajectories a way to compute the negative gradient of the potential energy has to be established.

DFT Forces

One way to compute the potential energy's negative gradient is through 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

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

where ${\displaystyle n(\mathbf{r}) }$ denotes the electronic ground state density and ${\displaystyle \psi_{i} }$ are the Kohn-Sham orbitals. ${\displaystyle E_{XC} }$ is the exchange correlation energy. To obtain the force acting on ion A the Hellmann-Feynman theorem has to be used.

${\displaystyle \mathbf{F}_{A}=-\nabla_{A} E_{tot}^{KS-DFT}=\nabla_{A}\sum_{A}\int\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n^{KS-DFT}({\bf r})d^{3}r -\nabla_{A}\frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert} }$

where ${\displaystyle \nabla_{A} }$ denotes the gradient with respect to ionic position ${\displaystyle \mathbf{R}_{A}}$. The DFT forces will depend on the chosen exchange correlation functional via the electronic ground state density ${\displaystyle n^{KS-DFT}({\bf r}) }$. Therefore the choice of the proper exchange correlation functional for the system of interest is crucial for obtaining proper forces and hence the correct material properties.

RPA-forces

To obtain more accurate forces the RPA approximation can be used to get better estimates for the exchange correlation energy. The RPA adds the following correlation energy to the DFT ground state energy

${\displaystyle E_{c} = \frac{1}{2 \pi} \int_{0}^{\infty} \mathrm{Tr} \, \ln(1-\chi( i \omega) V) + \chi( i \omega) V \, d \omega }$.

With this, the Helmann-Feynman theorem can be rewritten as

${\displaystyle \mathbf{F}_{A}=-\nabla_{A} E_{tot}^{DFT-RPA}=\nabla_{A}\sum_{A}\int\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n^{DFT-RPA}({\bf r})d^{3}r -\nabla_{A}\frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert} }$

Note that the RPA is a correction to the underlying DFT approximation of the forces. Therefore the choice of the proper exchange correlation functional is still crucial in the RPA approach for obtaining forces.

Machine-learning forces

A speedy but less accurate approach for obtaining forces is the machine learning approach. In this approach, a machine learning model is first fitted to either the DFT or RPA approach. During training the MLFF the forces, energies, and stresses are fitted with respect to a DFT or RPA approach. In the case of the RPA the stress tensor is not computed. The machine learning approach will be an approximation to the underlying method against which it was fitted. The machine learning force field decomposes the total DFT energy into local atomic contributions ${\displaystyle E_{B}(\{\mathbf{R}_{C}\}) }$ depending on all atomic positions in the system. Therefore the force acting on ion A is computed by

${\displaystyle \mathbf{F}_{A} = -\nabla_{A}\sum_{B=1}^{N}E_{B}(\{\mathbf{R}_{C}\})=-\sum_{B}^{N}w_{B}\frac{dK(\{\mathbf{R}_{C}\})}{d\mathbf{R}_{A}}, }$

where ${\displaystyle K(\mathbf{R}_{C}) }$ is the kernel matrix which can be found on the machine learning theory page. The kernel matrix as the local energies depends on the positions of all atoms ${\displaystyle \{\mathbf{R}_{C}\}}$ in the actual atomic configuration.

Related concepts

Stress and pressure

The stress tensor provides valuable information about how forces are distributed throughout a material, both in magnitude and direction. It includes normal stresses, which act perpendicular to a given plane, and shear stresses, which act parallel to the plane. Together, these components allow engineers and scientists to predict how materials will behave under various conditions, such as tension, compression, or shear. The stress tensor can be computed from a viral theorem including pair forces or with a finite difference approach deforming the simulation box. Pressure, often denoted as P, is a scalar component of the stress tensor. It represents the normal force per unit area acting on a surface within the material. In the stress tensor, pressure is related to the diagonal components σxx, σyy, and σzz:

${\displaystyle P = \frac{1}{3}(\sigma_{xx} + \sigma_{yy} + \sigma_{zz}) }$

So, the pressure is the average of the normal components of the stress tensor in the three spatial directions. In electronic structure calculations, finite basis sets are used to express the electron density. Due to this finiteness of the basis set errors on the stress tensor and the pressure are introduced. The error in the pressure is referred to as Pulay stress and can be corrected for with the tag PSTRESS.

Force constants and phonons

The forces are defined by the negative gradient of the potential energy. The force constant matrix is defined by

${\displaystyle \Phi_{I\alpha J\beta} (\{\mathbf{R}^0\}) = \left. \frac{\partial E(\{\mathbf{R}\})}{\partial R_{I\alpha} \partial R_{J\beta}} \right|_{\mathbf{R} =\mathbf{R^0}} = - \left. \frac{\partial F_{I\alpha}(\{\mathbf{R}\})}{\partial R_{J\beta}} \right|_{\mathbf{R} =\mathbf{R^0}} }$

and is therefore the gradient of the force. The force constant matrix is a fundamental concept in solid-state physics and materials science, especially in the context of understanding the vibrational properties of crystals. It is a mathematical representation of the interatomic forces and their interactions within a crystal lattice. This matrix is used to describe the relationships between atomic displacements and the resulting forces that occur in a crystal. By Fourier transforming the Force constant matrix the dynamical matrix is obtained. By computing the eigenvalues of the dynamic matrix on various reciprocal lattice points the phonon dispersion relation can be obtained. Understanding phonons is essential as they influence materials properties such as the electrical conductivity, thermal conductivity, and mechanical properties of materials.

How To

• Structure optimization
• Volume optimization
• Phonons from finite differences
• Phonons from perturbation theory
• Phonon dispersion
• Machine learning Basics
• Machine learning best practice
• Molecular dynamics simulation

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